Explanation:
Okay, so I need to figure out the minimum possible value of the expression 1 + 2 + 3 - 4 + 5 + 6 by adding parentheses in different places. The original value is 13, but by changing where the parentheses are, I can get different results. The key here is that I can't rearrange the order of the terms or change the operations; I can only add parentheses. Also, multiplication isn't allowed, so I can't do things like (1+2+3)(-4) or anything like that.

First, let me make sure I understand the problem correctly. The original expression is 1 + 2 + 3 - 4 + 5 + 6. The operations are addition and subtraction. Parentheses can be added to change the order of operations, but the sequence of numbers and operations must stay the same. So, for example, if I put parentheses around 1 + 2, that would be (1 + 2) + 3 - 4 + 5 + 6, which is the same as the original because addition is associative. But if I put parentheses around 1 + 2 + 3, that would be (1 + 2 + 3) - 4 + 5 + 6, which is still the same as the original. However, if I put parentheses around something like 4 + 5, then that would change the value, as in the example given: 1 + 2 + 3 - (4 + 5) + 6 = 1 + 2 + 3 - 9 + 6 = 3.

So the challenge is to figure out how to place parentheses in such a way that the overall value is minimized. Let me think about how parentheses affect the expression. Parentheses can change the order in which operations are performed, so they can prioritize certain operations over others. Since subtraction is involved, which is not associative, parentheses can significantly affect the result.

Let me recall that subtraction is not associative, meaning that (a - b) - c is not the same as a - (b - c). So, by adding parentheses, we can change which subtractions are performed first, which can lead to different results.

The original expression is 1 + 2 + 3 - 4 + 5 + 6. Let me write down the original order of operations without parentheses:

Starting from left to right, since addition and subtraction have the same precedence and are evaluated left to right. So:

1 + 2 = 3

3 + 3 = 6

6 - 4 = 2

2 + 5 = 7

7 + 6 = 13.

So that's the original value. Now, if I add parentheses, I can change the order. For example, if I group the subtraction with more numbers, maybe I can subtract a larger number, thereby reducing the total.

Let me think about how to maximize the amount subtracted. Since subtraction is involved, if I can make a larger number to subtract, that would decrease the total. So, perhaps grouping terms in such a way that a large positive number is subtracted.

But how can I do that? Let me consider the expression again: 1 + 2 + 3 - 4 + 5 + 6.

The operations are: +, +, -, +, +.

So the expression can be thought of as:

((((1 + 2) + 3) - 4) + 5) + 6 = 13.

But if I can change the order of operations, maybe I can subtract more. For example, if I can subtract a larger number by grouping. Let me try different groupings.

First, let me note that all the operations are addition and subtraction. So parentheses can only change the order of these operations.

Let me think of the expression as a sequence of numbers with operations between them: 1 + 2 + 3 - 4 + 5 + 6.

Each operation is either + or -, and parentheses can change the order.

One approach is to consider that parentheses can be used to create a larger negative term. For example, if I can make a term like -(4 + 5 + 6), that would be -15, which would significantly reduce the total. However, can I do that? Let me check.

Original expression: 1 + 2 + 3 - 4 + 5 + 6.

If I want to subtract (4 + 5 + 6), that would require that the subtraction applies to all three terms. But in the original expression, the subtraction is only on 4, and then addition on 5 and 6. So how can I make the subtraction apply to 4, 5, and 6?

Let me try to see. If I can group 4 + 5 + 6 as a single term, but how? Let me think.

Suppose I have 1 + 2 + 3 - (4 + 5 + 6). That would be 1 + 2 + 3 - 15 = 6 - 15 = -9. But is that a valid placement of parentheses? Let me check the original expression. The original expression is 1 + 2 + 3 - 4 + 5 + 6. To get 1 + 2 + 3 - (4 + 5 + 6), I need to add parentheses around 4 + 5 + 6. But in the original expression, after the subtraction sign, there is a 4, then +5, then +6. So to group 4 + 5 + 6, I need to have parentheses starting after the subtraction sign.

Wait, but the original expression is 1 + 2 + 3 - 4 + 5 + 6. So the subtraction is only on 4, and then addition on 5 and 6. If I want to group 4 + 5 + 6, I need to have parentheses that start after the subtraction. Let me think.

If I write it as 1 + 2 + 3 - (4 + 5 + 6), that would require that the subtraction applies to the entire group. However, in the original expression, the subtraction is only on 4, and then the rest are additions. So to make that happen, I need to have parentheses that start after the subtraction sign.

But how is that possible? Let me think about the structure. The original expression is:

1 + 2 + 3 - 4 + 5 + 6

So the operations are:

Between 1 and 2: +

Between 2 and 3: +

Between 3 and 4: -

Between 4 and 5: +

Between 5 and 6: +

So if I want to group 4 + 5 + 6, I need to have parentheses that start at 4 and include 5 and 6. However, the subtraction is between 3 and 4. So if I put parentheses around 4 + 5 + 6, that would be:

1 + 2 + 3 - (4 + 5 + 6)

But in the original expression, the subtraction is only on 4. So to do that, I need to have the parentheses start after the subtraction. But how does that work? Let me think of the expression as:

1 + 2 + 3 - 4 + 5 + 6

If I want to group 4 + 5 + 6, I need to have parentheses that start at the 4, but since the 4 is preceded by a subtraction, I need to make sure that the parentheses are placed correctly.

Wait, actually, in terms of operator precedence, if I have 1 + 2 + 3 - (4 + 5 + 6), that is valid. Because the parentheses override the default left-to-right evaluation. So even though the original expression is 1 + 2 + 3 - 4 + 5 + 6, adding parentheses around 4 + 5 + 6 would change the expression to 1 + 2 + 3 - (4 + 5 + 6).

But is that allowed? The problem says we can add parentheses in various locations. So as long as we don't rearrange the terms, just add parentheses. So yes, that should be allowed.

So that would be 1 + 2 + 3 - (4 + 5 + 6) = 6 - 15 = -9.

Is that a valid expression? Let me check. The original expression is 1 + 2 + 3 - 4 + 5 + 6. If I add parentheses around 4 + 5 + 6, then the expression becomes 1 + 2 + 3 - (4 + 5 + 6). That is allowed.

So that would be -9. Is that the minimum? Maybe not. Let me check if there are other ways to group terms to get a lower value.

Alternatively, maybe grouping more terms. For example, if I can subtract more numbers. Let me think.

Another idea: if I can have multiple subtractions. For example, if I can have something like 1 + (2 + 3 - 4) + 5 + 6. Wait, but that would be 1 + (2 + 3 - 4) + 5 + 6 = 1 + 1 + 5 + 6 = 13, same as original.

Alternatively, maybe grouping the first few terms. For example, (1 + 2 + 3 - 4) + 5 + 6. That would be (6 - 4) + 5 + 6 = 2 + 5 + 6 = 13. Still same.

Alternatively, if I group 1 + 2 + (3 - 4) + 5 + 6. That would be 1 + 2 + (-1) + 5 + 6 = 1 + 2 -1 + 5 + 6 = 13.

Hmm. So that doesn't change the value.

Alternatively, maybe grouping more terms. Let me think. What if I group 1 + 2 + 3 - (4 + 5) + 6? That was the example given, which equals 3.

Alternatively, grouping 1 + 2 + (3 - 4 + 5) + 6. Let's compute that: 1 + 2 + (4) + 6 = 1 + 2 + 4 + 6 = 13.

Same as original.

Alternatively, grouping 1 + (2 + 3 - 4 + 5) + 6. That would be 1 + (6) + 6 = 13.

Same.

Alternatively, grouping (1 + 2 + 3 - 4 + 5) + 6. That would be (7) + 6 = 13.

Same.

So maybe grouping the first part.

But if I want to get a lower value, I need to have more subtractions.

Wait, so earlier I thought of grouping 4 + 5 + 6 as a single term to subtract, which gives -9. Is that the minimum? Let me check if there are other groupings that can lead to a lower value.

Another idea: maybe grouping some additions and subtractions in a way that creates a larger negative number. For example, if I can have something like 1 + 2 + (3 - 4 + 5 + 6). Wait, that would be 1 + 2 + (10) = 13.

Alternatively, 1 + (2 + 3 - 4 + 5 + 6). That would be 1 + (12) = 13.

Alternatively, maybe grouping the first three terms and then subtracting something else.

Wait, let me think again. If I can have more subtractions. For example, if I can have 1 + 2 + 3 - 4 - 5 - 6. But how can I do that? Because in the original expression, after the subtraction of 4, there are additions of 5 and 6. To make them subtractions, I need to add parentheses.

Wait, but how? Let me think.

Original expression: 1 + 2 + 3 - 4 + 5 + 6.

If I want to make the +5 and +6 into -5 and -6, I need to have parentheses that change the sign. For example, if I can have something like 1 + 2 + 3 - 4 - (5 + 6). But how can I do that?

Let me check. If I have 1 + 2 + 3 - 4 - (5 + 6). That would be 6 - 4 - 11 = 2 - 11 = -9. Wait, same as before.

But how is that achieved by adding parentheses? Let me see.

Original expression: 1 + 2 + 3 - 4 + 5 + 6.

To get 1 + 2 + 3 - 4 - (5 + 6), I need to add parentheses around 5 + 6, but with a negative sign. However, in the original expression, after the subtraction of 4, there is a +5 and +6. So to make them -5 and -6, I need to have parentheses that change the sign.

But how? Because the original expression is 1 + 2 + 3 - 4 + 5 + 6. If I want to have - (5 + 6), that would require that the +5 and +6 are inside a parenthesis with a negative sign. But the original expression has a +5 and +6 after the subtraction.

Wait, maybe if I group the 5 + 6 as a negative term. Let me think.

If I have 1 + 2 + 3 - 4 + (5 + 6). That is the original expression. But if I want to have 1 + 2 + 3 - 4 - (5 + 6), that would require that the +5 and +6 are turned into -5 and -6. How can I do that?

Is there a way to add parentheses to achieve that?

Let me think. If I have 1 + 2 + 3 - 4 + 5 + 6. If I add parentheses around 5 + 6, but with a negative sign. Wait, but the original expression has a +5 and +6. To make them negative, I need to have a negative sign in front of them.

But how? Unless I can group them with a subtraction.

Wait, maybe if I have 1 + 2 + 3 - (4 + 5 + 6). Which is what I thought before. That would be 1 + 2 + 3 - (4 + 5 + 6) = 6 - 15 = -9.

Alternatively, if I have 1 + 2 + 3 - 4 - (5 + 6). But how is that different?

Wait, in the original expression, after the subtraction of 4, there is a +5 and +6. If I want to make them -5 and -6, I need to have a negative sign in front of them. But how can I do that?

Is there a way to add parentheses to create that? For example, if I have 1 + 2 + 3 - 4 - (5 + 6). But in the original expression, the operations are +, +, -, +, +. So to get a negative sign before 5 + 6, I need to have a subtraction operation that applies to 5 + 6.

But in the original expression, after the subtraction of 4, there is a +5. So if I want to have a subtraction of 5 and 6, I need to have parentheses that start after the subtraction of 4.

Wait, maybe like this: 1 + 2 + 3 - 4 - (5 + 6). But how is that different from 1 + 2 + 3 - (4 + 5 + 6)?

Let me compute both:

First: 1 + 2 + 3 - (4 + 5 + 6) = 6 - 15 = -9.

Second: 1 + 2 + 3 - 4 - (5 + 6) = 6 - 4 - 11 = 2 - 11 = -9.

Same result. So both ways give -9.

So that seems like a valid way.

But is there a way to get even lower? Let me think.

What if I can subtract more? For example, if I can have something like 1 + 2 + 3 - 4 - 5 - 6. That would be 6 - 4 - 5 - 6 = 6 - 15 = -9. Same as before.

But how can I get that?

Original expression: 1 + 2 + 3 - 4 + 5 + 6.

To get 1 + 2 + 3 - 4 - 5 - 6, I need to change the +5 and +6 into -5 and -6. How can I do that with parentheses?

If I can group the 5 and 6 with a negative sign. For example, if I have 1 + 2 + 3 - 4 - (5 + 6). Which is what I did before.

Alternatively, if I have 1 + 2 + 3 - (4 + 5 + 6).

Either way, same result.

So maybe -9 is the minimum? But let me check if there are other groupings.

Another idea: maybe grouping some terms in a way that creates more subtractions. For example, if I can have something like (1 + 2 + 3 - 4) - (5 + 6). Let me compute that.

