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 key here is that I can't rearrange the order of the terms or change the operations, just add parentheses. Let me start by understanding the original expression and how parentheses can affect the result.

First, the original expression without any parentheses is 1 + 2 + 3 - 4 + 5 + 6. Let me compute that to confirm it's 13.

Calculating step by step:
1 + 2 = 3
3 + 3 = 6
6 - 4 = 2
2 + 5 = 7
7 + 6 = 13. Yep, that checks out.

Now, the problem says that by adding parentheses, we can change the value. For example, they showed that 1+2+3-(4+5)+6 equals 3. Let me verify that:

1 + 2 + 3 = 6
4 + 5 = 9
So, 6 - 9 + 6 = (6 - 9) + 6 = (-3) + 6 = 3. Correct.

So, parentheses can change the order of operations. Since addition and subtraction are left-associative by default, but parentheses can override that. The goal is to find the minimum possible value by strategically placing parentheses.

First, I need to recall that parentheses can group operations, so changing the order in which operations are performed. Since subtraction is involved, grouping terms can lead to different results. For example, if I can make a larger negative number by grouping terms, that might decrease the overall value.

But how do I approach this systematically? Let me think.

The original expression is a sequence of numbers with alternating addition and subtraction. The expression is:

1 + 2 + 3 - 4 + 5 + 6

So, the operations are: +, +, -, +, +. The numbers are 1, 2, 3, 4, 5, 6.

Since we can't rearrange the terms, the order is fixed. So, the challenge is to group these operations with parentheses to change the result.

Let me think about how parentheses can affect the result. Parentheses can change the order of operations, so for example, if I have something like a - (b + c), that's different from (a - b) + c. Similarly, grouping multiple additions or subtractions can change the result.

Since we want the minimum value, we need to maximize the negative contributions. So, perhaps grouping terms in such a way that we subtract larger numbers, or have more negative terms.

But how?

Let me consider that subtraction is equivalent to adding a negative. So, maybe if I can create larger negative numbers by grouping terms. For example, if I can have a large number subtracted from a smaller one, that would be bad. Wait, but how?

Alternatively, maybe grouping terms so that more subtractions are applied to larger numbers. Let me think.

Let me try to think of all possible ways to add parentheses. However, since there are multiple operations, the number of possible parenthetical groupings could be quite large. Maybe there's a smarter way than brute-forcing all possibilities.

Alternatively, think about the expression as a sequence of operations and how parentheses can change the grouping. Let me note that the expression has five operations: +, +, -, +, +. So, between the numbers, there are five operations. The parentheses can group these operations in different ways.

But since we can't change the order of the terms, the structure is fixed. So, the problem is similar to inserting parentheses in the expression to change the order of operations.

This seems similar to the problem of evaluating expressions with different parenthetical groupings. For example, in the expression a + b - c + d + e, how can parentheses change the result?

But since we can't rearrange terms, the structure is fixed. So, perhaps