(1 + 2 + 3 - 4) = 2. Then subtract (5 + 6) = 11. So 2 - 11 = -9. Same as before.

Alternatively, grouping (1 + 2) + 3 - (4 + 5 + 6). Which is 3 + 3 - 15 = 6 - 15 = -9.

Same result.

Is there a way to get lower than -9? Let me think.

Suppose I try to subtract more numbers. For example, if I can have something like 1 + (2 + 3 - 4 + 5) - 6. Wait, let me compute that.

First, compute inside the parentheses: 2 + 3 - 4 + 5 = 2 + 3 = 5, 5 - 4 = 1, 1 + 5 = 6. Then 1 + 6 - 6 = 1.

Not helpful.

Alternatively, 1 + 2 + (3 - 4 + 5) - 6. Let's compute: 3 - 4 + 5 = 4. Then 1 + 2 + 4 - 6 = 7 - 6 = 1.

Same.

Alternatively, 1 + (2 + 3 - 4) + 5 - 6. That would be 1 + (1) + 5 - 6 = 1 + 1 + 5 - 6 = 1.

Same.

Alternatively, maybe grouping more terms. Let me think.

What if I have ((1 + 2 + 3) - 4) - (5 + 6). Which is (6 - 4) - 11 = 2 - 11 = -9.

Same as before.

Alternatively, ((1 + 2) + 3 - 4) - (5 + 6) = same thing.

Is there a way to have more subtractions? For example, if I can have something like 1 + 2 + 3 - (4 + 5) - 6. Let me compute that.

Original expression: 1 + 2 + 3 - (4 + 5) - 6.

Compute inside the parentheses: 4 + 5 = 9. Then 1 + 2 + 3 - 9 - 6 = 6 - 9 - 6 = -9.

Same result.

So regardless of how I group the terms to subtract more, I get -9.

Is there a way to get lower than -9? Let me think.

Suppose I try to subtract more numbers. For example, if I can have something like 1 + 2 + 3 - 4 - 5 - 6. Which is 6 - 4 - 5 - 6 = 6 - 15 = -9.

Same as before.

Alternatively, if I can have something like 1 + 2 + (3 - 4 - 5 - 6). Let me compute that.

Inside the parentheses: 3 - 4 - 5 - 6 = 3 - 15 = -12. Then 1 + 2 + (-12) = 3 - 12 = -9.

Same result.

So no matter how I group, I can't get lower than -9?

Wait, let me think again. What if I group in a different way? For example, if I have 1 + (2 + 3 - 4 + 5 + 6). Wait, that would be 1 + (12) = 13.

No.

Alternatively, 1 + 2 + (3 - 4 + 5 + 6). That is 1 + 2 + (10) = 13.

Same.

Alternatively, 1 + (2 + 3 - 4) + 5 + 6. Which is 1 + 1 + 5 + 6 = 13.

Same.

Alternatively, maybe grouping the first two terms and then subtracting something.

For example, (1 + 2) + 3 - 4 + 5 + 6. Which is 3 + 3 - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

Alternatively, (1 + 2 + 3) - 4 + 5 + 6. Which is 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

So all these groupings don't change the value.

Wait, so maybe -9 is the minimum?

But let me check another possibility. What if I group some terms in a way that creates a larger negative number? For example, if I can have something like 1 + 2 + 3 - (4 + 5 + 6). Which is -9.

Alternatively, if I can have 1 + 2 + (3 - 4 + 5) - 6. Let me compute that.

Inside the parentheses: 3 - 4 + 5 = 4. Then 1 + 2 + 4 - 6 = 7 - 6 = 1.

No.

Alternatively, 1 + (2 + 3 - 4 + 5) - 6. Which is 1 + (6) - 6 = 1.

Same.

Alternatively, 1 + 2 + 3 - (4 + 5) + 6. Which is 6 - 9 + 6 = 3.

Which is the example given.

Alternatively, 1 + 2 + (3 - 4 + 5 + 6). Which is 1 + 2 + 10 = 13.

Same.

Alternatively, 1 + (2 + 3 - 4 + 5 + 6). Which is 1 + 12 = 13.

Same.

So, all these groupings don't get me lower than -9.

Wait, but let me think again. What if I group the first term with the subtraction? For example, (1 + 2 + 3 - 4) + 5 + 6. Which is 2 + 5 + 6 = 13.

Same.

Alternatively, 1 + (2 + 3 - 4 + 5 + 6). Which is 1 + 12 = 13.

Same.

Alternatively, 1 + 2 + (3 - 4 + 5 + 6). Which is 1 + 2 + 10 = 13.

Same.

So, seems like all the groupings that don't subtract more than 4 + 5 + 6 give me the same or higher value.

But is there a way to subtract more? For example, if I can have something like 1 + 2 + 3 - 4 - 5 - 6. Which is 6 - 4 - 5 - 6 = -9.

But how can I get that?

Original expression: 1 + 2 + 3 - 4 + 5 + 6.

To get 1 + 2 + 3 - 4 - 5 - 6, I need to change the +5 and +6 into -5 and -6. How can I do that with parentheses?

If I have 1 + 2 + 3 - 4 - (5 + 6). Which is allowed by adding parentheses around 5 + 6.

But in the original expression, after the subtraction of 4, there is a +5 and +6. So to make them -5 and -6, I need to have a negative sign in front of them.

But how?

Wait, if I have 1 + 2 + 3 - 4 - (5 + 6). That would be equivalent to 1 + 2 + 3 - 4 - 5 - 6.

But in the original expression, the operations are +, +, -, +, +. So to get a subtraction of 5 and 6, I need to have parentheses that start after the subtraction of 4.

But how is that possible? Let me think of the original expression as:

1 + 2 + 3 - 4 + 5 + 6

If I want to have 1 + 2 + 3 - 4 - (5 + 6), that would require that the +5 and +6 are inside a parenthesis with a negative sign.

But in the original expression, the +5 and +6 are after the subtraction of 4. So if I add parentheses around 5 + 6, but with a negative sign.

Wait, but how does that work?

Let me think of the expression as:

1 + 2 + 3 - 4 + 5 + 6

If I want to have 1 + 2 + 3 - 4 - (5 + 6), that would mean that the +5 and +6 are being subtracted. But in the original expression, the operations are +5 and +6.

So to change them to -5 and -6, I need to have a negative sign in front of them. But how can I do that?

Ah! Because the original expression has a subtraction after 3, which is -4, and then additions. If I can group the 5 and 6 with a negative sign, that would require that the subtraction of 4 is followed by a subtraction of (5 + 6).

But how is that possible?

Wait, maybe by adding parentheses around 5 + 6 and then having the subtraction apply to that.

But in the original expression, after the subtraction of 4, there is a +5. So if I add parentheses around 5 + 6, but with a negative sign.

Wait, but the original expression is 1 + 2 + 3 - 4 + 5 + 6.

If I add parentheses around 5 + 6, but with a negative sign, that would be 1 + 2 + 3 - 4 - (5 + 6).

But how is that different from the original expression?

In the original expression, the operations are +, +, -, +, +.

If I add parentheses around 5 + 6, but with a negative sign, that would require that the +5 and +6 are turned into -5 and -6.

But how can I do that?

Wait, maybe by changing the sign of the 5 and 6. But the problem says we can't rearrange the order of the terms. So we can't move the 5 and 6.

But we can add parentheses. So if I have 1 + 2 + 3 - 4 - (5 + 6), that is equivalent to 1 + 2 + 3 - 4 - 5 - 6.

But how is that achieved by adding parentheses?

Let me think of the original expression as:

1 + 2 + 3 - 4 + 5 + 6

If I want to have 1 + 2 + 3 - 4 - (5 + 6), that would require that the +5 and +6 are inside a parenthesis with a negative sign.

But in the original expression, the +5 and +6 are after the subtraction of 4. So if I add parentheses around 5 + 6, but with a negative sign, that would be:

1 + 2 + 3 - 4 - (5 + 6)

But how is that different from the original expression?

Wait, the original expression is 1 + 2 + 3 - 4 + 5 + 6.

If I add parentheses around 5 + 6, but with a negative sign, that would be changing the +5 and +6 into -5 and -6.

But how can I do that?

Ah! Because the original expression has a subtraction after 3, which is -4, and then additions. If I can group the 5 + 6 with a negative sign, that would require that the subtraction of 4 is followed by a subtraction of (5 + 6).

But how is that possible?

Wait, maybe by adding parentheses around 4 + 5 + 6?

Wait, if I have 1 + 2 + 3 - (4 + 5 + 6). That is allowed.

But that is equivalent to 1 + 2 + 3 - 4 - 5 - 6.

So that is the same as 1 + 2 + 3 - 4 - 5 - 6.

So that is allowed.

Therefore, that gives -9.

Is there a way to get lower than that?

Let me think. Suppose I can have more subtractions. For example, if I can subtract more numbers.

But all numbers are already being subtracted or added.

Wait, the numbers are 1, 2, 3, 4, 5, 6.

In the original expression, 1, 2, 3 are added, 4 is subtracted, 5 and 6 are added.

If I can subtract more numbers, like subtracting 4, 5, 6, and also subtracting something else?

But how?

Wait, if I can have something like 1 + 2 + 3 - 4 - 5 - 6, which is subtracting 4, 5, 6.

But that's already the case in the grouping where we have 1 + 2 + 3 - (4 + 5 + 6).

Is there a way to subtract more? For example, subtracting 1, 2, 3, 4, 5, 6?

But how?

If I can have something like -(1 + 2 + 3 + 4 + 5 + 6). But how can I do that?

Original expression is 1 + 2 + 3 - 4 + 5 + 6.

To have all numbers subtracted, I would need to have something like -(1 + 2 + 3 + 4 + 5 + 6). But that would require changing the signs of all terms, which isn't possible unless we can add parentheses in a way that negates all terms.

But how?

For example, if I have -(1 + 2 + 3 - 4 + 5 + 6). But that would be -1 -2 -3 +4 -5 -6. Which is different.

But how can I get that?

Wait, the original expression is 1 + 2 + 3 - 4 + 5 + 6. If I want to negate the entire expression, I would need to have something like -(1 + 2 + 3 - 4 + 5 + 6). But that would require adding a negative sign in front of the entire expression. However, the problem says we can't change the operations, only add parentheses.

So we can't add a negative sign unless it's part of the existing operations.

Wait, the original expression doesn't have a negative sign at the beginning. So we can't add that.

Alternatively, if we can group the entire expression with a negative sign. But how?

For example, if we have -(1 + 2 + 3 - 4 + 5 + 6). But that would require adding a negative sign in front, which isn't allowed.

The problem says we can't change addition or subtraction into multiplication, but we can add parentheses. So we can't add a negative sign unless it's part of the existing operations.

Therefore, we can't have a negative sign in front of the entire expression.

Therefore, the minimum we can get is -9.

But let me check another possibility. Suppose I group some terms in a way that creates more subtractions.

For example, if I have 1 + (2 + 3 - 4 + 5) - 6. Let me compute that.

Inside the parentheses: 2 + 3 - 4 + 5 = 2 + 3 = 5, 5 - 4 = 1, 1 + 5 = 6. Then 1 + 6 - 6 = 1.

No.

Alternatively, 1 + 2 + (3 - 4 + 5) - 6.

Inside the parentheses: 3 - 4 + 5 = 4. Then 1 + 2 + 4 - 6 = 7 - 6 = 1.

Same.

Alternatively, 1 + (2 + 3 - 4) + 5 - 6.

Inside the parentheses: 2 + 3 - 4 = 1. Then 1 + 1 + 5 - 6 = 1 + 1 + 5 - 6 = 1.

Same.

Alternatively, ((1 + 2) + 3 - 4) - (5 + 6). Which is (6 - 4) - 11 = 2 - 11 = -9.

Same as before.

So, no improvement.

Another idea: what if I group the first two terms and then subtract something?

For example, (1 + 2) + 3 - 4 + 5 + 6. Which is 3 + 3 - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

Alternatively, (1 + 2 + 3) - 4 + 5 + 6. Which is 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

Alternatively, 1 + (2 + 3) - 4 + 5 + 6. Which is 1 + 5 - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

So, no improvement.

Another thought: maybe grouping the subtraction with some additions. For example, 1 + 2 + (3 - 4 + 5 + 6). Which is 1 + 2 + 10 = 13.

Same.

Alternatively, 1 + (2 + 3 - 4 + 5 + 6). Which is 1 + 12 = 13.

Same.

Alternatively, 1 + 2 + 3 - (4 + 5 + 6). Which is -9.

Same as before.

So, seems like -9 is the minimum.

But let me check if there's another grouping.

What if I group 1 + 2 + 3 - 4 + (5 + 6). Which is 6 - 4 + 11 = 2 + 11 = 13.

Same.

Alternatively, 1 + 2 + (3 - 4 + 5 + 6). Which is 1 + 2 + 10 = 13.

Same.

Alternatively, 1 + (2 + 3 - 4 + 5 + 6). Which is 1 + 12 = 13.

Same.

Alternatively, ((1 + 2 + 3) - 4) + (5 + 6). Which is 2 + 11 = 13.

Same.

So, no improvement.

Wait, but what if I group in a way that subtracts more than just 4, 5, 6?

But all numbers are already in the expression.

Wait, for example, if I can have something like 1 + 2 + 3 - (4 + 5) - 6. Which is 6 - 9 - 6 = -9.

Same as before.

Alternatively, 1 + 2 + (3 - 4) - (5 + 6). Which is 1 + 2 - 1 - 11 = 3 - 1 - 11 = -9.

Same.

Alternatively, 1 + (2 + 3 - 4) - (5 + 6). Which is 1 + 1 - 11 = -9.

Same.

So, no matter how I group, I can't get lower than -9.

Wait, but let me think again. What if I group the first term with the subtraction?

For example, (1 + 2 + 3 - 4) - 5 - 6. Which is 2 - 5 - 6 = -9.

Same.

Alternatively, (1 + 2 + 3 - 4 - 5) - 6. Which is (6 - 4 - 5) - 6 = (-3) - 6 = -9.

Same.

Alternatively, 1 + (2 + 3 - 4 - 5) - 6. Which is 1 + (-4) - 6 = -9.

Same.

So, all these ways give me -9.

Is there a way to get lower?

Let me think. Suppose I have something like 1 + 2 + 3 - 4 - 5 - 6. Which is 6 - 4 - 5 - 6 = -9.

Same.

Alternatively, if I can have more subtractions. For example, if I can subtract 1, 2, 3, 4, 5, 6. But how?

Original expression is 1 + 2 + 3 - 4 + 5 + 6.

If I could subtract all numbers, that would be -1 -2 -3 -4 -5 -6 = -21. But how can I do that?

To do that, I would need to have all numbers subtracted. But in the original expression, the first three numbers are added.

To subtract them, I would need to have parentheses that negate them.

For example, if I have -(1 + 2 + 3) - 4 - 5 - 6. But that would require adding a negative sign in front of the entire expression, which isn't allowed.

Alternatively, if I can have something like (1 + 2 + 3) - (4 + 5 + 6). Which is 6 - 15 = -9.

Same as before.

Alternatively, if I can have something like 1 + 2 + 3 - 4 - 5 - 6. Which is same as above.

So, no way to get lower than -9.

Wait, but let me think again. What if I group the first term with the subtraction?

For example, 1 + (2 + 3 - 4 + 5 + 6). Which is 1 + 12 = 13.

No.

Alternatively, 1 + 2 + (3 - 4 + 5 + 6). Which is 1 + 2 + 10 = 13.

Same.

Alternatively, 1 + (2 + 3 - 4) + 5 + 6. Which is 1 + 1 + 5 + 6 = 13.

Same.

Alternatively, 1 + 2 + 3 - (4 + 5 + 6). Which is -9.

Same.

So, I think that -9 is the minimum.

But let me check if there's another grouping.

What if I group 1 + 2 + 3 - 4 + 5 - 6. Wait, that would be 6 - 4 + 5 - 6 = 2 + 5 - 6 = 1.

But how can I get that?

Original expression is 1 + 2 + 3 - 4 + 5 + 6. To get 1 + 2 + 3 - 4 + 5 - 6, I need to change the +6 into -6. How?

By adding parentheses around 6 with a negative sign.

But how?

If I have 1 + 2 + 3 - 4 + 5 - 6. Which is equivalent to 1 + 2 + 3 - 4 + 5 - 6.

But in the original expression, the last operation is +6. To make it -6, I need to have a negative sign in front of 6.

But how?

If I have 1 + 2 + 3 - 4 + 5 - (6). Which is allowed by adding parentheses around 6.

But in the original expression, the operations are +, +, -, +, +.

So, adding parentheses around 6 would be 1 + 2 + 3 - 4 + 5 - (6).

But that would be changing the +6 into -6.

But is that allowed?

Yes, because we can add parentheses.

So, that would be 1 + 2 + 3 - 4 + 5 - 6 = 6 - 4 + 5 - 6 = 2 + 5 - 6 = 1.

But that's not lower than -9.

Alternatively, if I can have something like 1 + 2 + 3 - 4 - 5 - 6. Which is -9.

So, that's better.

But how is that achieved?

By grouping 5 + 6 as a negative term.

So, in conclusion, after checking all possible groupings, it seems that the minimum value achievable is -9.

But let me check once more for any possible grouping that I might have missed.

Another idea: what if I group the first two terms and then subtract something?

For example, (1 + 2) + 3 - 4 + 5 + 6. Which is 3 + 3 - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

Alternatively, (1 + 2 + 3) - 4 + 5 + 6. Which is 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

Alternatively, 1 + (2 + 3) - 4 + 5 + 6. Which is 1 + 5 - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

Alternatively, 1 + 2 + (3 - 4) + 5 + 6. Which is 1 + 2 - 1 + 5 + 6 = 3 - 1 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

Alternatively, 1 + 2 + 3 - (4 + 5) + 6. Which is 6 - 9 + 6 = 3.

Same as example.

Alternatively, 1 + 2 + 3 - 4 + (5 + 6). Which is 6 - 4 + 11 = 13.

Same.

Alternatively, 1 + 2 + (3 - 4 + 5 + 6). Which is 1 + 2 + 10 = 13.

Same.

Alternatively, 1 + (2 + 3 - 4 + 5 + 6). Which is 1 + 12 = 13.

Same.

So, no improvement.

Therefore, after checking all possible groupings, I think the minimum value is -9.

But just to be thorough, let me think if there's a way to have more subtractions.

Suppose I have something like 1 + 2 + 3 - 4 - 5 - 6. Which is -9.

Alternatively, if I can have something like 1 + 2 + 3 - 4 - 5 - 6.

But that's the same as before.

Alternatively, if I can have 1 + 2 + 3 - (4 + 5 + 6). Which is same.

Alternatively, if I can have 1 + 2 + (3 - 4 - 5 - 6). Which is 1 + 2 + (-12) = -9.

Same.

So, no way to get lower.

Therefore, I think the minimum possible value is -9.

**Final Answer**
\boxed{-9}

To determine the **minimum possible value** of the expression
$$
1 + 2 + 3 - 4 + 5 + 6
$$
by **adding parentheses** (but **not rearranging terms** or changing operations), we must explore how parentheses can alter the **order of operations** to reduce the final result.

---

### Step 1: Understand the Original Expression

The original expression, evaluated left to right (since addition and subtraction have equal precedence), is:

$$
(1 + 2) + 3 - 4 + 5 + 6 = 3 + 3 - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13
$$

So, the **original value** is **13**.

---

### Step 2: Use Parentheses to Change the Order of Operations

We are allowed to **add parentheses** to change the **grouping** of terms, but **not the order** of the numbers or the operations. That is, we can't rearrange the sequence of numbers or change a '+' to a '-' or vice versa.

The key idea is that **parentheses can change the order in which subtractions are applied**, potentially **increasing the amount subtracted**, which would **reduce the total value**.

---

### Step 3: Try to Maximize the Amount Subtracted

To minimize the value, we want to **subtract as much as possible**. The most effective way to do this is to **group the largest possible positive numbers** and **subtract them all at once**.

Let’s try grouping the last three terms:

$$
1 + 2 + 3 - (4 + 5 + 6)
$$

Compute the grouped part:

$$
4 + 5 + 6 = 15
$$

Now compute the full expression:

$$
1 + 2 + 3 - 15 = 6 - 15 = -9
$$

This is a **valid** placement of parentheses, and it results in **-9**.

---

### Step 4: Check for Other Groupings That Could Yield a Lower Value

Let’s consider other groupings to see if we can get a value **less than -9**.

- **Grouping 4 + 5 + 6** as a single term to subtract:
  $$
  1 + 2 + 3 - (4 + 5 + 6) = -9
  $$

- **Grouping 5 + 6** and subtracting them:
  $$
  1 + 2 + 3 - 4 - (5 + 6) = 6 - 4 - 11 = -9
  $$

- **Grouping 3 - 4 + 5 + 6** and subtracting the result:
  $$
  1 + 2 + (3 - 4 + 5 + 6) = 1 + 2 + 10 = 13
  $$

- **Grouping 2 + 3 - 4 + 5 + 6** and subtracting the result:
  $$
  1 + (2 + 3 - 4 + 5 + 6) = 1 + 12 = 13
  $$

All other groupings either **do not change the value** or **result in a higher value** than -9.

---

### Step 5: Conclusion

After exploring all valid groupings of the expression using parentheses, the **minimum possible value** is achieved when we subtract the **sum of the largest three positive numbers** (4, 5, and 6), resulting in:

$$
1 + 2 + 3 - (4 + 5 + 6) = -9
$$

This is the **lowest possible value** under the given constraints.

---

### Final Answer

$$
\boxed{-9}
$$

[Assistant]
Explanation:
Okay, so I need to figure out the minimum possible value of the expression 1 + 2 + 3 - 4 + 5 + 6 by adding parentheses in different places. The original value is 13, but by changing where the parentheses are, I can get different results. The key here is that I can't rearrange the order of the terms or change the operations; I can only add parentheses. Also, multiplication isn't allowed, so I can't do things like (1+2+3)(-4) or anything like that.

First, let me make sure I understand the problem correctly. The original expression is 1 + 2 + 3 - 4 + 5 + 6. The operations are addition and subtraction. Parentheses can be added to change the order of operations, but the sequence of numbers and operations must stay the same. So, for example, if I put parentheses around 1 + 2, that would be (1 + 2) + 3 - 4 + 5 + 6, which is the same as the original because addition is associative. But if I put parentheses around 1 + 2 + 3, that would be (1 + 2 + 3) - 4 + 5 + 6, which is still the same as the original. However, if I put parentheses around something like 4 + 5, then that would change the value, as in the example given: 1 + 2 + 3 - (4 + 5) + 6 = 1 + 2 + 3 - 9 + 6 = 3.

So the challenge is to figure out how to place parentheses in such a way that the overall value is minimized. Let me think about how parentheses affect the expression. Parentheses can change the order in which operations are performed, so they can prioritize certain operations over others. Since subtraction is involved, which is not associative, parentheses can significantly affect the result.

Let me recall that subtraction is not associative, meaning that (a - b) - c is not the same as a - (b - c). So, by adding parentheses, we can change which subtractions are performed first, which can lead to different results.

The original expression is 1 + 2 + 3 - 4 + 5 + 6. Let me write down the original order of operations without parentheses:

Starting from left to right, since addition and subtraction have the same precedence and are evaluated left to right. So:

1 + 2 = 3

3 + 3 = 6

6 - 4 = 2

2 + 5 = 7

7 + 6 = 13.

So that's the original value. Now, if I add parentheses, I can change the order. For example, if I group the subtraction with more numbers, maybe I can subtract a larger number, thereby reducing the total.

Let me think about how to maximize the amount subtracted. Since subtraction is involved, if I can make a larger number to subtract, that would decrease the total. So, perhaps grouping terms in such a way that a large positive number is subtracted.

But how can I do that? Let me consider the expression again: 1 + 2 + 3 - 4 + 5 + 6.

The operations are: +, +, -, +, +.

So the expression can be thought of as:

((((1 + 2) + 3) - 4) + 5) + 6 = 13.

But if I can change the order of operations, maybe I can subtract more. For example, if I can subtract a larger number by grouping. Let me try different groupings.

First, let me note that all the operations are addition and subtraction. So parentheses can only change the order of these operations.

Let me think of the expression as a sequence of numbers with operations between them: 1 + 2 + 3 - 4 + 5 + 6.

Each operation is either + or -, and parentheses can change the order.

One approach is to consider that parentheses can be used to create a larger negative term. For example, if I can make a term like -(4 + 5 + 6), that would be -15, which would significantly reduce the total. However, can I do that? Let me check.

Original expression: 1 + 2 + 3 - 4 + 5 + 6.

If I want to subtract (4 + 5 + 6), that would require that the subtraction applies to all three terms. But in the original expression, the subtraction is only on 4, and then addition on 5 and 6. So how can I make the subtraction apply to 4, 5, and 6?

Let me try to see. If I can group 4 + 5 + 6 as a single term, but how? Let me think.

Suppose I have 1 + 2 + 3 - (4 + 5 + 6). That would be 1 + 2 + 3 - 15 = 6 - 15 = -9. But is that a valid placement of parentheses? Let me check the original expression. The original expression is 1 + 2 + 3 - 4 + 5 + 6. To get 1 + 2 + 3 - (4 + 5 + 6), I need to add parentheses around 4 + 5 + 6. But in the original expression, after the subtraction sign, there is a 4, then +5, then +6. So to group 4 + 5 + 6, I need to have parentheses starting after the subtraction sign.

Wait, but the original expression is 1 + 2 + 3 - 4 + 5 + 6. So the subtraction is only on 4, and then addition on 5 and 6. If I want to group 4 + 5 + 6, I need to have parentheses that start after the subtraction. Let me think.

If I write it as 1 + 2 + 3 - (4 + 5 + 6), that would require that the subtraction applies to the entire group. However, in the original expression, the subtraction is only on 4, and then the rest are additions. So to make that happen, I need to have parentheses that start after the subtraction sign.

But how is that possible? Let me think about the structure. The original expression is:

1 + 2 + 3 - 4 + 5 + 6

So the operations are:

Between 1 and 2: +

Between 2 and 3: +

Between 3 and 4: -

Between 4 and 5: +

Between 5 and 6: +

So if I want to group 4 + 5 + 6, I need to have parentheses that start at 4 and include 5 and 6. However, the subtraction is between 3 and 4. So if I put parentheses around 4 + 5 + 6, that would be:

1 + 2 + 3 - (4 + 5 + 6)

But in the original expression, the subtraction is only on 4. So to do that, I need to have the parentheses start after the subtraction. But how does that work? Let me think of the expression as:

1 + 2 + 3 - 4 + 5 + 6

If I want to group 4 + 5 + 6, I need to have parentheses that start at the 4, but since the 4 is preceded by a subtraction, I need to make sure that the parentheses are placed correctly.

Wait, actually, in terms of operator precedence, if I have 1 + 2 + 3 - (4 + 5 + 6), that is valid. Because the parentheses override the default left-to-right evaluation. So even though the original expression is 1 + 2 + 3 - 4 + 5 + 6, adding parentheses around 4 + 5 + 6 would change the expression to 1 + 2 + 3 - (4 + 5 + 6).

But is that allowed? The problem says we can add parentheses in various locations. So as long as we don't rearrange the terms, just add parentheses. So yes, that should be allowed.

So that would be 1 + 2 + 3 - (4 + 5 + 6) = 6 - 15 = -9.

Is that a valid expression? Let me check. The original expression is 1 + 2 + 3 - 4 + 5 + 6. If I add parentheses around 4 + 5 + 6, then the expression becomes 1 + 2 + 3 - (4 + 5 + 6). That is allowed.

So that would be -9. Is that the minimum? Maybe not. Let me check if there are other ways to group terms to get a lower value.

Alternatively, maybe grouping more terms. For example, if I can subtract more numbers. Let me think.

Another idea: if I can have multiple subtractions. For example, if I can have something like 1 + (2 + 3 - 4) + 5 + 6. Wait, but that would be 1 + (2 + 3 - 4) + 5 + 6 = 1 + 1 + 5 + 6 = 13, same as original.

Alternatively, maybe grouping the first few terms. For example, (1 + 2 + 3 - 4) + 5 + 6. That would be (6 - 4) + 5 + 6 = 2 + 5 + 6 = 13. Still same.

Alternatively, if I group 1 + 2 + (3 - 4) + 5 + 6. That would be 1 + 2 + (-1) + 5 + 6 = 1 + 2 -1 + 5 + 6 = 13.

Hmm. So that doesn't change the value.

Alternatively, maybe grouping more terms. Let me think. What if I group 1 + 2 + 3 - (4 + 5) + 6? That was the example given, which equals 3.

Alternatively, grouping 1 + 2 + (3 - 4 + 5) + 6. Let's compute that: 1 + 2 + (4) + 6 = 1 + 2 + 4 + 6 = 13.

Same as original.

Alternatively, grouping 1 + (2 + 3 - 4 + 5) + 6. That would be 1 + (6) + 6 = 13.

Same.

Alternatively, grouping (1 + 2 + 3 - 4 + 5) + 6. That would be (7) + 6 = 13.

Same.

So maybe grouping the first part.

But if I want to get a lower value, I need to have more subtractions.

Wait, so earlier I thought of grouping 4 + 5 + 6 as a single term to subtract, which gives -9. Is that the minimum? Let me check if there are other groupings that can lead to a lower value.

Another idea: maybe grouping some additions and subtractions in a way that creates a larger negative number. For example, if I can have something like 1 + 2 + (3 - 4 + 5 + 6). Wait, that would be 1 + 2 + (10) = 13.

Alternatively, 1 + (2 + 3 - 4 + 5 + 6). That would be 1 + (12) = 13.

Alternatively, maybe grouping the first three terms and then subtracting something else.

Wait, let me think again. If I can have more subtractions. For example, if I can have 1 + 2 + 3 - 4 - 5 - 6. But how can I do that? Because in the original expression, after the subtraction of 4, there are additions of 5 and 6. To make them subtractions, I need to add parentheses.

Wait, but how? Let me think.

Original expression: 1 + 2 + 3 - 4 + 5 + 6.

If I want to make the +5 and +6 into -5 and -6, I need to have parentheses that change the sign. For example, if I can have something like 1 + 2 + 3 - 4 - (5 + 6). But how can I do that?

Let me check. If I have 1 + 2 + 3 - 4 - (5 + 6). That would be 6 - 4 - 11 = 2 - 11 = -9. Wait, same as before.

But how is that achieved by adding parentheses? Let me see.

Original expression: 1 + 2 + 3 - 4 + 5 + 6.

To get 1 + 2 + 3 - 4 - (5 + 6), I need to add parentheses around 5 + 6, but with a negative sign. However, in the original expression, after the subtraction of 4, there is a +5 and +6. So to make them -5 and -6, I need to have parentheses that change the sign.

But how? Because the original expression is 1 + 2 + 3 - 4 + 5 + 6. If I want to have - (5 + 6), that would require that the +5 and +6 are inside a parenthesis with a negative sign. But the original expression has a +5 and +6 after the subtraction.

Wait, maybe if I group the 5 + 6 as a negative term. Let me think.

If I have 1 + 2 + 3 - 4 + (5 + 6). That is the original expression. But if I want to have 1 + 2 + 3 - 4 - (5 + 6), that would require that the +5 and +6 are turned into -5 and -6. How can I do that?

Is there a way to add parentheses to achieve that?

Let me think. If I have 1 + 2 + 3 - 4 + 5 + 6. If I add parentheses around 5 + 6, but with a negative sign. Wait, but the original expression has a +5 and +6. To make them negative, I need to have a negative sign in front of them.

But how? Unless I can group them with a subtraction.

Wait, maybe if I have 1 + 2 + 3 - (4 + 5 + 6). Which is what I thought before. That would be 1 + 2 + 3 - (4 + 5 + 6) = 6 - 15 = -9.

Alternatively, if I have 1 + 2 + 3 - 4 - (5 + 6). But how is that different?

Wait, in the original expression, after the subtraction of 4, there is a +5 and +6. If I want to make them -5 and -6, I need to have a negative sign in front of them. But how can I do that?

Is there a way to add parentheses to create that? For example, if I have 1 + 2 + 3 - 4 - (5 + 6). But in the original expression, the operations are +, +, -, +, +. So to get a negative sign before 5 + 6, I need to have a subtraction operation that applies to 5 + 6.

But in the original expression, after the subtraction of 4, there is a +5. So if I want to have a subtraction of 5 and 6, I need to have parentheses that start after the subtraction of 4.

Wait, maybe like this: 1 + 2 + 3 - 4 - (5 + 6). But how is that different from 1 + 2 + 3 - (4 + 5 + 6)?

Let me compute both:

First: 1 + 2 + 3 - (4 + 5 + 6) = 6 - 15 = -9.

Second: 1 + 2 + 3 - 4 - (5 + 6) = 6 - 4 - 11 = 2 - 11 = -9.

Same result. So both ways give -9.

So that seems like a valid way.

But is there a way to get even lower? Let me think.

What if I can subtract more? For example, if I can have something like 1 + 2 + 3 - 4 - 5 - 6. That would be 6 - 4 - 5 - 6 = 6 - 15 = -9. Same as before.

But how can I get that?

Original expression: 1 + 2 + 3 - 4 + 5 + 6.

To get 1 + 2 + 3 - 4 - 5 - 6, I need to change the +5 and +6 into -5 and -6. How can I do that with parentheses?

If I can group the 5 and 6 with a negative sign. For example, if I have 1 + 2 + 3 - 4 - (5 + 6). Which is what I did before.

Alternatively, if I have 1 + 2 + 3 - (4 + 5 + 6).

Either way, same result.

So maybe -9 is the minimum? But let me check if there are other groupings.

Another idea: maybe grouping some terms in a way that creates more subtractions. For example, if I can have something like (1 + 2 + 3 - 4) - (5 + 6). Let me compute that.

(1 + 2 + 3 - 4) = 2. Then subtract (5 + 6) = 11. So 2 - 11 = -9. Same as before.

Alternatively, grouping (1 + 2) + 3 - (4 + 5 + 6). Which is 3 + 3 - 15 = 6 - 15 = -9.

Same result.

Is there a way to get lower than -9? Let me think.

Suppose I try to subtract more numbers. For example, if I can have something like 1 + (2 + 3 - 4 + 5) - 6. Wait, let me compute that.

First, compute inside the parentheses: 2 + 3 - 4 + 5 = 2 + 3 = 5, 5 - 4 = 1, 1 + 5 = 6. Then 1 + 6 - 6 = 1.

Not helpful.

Alternatively, 1 + 2 + (3 - 4 + 5) - 6. Let's compute: 3 - 4 + 5 = 4. Then 1 + 2 + 4 - 6 = 7 - 6 = 1.

Same.

Alternatively, 1 + (2 + 3 - 4) + 5 - 6. That would be 1 + (1) + 5 - 6 = 1 + 1 + 5 - 6 = 1.

Same.

Alternatively, maybe grouping more terms. Let me think.

What if I have ((1 + 2 + 3) - 4) - (5 + 6). Which is (6 - 4) - 11 = 2 - 11 = -9.

Same as before.

Alternatively, ((1 + 2) + 3 - 4) - (5 + 6) = same thing.

Is there a way to have more subtractions? For example, if I can have something like 1 + 2 + 3 - (4 + 5) - 6. Let me compute that.

Original expression: 1 + 2 + 3 - (4 + 5) - 6.

Compute inside the parentheses: 4 + 5 = 9. Then 1 + 2 + 3 - 9 - 6 = 6 - 9 - 6 = -9.

Same result.

So regardless of how I group the terms to subtract more, I get -9.

Is there a way to get lower than -9? Let me think.

Suppose I try to subtract more numbers. For example, if I can have something like 1 + 2 + 3 - 4 - 5 - 6. Which is 6 - 4 - 5 - 6 = 6 - 15 = -9.

Same as before.

Alternatively, if I can have something like 1 + 2 + (3 - 4 - 5 - 6). Let me compute that.

Inside the parentheses: 3 - 4 - 5 - 6 = 3 - 15 = -12. Then 1 + 2 + (-12) = 3 - 12 = -9.

Same result.

So no matter how I group, I can't get lower than -9?

Wait, let me think again. What if I group in a different way? For example, if I have 1 + (2 + 3 - 4 + 5 + 6). Wait, that would be 1 + (12) = 13.

No.

Alternatively, 1 + 2 + (3 - 4 + 5 + 6). That is 1 + 2 + (10) = 13.

Same.

Alternatively, 1 + (2 + 3 - 4) + 5 + 6. Which is 1 + 1 + 5 + 6 = 13.

Same.

Alternatively, maybe grouping the first two terms and then subtracting something.

For example, (1 + 2) + 3 - 4 + 5 + 6. Which is 3 + 3 - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

Alternatively, (1 + 2 + 3) - 4 + 5 + 6. Which is 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

So all these groupings don't change the value.

Wait, so maybe -9 is the minimum?

But let me check another possibility. What if I group some terms in a way that creates a larger negative number? For example, if I can have something like 1 + 2 + 3 - (4 + 5 + 6). Which is -9.

Alternatively, if I can have 1 + 2 + (3 - 4 + 5) - 6. Let me compute that.

Inside the parentheses: 3 - 4 + 5 = 4. Then 1 + 2 + 4 - 6 = 7 - 6 = 1.

No.

Alternatively, 1 + (2 + 3 - 4 + 5) - 6. Which is 1 + (6) - 6 = 1.

Same.

Alternatively, 1 + 2 + 3 - (4 + 5) + 6. Which is 6 - 9 + 6 = 3.

Which is the example given.

Alternatively, 1 + 2 + (3 - 4 + 5 + 6). Which is 1 + 2 + 10 = 13.

Same.

Alternatively, 1 + (2 + 3 - 4 + 5 + 6). Which is 1 + 12 = 13.

Same.

So, all these groupings don't get me lower than -9.

Wait, but let me think again. What if I group the first term with the subtraction? For example, (1 + 2 + 3 - 4) + 5 + 6. Which is 2 + 5 + 6 = 13.

Same.

Alternatively, 1 + (2 + 3 - 4 + 5 + 6). Which is 1 + 12 = 13.

Same.

Alternatively, 1 + 2 + (3 - 4 + 5 + 6). Which is 1 + 2 + 10 = 13.

Same.

So, seems like all the groupings that don't subtract more than 4 + 5 + 6 give me the same or higher value.

But is there a way to subtract more? For example, if I can have something like 1 + 2 + 3 - 4 - 5 - 6. Which is 6 - 4 - 5 - 6 = -9.

But how can I get that?

Original expression: 1 + 2 + 3 - 4 + 5 + 6.

To get 1 + 2 + 3 - 4 - 5 - 6, I need to change the +5 and +6 into -5 and -6. How can I do that with parentheses?

If I have 1 + 2 + 3 - 4 - (5 + 6). Which is allowed by adding parentheses around 5 + 6.

But in the original expression, after the subtraction of 4, there is a +5 and +6. So to make them -5 and -6, I need to have a negative sign in front of them.

But how?

Wait, if I have 1 + 2 + 3 - 4 - (5 + 6). That would be equivalent to 1 + 2 + 3 - 4 - 5 - 6.

But in the original expression, the operations are +, +, -, +, +. So to get a subtraction of 5 and 6, I need to have parentheses that start after the subtraction of 4.

But how is that possible? Let me think of the original expression as:

1 + 2 + 3 - 4 + 5 + 6

If I want to have 1 + 2 + 3 - 4 - (5 + 6), that would require that the +5 and +6 are inside a parenthesis with a negative sign.

But in the original expression, the +5 and +6 are after the subtraction of 4. So if I add parentheses around 5 + 6, but with a negative sign.

Wait, but how does that work?

Let me think of the expression as:

1 + 2 + 3 - 4 + 5 + 6

If I want to have 1 + 2 + 3 - 4 - (5 + 6), that would mean that the +5 and +6 are being subtracted. But in the original expression, the operations are +5 and +6.

So to change them to -5 and -6, I need to have a negative sign in front of them. But how can I do that?

Ah! Because the original expression has a subtraction after 3, which is -4, and then additions. If I can group the 5 and 6 with a negative sign, that would require that the subtraction of 4 is followed by a subtraction of (5 + 6).

But how is that possible?

Wait, maybe by adding parentheses around 5 + 6 and then having the subtraction apply to that.

But in the original expression, after the subtraction of 4, there is a +5. So if I add parentheses around 5 + 6, but with a negative sign.

Wait, but the original expression is 1 + 2 + 3 - 4 + 5 + 6.

If I add parentheses around 5 + 6, but with a negative sign, that would be 1 + 2 + 3 - 4 - (5 + 6).

But how is that different from the original expression?

In the original expression, the operations are +, +, -, +, +.

If I add parentheses around 5 + 6, but with a negative sign, that would require that the +5 and +6 are turned into -5 and -6.

But how can I do that?

Wait, maybe by changing the sign of the 5 and 6. But the problem says we can't rearrange the order of the terms. So we can't move the 5 and 6.

But we can add parentheses. So if I have 1 + 2 + 3 - 4 - (5 + 6), that is equivalent to 1 + 2 + 3 - 4 - 5 - 6.

But how is that achieved by adding parentheses?

Let me think of the original expression as:

1 + 2 + 3 - 4 + 5 + 6

If I want to have 1 + 2 + 3 - 4 - (5 + 6), that would require that the +5 and +6 are inside a parenthesis with a negative sign.

But in the original expression, the +5 and +6 are after the subtraction of 4. So if I add parentheses around 5 + 6, but with a negative sign, that would be:

1 + 2 + 3 - 4 - (5 + 6)

But how is that different from the original expression?

Wait, the original expression is 1 + 2 + 3 - 4 + 5 + 6.

If I add parentheses around 5 + 6, but with a negative sign, that would be changing the +5 and +6 into -5 and -6.

But how can I do that?

Ah! Because the original expression has a subtraction after 3, which is -4, and then additions. If I can group the 5 + 6 with a negative sign, that would require that the subtraction of 4 is followed by a subtraction of (5 + 6).

But how is that possible?

Wait, maybe by adding parentheses around 4 + 5 + 6?

Wait, if I have 1 + 2 + 3 - (4 + 5 + 6). That is allowed.

But that is equivalent to 1 + 2 + 3 - 4 - 5 - 6.

So that is the same as 1 + 2 + 3 - 4 - 5 - 6.

So that is allowed.

Therefore, that gives -9.

Is there a way to get lower than that?

Let me think. Suppose I can have more subtractions. For example, if I can subtract more numbers.

But all numbers are already being subtracted or added.

Wait, the numbers are 1, 2, 3, 4, 5, 6.

In the original expression, 1, 2, 3 are added, 4 is subtracted, 5 and 6 are added.

If I can subtract more numbers, like subtracting 4, 5, 6, and also subtracting something else?

But how?

Wait, if I can have something like 1 + 2 + 3 - 4 - 5 - 6, which is subtracting 4, 5, 6.

But that's already the case in the grouping where we have 1 + 2 + 3 - (4 + 5 + 6).

Is there a way to subtract more? For example, subtracting 1, 2, 3, 4, 5, 6?

But how?

If I can have something like -(1 + 2 + 3 + 4 + 5 + 6). But how can I do that?

Original expression is 1 + 2 + 3 - 4 + 5 + 6.

To have all numbers subtracted, I would need to have something like -(1 + 2 + 3 + 4 + 5 + 6). But that would require changing the signs of all terms, which isn't possible unless we can add parentheses in a way that negates all terms.

But how?

For example, if I have -(1 + 2 + 3 - 4 + 5 + 6). But that would be -1 -2 -3 +4 -5 -6. Which is different.

But how can I get that?

Wait, the original expression is 1 + 2 + 3 - 4 + 5 + 6. If I want to negate the entire expression, I would need to have something like -(1 + 2 + 3 - 4 + 5 + 6). But that would require adding a negative sign in front of the entire expression. However, the problem says we can't change the operations, only add parentheses.

So we can't add a negative sign unless it's part of the existing operations.

Wait, the original expression doesn't have a negative sign at the beginning. So we can't add that.

Alternatively, if we can group the entire expression with a negative sign. But how?

For example, if we have -(1 + 2 + 3 - 4 + 5 + 6). But that would require adding a negative sign in front, which isn't allowed.

The problem says we can't change addition or subtraction into multiplication, but we can add parentheses. So we can't add a negative sign unless it's part of the existing operations.

Therefore, we can't have a negative sign in front of the entire expression.

Therefore, the minimum we can get is -9.

But let me check another possibility. Suppose I group some terms in a way that creates more subtractions.

For example, if I have 1 + (2 + 3 - 4 + 5) - 6. Let me compute that.

Inside the parentheses: 2 + 3 - 4 + 5 = 2 + 3 = 5, 5 - 4 = 1, 1 + 5 = 6. Then 1 + 6 - 6 = 1.

No.

Alternatively, 1 + 2 + (3 - 4 + 5) - 6.

Inside the parentheses: 3 - 4 + 5 = 4. Then 1 + 2 + 4 - 6 = 7 - 6 = 1.

Same.

Alternatively, 1 + (2 + 3 - 4) + 5 - 6.

Inside the parentheses: 2 + 3 - 4 = 1. Then 1 + 1 + 5 - 6 = 1 + 1 + 5 - 6 = 1.

Same.

Alternatively, ((1 + 2) + 3 - 4) - (5 + 6). Which is (6 - 4) - 11 = 2 - 11 = -9.

Same as before.

So, no improvement.

Another idea: what if I group the first two terms and then subtract something?

For example, (1 + 2) + 3 - 4 + 5 + 6. Which is 3 + 3 - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

Alternatively, (1 + 2 + 3) - 4 + 5 + 6. Which is 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

Alternatively, 1 + (2 + 3) - 4 + 5 + 6. Which is 1 + 5 - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

So, no improvement.

Another thought: maybe grouping the subtraction with some additions. For example, 1 + 2 + (3 - 4 + 5 + 6). Which is 1 + 2 + 10 = 13.

Same.

Alternatively, 1 + (2 + 3 - 4 + 5 + 6). Which is 1 + 12 = 13.

Same.

Alternatively, 1 + 2 + 3 - (4 + 5 + 6). Which is -9.

Same as before.

So, seems like -9 is the minimum.

But let me check if there's another grouping.

What if I group 1 + 2 + 3 - 4 + (5 + 6). Which is 6 - 4 + 11 = 2 + 11 = 13.

Same.

Alternatively, 1 + 2 + (3 - 4 + 5 + 6). Which is 1 + 2 + 10 = 13.

Same.

Alternatively, 1 + (2 + 3 - 4 + 5 + 6). Which is 1 + 12 = 13.

Same.

Alternatively, ((1 + 2 + 3) - 4) + (5 + 6). Which is 2 + 11 = 13.

Same.

So, no improvement.

Wait, but what if I group in a way that subtracts more than just 4, 5, 6?

But all numbers are already in the expression.

Wait, for example, if I can have something like 1 + 2 + 3 - (4 + 5) - 6. Which is 6 - 9 - 6 = -9.

Same as before.

Alternatively, 1 + 2 + (3 - 4) - (5 + 6). Which is 1 + 2 - 1 - 11 = 3 - 1 - 11 = -9.

Same.

Alternatively, 1 + (2 + 3 - 4) - (5 + 6). Which is 1 + 1 - 11 = -9.

Same.

So, no matter how I group, I can't get lower than -9.

Wait, but let me think again. What if I group the first term with the subtraction?

For example, (1 + 2 + 3 - 4) - 5 - 6. Which is 2 - 5 - 6 = -9.

Same.

Alternatively, (1 + 2 + 3 - 4 - 5) - 6. Which is (6 - 4 - 5) - 6 = (-3) - 6 = -9.

Same.

Alternatively, 1 + (2 + 3 - 4 - 5) - 6. Which is 1 + (-4) - 6 = -9.

Same.

So, all these ways give me -9.

Is there a way to get lower?

Let me think. Suppose I have something like 1 + 2 + 3 - 4 - 5 - 6. Which is 6 - 4 - 5 - 6 = -9.

Same.

Alternatively, if I can have more subtractions. For example, if I can subtract 1, 2, 3, 4, 5, 6. But how?

Original expression is 1 + 2 + 3 - 4 + 5 + 6.

If I could subtract all numbers, that would be -1 -2 -3 -4 -5 -6 = -21. But how can I do that?

To do that, I would need to have all numbers subtracted. But in the original expression, the first three numbers are added.

To subtract them, I would need to have parentheses that negate them.

For example, if I have -(1 + 2 + 3) - 4 - 5 - 6. But that would require adding a negative sign in front of the entire expression, which isn't allowed.

Alternatively, if I can have something like (1 + 2 + 3) - (4 + 5 + 6). Which is 6 - 15 = -9.

Same as before.

Alternatively, if I can have something like 1 + 2 + 3 - 4 - 5 - 6. Which is same as above.

So, no way to get lower than -9.

Wait, but let me think again. What if I group the first term with the subtraction?

For example, 1 + (2 + 3 - 4 + 5 + 6). Which is 1 + 12 = 13.

No.

Alternatively, 1 + 2 + (3 - 4 + 5 + 6). Which is 1 + 2 + 10 = 13.

Same.

Alternatively, 1 + (2 + 3 - 4) + 5 + 6. Which is 1 + 1 + 5 + 6 = 13.

Same.

Alternatively, 1 + 2 + 3 - (4 + 5 + 6). Which is -9.

Same.

So, I think that -9 is the minimum.

But let me check if there's another grouping.

What if I group 1 + 2 + 3 - 4 + 5 - 6. Wait, that would be 6 - 4 + 5 - 6 = 2 + 5 - 6 = 1.

But how can I get that?

Original expression is 1 + 2 + 3 - 4 + 5 + 6. To get 1 + 2 + 3 - 4 + 5 - 6, I need to change the +6 into -6. How?

By adding parentheses around 6 with a negative sign.

But how?

If I have 1 + 2 + 3 - 4 + 5 - 6. Which is equivalent to 1 + 2 + 3 - 4 + 5 - 6.

But in the original expression, the last operation is +6. To make it -6, I need to have a negative sign in front of 6.

But how?

If I have 1 + 2 + 3 - 4 + 5 - (6). Which is allowed by adding parentheses around 6.

But in the original expression, the operations are +, +, -, +, +.

So, adding parentheses around 6 would be 1 + 2 + 3 - 4 + 5 - (6).

But that would be changing the +6 into -6.

But is that allowed?

Yes, because we can add parentheses.

So, that would be 1 + 2 + 3 - 4 + 5 - 6 = 6 - 4 + 5 - 6 = 2 + 5 - 6 = 1.

But that's not lower than -9.

Alternatively, if I can have something like 1 + 2 + 3 - 4 - 5 - 6. Which is -9.

So, that's better.

But how is that achieved?

By grouping 5 + 6 as a negative term.

So, in conclusion, after checking all possible groupings, it seems that the minimum value achievable is -9.

But let me check once more for any possible grouping that I might have missed.

Another idea: what if I group the first two terms and then subtract something?

For example, (1 + 2) + 3 - 4 + 5 + 6. Which is 3 + 3 - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

Alternatively, (1 + 2 + 3) - 4 + 5 + 6. Which is 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

Alternatively, 1 + (2 + 3) - 4 + 5 + 6. Which is 1 + 5 - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

Alternatively, 1 + 2 + (3 - 4) + 5 + 6. Which is 1 + 2 - 1 + 5 + 6 = 3 - 1 + 5 + 6 = 2 + 5 + 6 = 13.

Same.

Alternatively, 1 + 2 + 3 - (4 + 5) + 6. Which is 6 - 9 + 6 = 3.

Same as example.

Alternatively, 1 + 2 + 3 - 4 + (5 + 6). Which is 6 - 4 + 11 = 13.

Same.

Alternatively, 1 + 2 + (3 - 4 + 5 + 6). Which is 1 + 2 + 10 = 13.

Same.

Alternatively, 1 + (2 + 3 - 4 + 5 + 6). Which is 1 + 12 = 13.

Same.

So, no improvement.

Therefore, after checking all possible groupings, I think the minimum value is -9.

But just to be thorough, let me think if there's a way to have more subtractions.

Suppose I have something like 1 + 2 + 3 - 4 - 5 - 6. Which is -9.

Alternatively, if I can have something like 1 + 2 + 3 - 4 - 5 - 6.

But that's the same as before.

Alternatively, if I can have 1 + 2 + 3 - (4 + 5 + 6). Which is same.

Alternatively, if I can have 1 + 2 + (3 - 4 - 5 - 6). Which is 1 + 2 + (-12) = -9.

Same.

So, no way to get lower.

Therefore, I think the minimum possible value is -9.

**Final Answer**
\boxed{-9}

To determine the **minimum possible value** of the expression
$$
1 + 2 + 3 - 4 + 5 + 6
$$
by **adding parentheses** (but **not rearranging terms** or changing operations), we must explore how parentheses can alter the **order of operations** to reduce the final result.

---

### Step 1: Understand the Original Expression

The original expression, evaluated left to right (since addition and subtraction have equal precedence), is:

$$
(1 + 2) + 3 - 4 + 5 + 6 = 3 + 3 - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 2 + 5 + 6 = 13
$$

So, the **original value** is **13**.

---

### Step 2: Use Parentheses to Change the Order of Operations

We are allowed to **add parentheses** to change the **grouping** of terms, but **not the order** of the numbers or the operations. That is, we can't rearrange the sequence of numbers or change a '+' to a '-' or vice versa.

The key idea is that **parentheses can change the order in which subtractions are applied**, potentially **increasing the amount subtracted**, which would **reduce the total value**.

---

### Step 3: Try to Maximize the Amount Subtracted

To minimize the value, we want to **subtract as much as possible**. The most effective way to do this is to **group the largest possible positive numbers** and **subtract them all at once**.

Let’s try grouping the last three terms:

$$
1 + 2 + 3 - (4 + 5 + 6)
$$

Compute the grouped part:

$$
4 + 5 + 6 = 15
$$

Now compute the full expression:

$$
1 + 2 + 3 - 15 = 6 - 15 = -9
$$

This is a **valid** placement of parentheses, and it results in **-9**.

---

### Step 4: Check for Other Groupings That Could Yield a Lower Value

Let’s consider other groupings to see if we can get a value **less than -9**.

- **Grouping 4 + 5 + 6** as a single term to subtract:
  $$
  1 + 2 + 3 - (4 + 5 + 6) = -9
  $$

- **Grouping 5 + 6** and subtracting them:
  $$
  1 + 2 + 3 - 4 - (5 + 6) = 6 - 4 - 11 = -9
  $$

- **Grouping 3 - 4 + 5 + 6** and subtracting the result:
  $$
  1 + 2 + (3 - 4 + 5 + 6) = 1 + 2 + 10 = 13
  $$

- **Grouping 2 + 3 - 4 + 5 + 6** and subtracting the result:
  $$
  1 + (2 + 3 - 4 + 5 + 6) = 1 + 12 = 13
  $$

All other groupings either **do not change the value** or **result in a higher value** than -9.

---

### Step 5: Conclusion

After exploring all valid groupings of the expression using parentheses, the **minimum possible value** is achieved when we subtract the **sum of the largest three positive numbers** (4, 5, and 6), resulting in:

$$
1 + 2 + 3 - (4 + 5 + 6) = -9
$$

This is the **lowest possible value** under the given constraints.

---

### Final Answer

$$
\boxed{-9}
$$

Solution: Because every term before the subtraction sign is added, and addition is associative (parentheses don't matter) placing parentheses on them will not change their value. However, subtraction is not associative. Since we are trying to minimize this expression's value, we want to subtract as much as possible. The expression is thus minimized when parentheses are placed in the following location: $1+2+3-(4+5+6)$. This simplifies to $1+2+3-15 = 6-15=\boxed{-9}$.


Explanation:
Okay, so Daniel says that the popularity of a television, which is measured by the number of sales, is inversely proportional to its cost. Hmm, inversely proportional... Let me recall what that means. If two quantities are inversely proportional, that means as one increases, the other decreases, and their product remains constant. So, if I have two variables, let's say sales (let me call that S) and cost (let me call that C), then according to Daniel's theory, S is inversely proportional to C.

So, mathematically, that would be S = k / C, where k is the constant of proportionality. Or maybe it's better to write it as S * C = k. Yeah, that's another way to think about it. So the product of the number of sales and the cost should be a constant.

Alright, so the problem says that 15 customers buy a television that costs $1500. So, using that information, I can find the constant k. Let me write that down.

Given:
When C = 1500, S = 15. So plugging into S * C = k, we get 15 * 1500 = k. Let me calculate that.

15 times 1500... Well, 15 times 1000 is 15,000, and 15 times 500 is 7,500. So adding those together, 15,000 + 7,500 = 22,500. So k is 22,500.

So according to Daniel's theory, the product of the number of sales and the cost is always 22,500. Now, the question is asking, how many customers would buy a television that costs $2500? Let me denote the new number of sales as S', and the new cost as C' = 2500.

So, using the same formula S' * C' = k. Therefore, S' = k / C'. Plugging in the numbers, S' = 22,500 / 2500. Let me compute that.

Dividing 22,500 by 2500. Hmm, 2500 goes into 22,500 how many times? Well, 2500 times 9 is 22,500 because 2500*10 is 25,000, so subtract 2500 once, which is 22,500. So 9 times. Therefore, S' = 9.

Wait, so according to Daniel's theory, if the cost increases to $2500, the number of customers would decrease to 9? Let me check my steps again to make sure I didn't make a mistake.

First, inverse proportionality: S ∝ 1/C, so S*C = k. Then, with S = 15 and C = 1500, k = 15*1500 = 22,500. Then, for C = 2500, S = 22,500 / 2500 = 9. That seems correct.

But let me think again if inverse proportionality is the right approach here. The problem says "popularity... is inversely proportional to its cost." So popularity is sales, so sales inversely proportional to cost. So yes, that's what I used.

Alternatively, maybe someone could misinterpret it as popularity is inversely proportional to the cost, meaning that popularity = k / cost. Which is the same thing. So, yeah, that seems right.

Alternatively, maybe there's a different way to think about it. For example, if the cost doubles, the sales would halve, but in this case, the cost isn't doubling, it's increasing from 1500 to 2500. Let me check the ratio of the costs.

Original cost: 1500, new cost: 2500. The ratio of new cost to original cost is 2500 / 1500 = 5/3 ≈ 1.666... So, since they are inversely proportional, the number of sales should be multiplied by 3/5.

Original sales: 15. So new sales would be 15 * (3/5) = 9. Which matches the previous result.

So that's another way to think about it. If the cost increases by a factor of 5/3, then sales decrease by a factor of 3/5.

So, 15 * 3/5 is indeed 9.

Therefore, the answer should be 9 customers.

But let me just make sure that there isn't any trick or anything in the question. The problem says "according to Daniel's theory," so we don't have to question the validity of the theory, just apply it.

So, given that, the answer is 9.

Alternatively, maybe the question is trying to test understanding of inverse proportionality. Let me think if there's another way someone might approach it incorrectly. For example, if someone thought it was directly proportional, they would do 15 * (2500 / 1500) = 15 * (5/3) = 25. But that's not the case here.

But since it's inversely proportional, it's the reciprocal. So, 15 * (1500 / 2500) = 15 * (3/5) = 9.

Yes, that's correct.

Alternatively, maybe someone might think that the number of customers is inversely proportional to the cost, so they might set up the proportion as 15 / 1500 = S / 2500, but that would be incorrect because if it's inversely proportional, the proportion should be 15 / S = 1500 / 2500.

Let me check that.

If S1 * C1 = S2 * C2, then 15 * 1500 = S2 * 2500. Therefore, S2 = (15 * 1500) / 2500 = same as before, 9.

Alternatively, if someone set up the proportion as S1 / S2 = C2 / C1, which is the same as S1 / S2 = 2500 / 1500, so S2 = S1 * (1500 / 2500) = 15 * 3/5 = 9.

So that's another way to think about it.

So, all methods lead to 9.

Therefore, I think the answer is 9.

**Final Answer**
The number of customers who would buy a television that costs $\$$2500 is \boxed{9}.

To solve the problem, we begin by interpreting Daniel's claim: the **popularity of a television (measured in number of sales)** is **inversely proportional** to its **cost**. This means that as the cost increases, the number of sales decreases, and their product remains constant.

---

### Step 1: Understanding Inverse Proportionality

If two quantities are inversely proportional, their product is constant. That is:

$$
S \cdot C = k
$$

Where:
- $ S $ is the number of sales,
- $ C $ is the cost of the television,
- $ k $ is the constant of proportionality.

---

### Step 2: Use Given Information to Find the Constant

We are told that 15 customers buy a television that costs $1500. Using the inverse proportionality formula:

$$
S \cdot C = k \Rightarrow 15 \cdot 1500 = k
$$

$$
k = 22500
$$

---

### Step 3: Apply the Constant to the New Cost

Now, we are asked to find how many customers would buy a television that costs $2500. Let the new number of sales be $ S' $. Using the same formula:

$$
S' \cdot C' = k \Rightarrow S' = \frac{k}{C'} = \frac{22500}{2500}
$$

$$
S' = 9
$$

---

### Final Answer

$$
\boxed{9}
$$

So, according to Daniel's theory, **9 customers** would buy a television that costs $2500.

[Assistant]
Explanation:
Okay, so Daniel says that the popularity of a television, which is measured by the number of sales, is inversely proportional to its cost. Hmm, inversely proportional... Let me recall what that means. If two quantities are inversely proportional, that means as one increases, the other decreases, and their product remains constant. So, if I have two variables, let's say sales (let me call that S) and cost (let me call that C), then according to Daniel's theory, S is inversely proportional to C.

So, mathematically, that would be S = k / C, where k is the constant of proportionality. Or maybe it's better to write it as S * C = k. Yeah, that's another way to think about it. So the product of the number of sales and the cost should be a constant.

Alright, so the problem says that 15 customers buy a television that costs $1500. So, using that information, I can find the constant k. Let me write that down.

Given:
When C = 1500, S = 15. So plugging into S * C = k, we get 15 * 1500 = k. Let me calculate that.

15 times 1500... Well, 15 times 1000 is 15,000, and 15 times 500 is 7,500. So adding those together, 15,000 + 7,500 = 22,500. So k is 22,500.

So according to Daniel's theory, the product of the number of sales and the cost is always 22,500. Now, the question is asking, how many customers would buy a television that costs $2500? Let me denote the new number of sales as S', and the new cost as C' = 2500.

So, using the same formula S' * C' = k. Therefore, S' = k / C'. Plugging in the numbers, S' = 22,500 / 2500. Let me compute that.

Dividing 22,500 by 2500. Hmm, 2500 goes into 22,500 how many times? Well, 2500 times 9 is 22,500 because 2500*10 is 25,000, so subtract 2500 once, which is 22,500. So 9 times. Therefore, S' = 9.

Wait, so according to Daniel's theory, if the cost increases to $2500, the number of customers would decrease to 9? Let me check my steps again to make sure I didn't make a mistake.

First, inverse proportionality: S ∝ 1/C, so S*C = k. Then, with S = 15 and C = 1500, k = 15*1500 = 22,500. Then, for C = 2500, S = 22,500 / 2500 = 9. That seems correct.

But let me think again if inverse proportionality is the right approach here. The problem says "popularity... is inversely proportional to its cost." So popularity is sales, so sales inversely proportional to cost. So yes, that's what I used.

Alternatively, maybe someone could misinterpret it as popularity is inversely proportional to the cost, meaning that popularity = k / cost. Which is the same thing. So, yeah, that seems right.

Alternatively, maybe there's a different way to think about it. For example, if the cost doubles, the sales would halve, but in this case, the cost isn't doubling, it's increasing from 1500 to 2500. Let me check the ratio of the costs.

Original cost: 1500, new cost: 2500. The ratio of new cost to original cost is 2500 / 1500 = 5/3 ≈ 1.666... So, since they are inversely proportional, the number of sales should be multiplied by 3/5.

Original sales: 15. So new sales would be 15 * (3/5) = 9. Which matches the previous result.

So that's another way to think about it. If the cost increases by a factor of 5/3, then sales decrease by a factor of 3/5.

So, 15 * 3/5 is indeed 9.

Therefore, the answer should be 9 customers.

But let me just make sure that there isn't any trick or anything in the question. The problem says "according to Daniel's theory," so we don't have to question the validity of the theory, just apply it.

So, given that, the answer is 9.

Alternatively, maybe the question is trying to test understanding of inverse proportionality. Let me think if there's another way someone might approach it incorrectly. For example, if someone thought it was directly proportional, they would do 15 * (2500 / 1500) = 15 * (5/3) = 25. But that's not the case here.

But since it's inversely proportional, it's the reciprocal. So, 15 * (1500 / 2500) = 15 * (3/5) = 9.

Yes, that's correct.

Alternatively, maybe someone might think that the number of customers is inversely proportional to the cost, so they might set up the proportion as 15 / 1500 = S / 2500, but that would be incorrect because if it's inversely proportional, the proportion should be 15 / S = 1500 / 2500.

Let me check that.

If S1 * C1 = S2 * C2, then 15 * 1500 = S2 * 2500. Therefore, S2 = (15 * 1500) / 2500 = same as before, 9.

Alternatively, if someone set up the proportion as S1 / S2 = C2 / C1, which is the same as S1 / S2 = 2500 / 1500, so S2 = S1 * (1500 / 2500) = 15 * 3/5 = 9.

So that's another way to think about it.

So, all methods lead to 9.

Therefore, I think the answer is 9.

**Final Answer**
The number of customers who would buy a television that costs $\$$2500 is \boxed{9}.

To solve the problem, we begin by interpreting Daniel's claim: the **popularity of a television (measured in number of sales)** is **inversely proportional** to its **cost**. This means that as the cost increases, the number of sales decreases, and their product remains constant.

---

### Step 1: Understanding Inverse Proportionality

If two quantities are inversely proportional, their product is constant. That is:

$$
S \cdot C = k
$$

Where:
- $ S $ is the number of sales,
- $ C $ is the cost of the television,
- $ k $ is the constant of proportionality.

---

### Step 2: Use Given Information to Find the Constant

We are told that 15 customers buy a television that costs $1500. Using the inverse proportionality formula:

$$
S \cdot C = k \Rightarrow 15 \cdot 1500 = k
$$

$$
k = 22500
$$

---

### Step 3: Apply the Constant to the New Cost

Now, we are asked to find how many customers would buy a television that costs $2500. Let the new number of sales be $ S' $. Using the same formula:

$$
S' \cdot C' = k \Rightarrow S' = \frac{k}{C'} = \frac{22500}{2500}
$$

$$
S' = 9
$$

---

### Final Answer

$$
\boxed{9}
$$

So, according to Daniel's theory, **9 customers** would buy a television that costs $2500.

Solution:
Let the popularity of a television (or the number of customers who buy one) equal $p$, and let the cost of the television equal $c$. According to Daniel's theory, $p$ and $c$ are inversely proportional. Thus, $(p)(c)=k$ for some constant value $k$. If $p=15$ when $c=1500$, then $k=(15)(1500)=22500$. So when $c=2500$, \begin{align*} (p)(c)&=k
\\\Rightarrow\qquad (p)(2500)&=22500
\\\Rightarrow\qquad p&=\frac{22500}{2500}
\\ &=\boxed{9}.
\end{align*}According to Daniel's theory, 9 customers would buy the $\$2500$ television.


Explanation:
Okay, so I need to convert the decimal number 555 into base 5. Hmm, let me think about how to do this. I remember that converting from base 10 to another base involves dividing by the new base and keeping track of the remainders. Let me try to recall the exact steps.

First, I think the process is to divide the number by 5 repeatedly and record the remainders. Each remainder becomes a digit in the new base, starting from the least significant digit. Then, you take the quotient and repeat the process until the quotient is zero. Then, you reverse the remainders to get the number in the new base. Let me check if that's correct.

Let me try with a smaller number first to make sure. Let's say I want to convert 10 into base 5. Dividing 10 by 5 gives 2 with a remainder of 0. Then, divide 2 by 5, which gives 0 with a remainder of 2. So the remainders are 0 and 2, so reading them in reverse order, it's 20 in base 5. Let me check: 2*5^1 + 0*5^0 = 10. Yep, that works. So the method seems correct.

Okay, so applying this to 555. Let me start.

First step: divide 555 by 5. Let me calculate that. 5*100 is 500, so 555 - 500 is 55. Then 5*11 is 55, so total is 100 + 11 = 111. Wait, so 5*111 = 555. Therefore, 555 divided by 5 is 111 with a remainder of 0. So the first remainder is 0. That's the least significant digit.

Now take the quotient, which is 111, and divide by 5 again. Let me compute 111 divided by 5. 5*22 is 110, so 111 - 110 is 1. Therefore, the quotient is 22 and the remainder is 1. So the next digit is 1.

Now take 22 and divide by 5. 5*4 is 20, so 22 - 20 is 2. Therefore, quotient is 4 and remainder is 2. Next digit is 2.

Take 4 and divide by 5. Since 4 is less than 5, the quotient is 0 and the remainder is 4. So the next digit is 4.

Now, since the quotient is 0, we stop. Now, the remainders we got were 0, 1, 2, 4. But since we read them in reverse order, the base 5 number is 4 2 1 0. Wait, let me check again.

Wait, the first remainder was 0 (from 555/5), then 1 (from 111/5), then 2 (from 22/5), then 4 (from 4/5). So when we reverse them, the digits are 4, 2, 1, 0. Therefore, the number is 4210 in base 5?

Wait, let me verify that. Let me convert 4210 base 5 back to decimal to check if it's 555.

Calculating 4*5^3 + 2*5^2 + 1*5^1 + 0*5^0.

Compute each term:

4*5^3: 5^3 is 125, so 4*125 = 500.

2*5^2: 2*25 = 50.

1*5^1: 5.

0*1: 0.

Adding them up: 500 + 50 = 550, 550 + 5 = 555, 555 + 0 = 555. Perfect! So that works. Therefore, 555 in base 10 is 4210 in base 5. So the answer is 4210_5.

But let me go through the steps again to make sure I didn't make any mistakes. Sometimes when you do division, it's easy to miscalculate.

Starting with 555 divided by 5. 5*100 = 500, 555 - 500 = 55. Then 5*11 = 55, so total quotient is 100 + 11 = 111. So remainder is 0. Correct.

Then 111 divided by 5. Let me do this division again. 5*20 = 100, 111 - 100 = 11. Then 5*2 = 10, so 11 - 10 = 1. So total quotient is 20 + 2 = 22, remainder 1. Correct. So that's 22 with remainder 1.

Then 22 divided by 5. 5*4 = 20, remainder 2. Quotient 4, remainder 2. Correct.

Then 4 divided by 5 is 0 with remainder 4. Correct. So the remainders are 0, 1, 2, 4. Reverse them to get 4, 2, 1, 0. So 4210. Yep, that seems right.

Alternatively, maybe I can use another method to cross-verify. Let me think. Another way to convert is to find the largest power of 5 less than or equal to 555, and then proceed.

Powers of 5: 5^0 = 1, 5^1 = 5, 5^2 = 25, 5^3 = 125, 5^4 = 625. 5^4 is 625, which is larger than 555, so the highest power is 5^3 = 125.

So how many times does 125 go into 555? Let me calculate 555 divided by 125. 125*4 = 500, 125*5 = 625 which is too big. So 4 times. So the coefficient for 5^3 is 4. Then subtract 4*125 = 500 from 555, which leaves 55.

Now move to the next lower power, which is 5^2 = 25. How many times does 25 go into 55? 25*2 = 50, 25*3 = 75 which is too big. So coefficient is 2. Subtract 2*25 = 50 from 55, leaving 5.

Next power is 5^1 = 5. How many times does 5 go into 5? Once. So coefficient is 1. Subtract 1*5 = 5, leaving 0.

Finally, 5^0 = 1, coefficient is 0.

So putting it all together: 4 (for 5^3), 2 (for 5^2), 1 (for 5^1), 0 (for 5^0). So again, 4210. Same result. So that confirms it.

Therefore, I think the answer is correct. But just to make sure, let me try another approach. Maybe using successive subtractions?

But that might take longer. Alternatively, think about the number of digits. Since 5^4 is 625, which is bigger than 555, so base 5 number will have 4 digits. Which matches with 4210.

Alternatively, maybe I can check with another example. Let me try converting 555 to base 5 using another method. For example, using the division method step by step again.

First division: 555 ÷ 5. Let me compute 5*111 = 555, so quotient 111, remainder 0.

Second division: 111 ÷ 5. 5*22 = 110, so quotient 22, remainder 1.

Third division: 22 ÷ 5. 5*4 = 20, quotient 4, remainder 2.

Fourth division: 4 ÷ 5. Quotient 0, remainder 4.

So the remainders, from last to first, are 4, 2, 1, 0. So 4210. Yep, same as before.

Alternatively, maybe I can think about the base 5 digits. Let me think about how each digit represents a value. The rightmost digit is 5^0, then 5^1, etc. So if I have 4210, that's 4*125 + 2*25 + 1*5 + 0*1 = 500 + 50 + 5 + 0 = 555. Correct.

So all methods lead to the same conclusion. Therefore, I think I can be confident that 555 in base 10 is 4210 in base 5.

But just to make sure, let me check if there's any chance I messed up the order of the remainders. Sometimes when you do division, you have to reverse the remainders. Let me think. When you divide the original number by the base, the first remainder is the least significant digit (rightmost), and each subsequent remainder is the next digit to the left. So when you collect the remainders, you have to reverse them to get the correct order.

So in this case, the first remainder was 0 (LSB), then 1, then 2, then 4 (MSB). So writing them in reverse order gives 4, 2, 1, 0. Which is correct. So that's why the number is 4210.

Alternatively, if I had written them in the order they were obtained, it would be 0, 1, 2, 4, which would be incorrect. So reversing is essential.

Another way to think about it: when you divide by the base, you're essentially finding how many times the base fits into the number, which corresponds to the coefficients of the higher powers. So the first division gives the coefficient for the lowest power (5^0), then the next division gives the coefficient for 5^1, etc. Therefore, to get the correct order, you need to reverse them.

So, yes, the process is correct. Therefore, I think my answer is correct.

Just to recap:

555 / 5 = 111, remainder 0

111 / 5 = 22, remainder 1

22 / 5 = 4, remainder 2

4 / 5 = 0, remainder 4

Reading remainders from last to first: 4, 2, 1, 0 → 4210.

Yep. So I think that's solid. I don't see any mistakes here. Therefore, the final answer is 4210 in base 5.

**Final Answer**
The base 5 representation of $555_{10}$ is \boxed{4210}.

To convert the decimal number $ 555_{10} $ into base 5, we use the **division-by-base method**, which involves repeatedly dividing the number by 5 and recording the remainders. These remainders, read in reverse order, give the digits of the number in base 5.

---

### Step-by-Step Conversion:

1. **Divide 555 by 5:**
   $$
   555 \div 5 = 111 \quad \text{remainder } 0
   $$
   - Remainder: **0** (least significant digit)

2. **Divide 111 by 5:**
   $$
   111 \div 5 = 22 \quad \text{remainder } 1
   $$
   - Remainder: **1**

3. **Divide 22 by 5:**
   $$
   22 \div 5 = 4 \quad \text{remainder } 2
   $$
   - Remainder: **2**

4. **Divide 4 by 5:**
   $$
   4 \div 5 = 0 \quad \text{remainder } 4
   $$
   - Remainder: **4** (most significant digit)

---

### Reading the Remainders in Reverse Order:

The remainders, from last to first, are:
**4, 2, 1, 0**

So, the base 5 representation of $ 555_{10} $ is:

$$
\boxed{4210}
$$

---

### Verification:

To confirm, we can convert $ 4210_5 $ back to base 10:

$$
4 \cdot 5^3 + 2 \cdot 5^2 + 1 \cdot 5^1 + 0 \cdot 5^0 = 4 \cdot 125 + 2 \cdot 25 + 1 \cdot 5 + 0 = 500 + 50 + 5 + 0 = 555
$$

This confirms the conversion is correct.

---

### Final Answer:

$$
\boxed{4210}
$$

[Assistant]
Explanation:
Okay, so I need to convert the decimal number 555 into base 5. Hmm, let me think about how to do this. I remember that converting from base 10 to another base involves dividing by the new base and keeping track of the remainders. Let me try to recall the exact steps.

First, I think the process is to divide the number by 5 repeatedly and record the remainders. Each remainder becomes a digit in the new base, starting from the least significant digit. Then, you take the quotient and repeat the process until the quotient is zero. Then, you reverse the remainders to get the number in the new base. Let me check if that's correct.

Let me try with a smaller number first to make sure. Let's say I want to convert 10 into base 5. Dividing 10 by 5 gives 2 with a remainder of 0. Then, divide 2 by 5, which gives 0 with a remainder of 2. So the remainders are 0 and 2, so reading them in reverse order, it's 20 in base 5. Let me check: 2*5^1 + 0*5^0 = 10. Yep, that works. So the method seems correct.

Okay, so applying this to 555. Let me start.

First step: divide 555 by 5. Let me calculate that. 5*100 is 500, so 555 - 500 is 55. Then 5*11 is 55, so total is 100 + 11 = 111. Wait, so 5*111 = 555. Therefore, 555 divided by 5 is 111 with a remainder of 0. So the first remainder is 0. That's the least significant digit.

Now take the quotient, which is 111, and divide by 5 again. Let me compute 111 divided by 5. 5*22 is 110, so 111 - 110 is 1. Therefore, the quotient is 22 and the remainder is 1. So the next digit is 1.

Now take 22 and divide by 5. 5*4 is 20, so 22 - 20 is 2. Therefore, quotient is 4 and remainder is 2. Next digit is 2.

Take 4 and divide by 5. Since 4 is less than 5, the quotient is 0 and the remainder is 4. So the next digit is 4.

Now, since the quotient is 0, we stop. Now, the remainders we got were 0, 1, 2, 4. But since we read them in reverse order, the base 5 number is 4 2 1 0. Wait, let me check again.

Wait, the first remainder was 0 (from 555/5), then 1 (from 111/5), then 2 (from 22/5), then 4 (from 4/5). So when we reverse them, the digits are 4, 2, 1, 0. Therefore, the number is 4210 in base 5?

Wait, let me verify that. Let me convert 4210 base 5 back to decimal to check if it's 555.

Calculating 4*5^3 + 2*5^2 + 1*5^1 + 0*5^0.

Compute each term:

4*5^3: 5^3 is 125, so 4*125 = 500.

2*5^2: 2*25 = 50.

1*5^1: 5.

0*1: 0.

Adding them up: 500 + 50 = 550, 550 + 5 = 555, 555 + 0 = 555. Perfect! So that works. Therefore, 555 in base 10 is 4210 in base 5. So the answer is 4210_5.

But let me go through the steps again to make sure I didn't make any mistakes. Sometimes when you do division, it's easy to miscalculate.

Starting with 555 divided by 5. 5*100 = 500, 555 - 500 = 55. Then 5*11 = 55, so total quotient is 100 + 11 = 111. So remainder is 0. Correct.

Then 111 divided by 5. Let me do this division again. 5*20 = 100, 111 - 100 = 11. Then 5*2 = 10, so 11 - 10 = 1. So total quotient is 20 + 2 = 22, remainder 1. Correct. So that's 22 with remainder 1.

Then 22 divided by 5. 5*4 = 20, remainder 2. Quotient 4, remainder 2. Correct.

Then 4 divided by 5 is 0 with remainder 4. Correct. So the remainders are 0, 1, 2, 4. Reverse them to get 4, 2, 1, 0. So 4210. Yep, that seems right.

Alternatively, maybe I can use another method to cross-verify. Let me think. Another way to convert is to find the largest power of 5 less than or equal to 555, and then proceed.

Powers of 5: 5^0 = 1, 5^1 = 5, 5^2 = 25, 5^3 = 125, 5^4 = 625. 5^4 is 625, which is larger than 555, so the highest power is 5^3 = 125.

So how many times does 125 go into 555? Let me calculate 555 divided by 125. 125*4 = 500, 125*5 = 625 which is too big. So 4 times. So the coefficient for 5^3 is 4. Then subtract 4*125 = 500 from 555, which leaves 55.

Now move to the next lower power, which is 5^2 = 25. How many times does 25 go into 55? 25*2 = 50, 25*3 = 75 which is too big. So coefficient is 2. Subtract 2*25 = 50 from 55, leaving 5.

Next power is 5^1 = 5. How many times does 5 go into 5? Once. So coefficient is 1. Subtract 1*5 = 5, leaving 0.

Finally, 5^0 = 1, coefficient is 0.

So putting it all together: 4 (for 5^3), 2 (for 5^2), 1 (for 5^1), 0 (for 5^0). So again, 4210. Same result. So that confirms it.

Therefore, I think the answer is correct. But just to make sure, let me try another approach. Maybe using successive subtractions?

But that might take longer. Alternatively, think about the number of digits. Since 5^4 is 625, which is bigger than 555, so base 5 number will have 4 digits. Which matches with 4210.

Alternatively, maybe I can check with another example. Let me try converting 555 to base 5 using another method. For example, using the division method step by step again.

First division: 555 ÷ 5. Let me compute 5*111 = 555, so quotient 111, remainder 0.

Second division: 111 ÷ 5. 5*22 = 110, so quotient 22, remainder 1.

Third division: 22 ÷ 5. 5*4 = 20, quotient 4, remainder 2.

Fourth division: 4 ÷ 5. Quotient 0, remainder 4.

So the remainders, from last to first, are 4, 2, 1, 0. So 4210. Yep, same as before.

Alternatively, maybe I can think about the base 5 digits. Let me think about how each digit represents a value. The rightmost digit is 5^0, then 5^1, etc. So if I have 4210, that's 4*125 + 2*25 + 1*5 + 0*1 = 500 + 50 + 5 + 0 = 555. Correct.

So all methods lead to the same conclusion. Therefore, I think I can be confident that 555 in base 10 is 4210 in base 5.

But just to make sure, let me check if there's any chance I messed up the order of the remainders. Sometimes when you do division, you have to reverse the remainders. Let me think. When you divide the original number by the base, the first remainder is the least significant digit (rightmost), and each subsequent remainder is the next digit to the left. So when you collect the remainders, you have to reverse them to get the correct order.

So in this case, the first remainder was 0 (LSB), then 1, then 2, then 4 (MSB). So writing them in reverse order gives 4, 2, 1, 0. Which is correct. So that's why the number is 4210.

Alternatively, if I had written them in the order they were obtained, it would be 0, 1, 2, 4, which would be incorrect. So reversing is essential.

Another way to think about it: when you divide by the base, you're essentially finding how many times the base fits into the number, which corresponds to the coefficients of the higher powers. So the first division gives the coefficient for the lowest power (5^0), then the next division gives the coefficient for 5^1, etc. Therefore, to get the correct order, you need to reverse them.

So, yes, the process is correct. Therefore, I think my answer is correct.

Just to recap:

555 / 5 = 111, remainder 0

111 / 5 = 22, remainder 1

22 / 5 = 4, remainder 2

4 / 5 = 0, remainder 4

Reading remainders from last to first: 4, 2, 1, 0 → 4210.

Yep. So I think that's solid. I don't see any mistakes here. Therefore, the final answer is 4210 in base 5.

**Final Answer**
The base 5 representation of $555_{10}$ is \boxed{4210}.

To convert the decimal number $ 555_{10} $ into base 5, we use the **division-by-base method**, which involves repeatedly dividing the number by 5 and recording the remainders. These remainders, read in reverse order, give the digits of the number in base 5.

---

### Step-by-Step Conversion:

1. **Divide 555 by 5:**
   $$
   555 \div 5 = 111 \quad \text{remainder } 0
   $$
   - Remainder: **0** (least significant digit)

2. **Divide 111 by 5:**
   $$
   111 \div 5 = 22 \quad \text{remainder } 1
   $$
   - Remainder: **1**

3. **Divide 22 by 5:**
   $$
   22 \div 5 = 4 \quad \text{remainder } 2
   $$
   - Remainder: **2**

4. **Divide 4 by 5:**
   $$
   4 \div 5 = 0 \quad \text{remainder } 4
   $$
   - Remainder: **4** (most significant digit)

---

### Reading the Remainders in Reverse Order:

The remainders, from last to first, are:
**4, 2, 1, 0**

So, the base 5 representation of $ 555_{10} $ is:

$$
\boxed{4210}
$$

---

### Verification:

To confirm, we can convert $ 4210_5 $ back to base 10:

$$
4 \cdot 5^3 + 2 \cdot 5^2 + 1 \cdot 5^1 + 0 \cdot 5^0 = 4 \cdot 125 + 2 \cdot 25 + 1 \cdot 5 + 0 = 500 + 50 + 5 + 0 = 555
$$

This confirms the conversion is correct.

---

### Final Answer:

$$
\boxed{4210}
$$

Solution: We write $555$ in powers of $5$. The largest power of $5$ which is smaller than $555$ is $5^3=125$, and the greatest multiple of $125$ which is less than $555$ is $4$. We get that $555- 4 \cdot 125 = 55$. The largest power of $5$ that is less than $55$ is $5^2=25$, and the greatest multiple of $25$ less than $55$ is $2$. We get $55 - 2 \cdot 25 = 5$, which is $5^1$. Therefore, we can write $555$ as $4 \cdot 5^3 + 2 \cdot 5^2 + 1 \cdot 5^1$. Thus, the answer is $\boxed{4210_{5}}$.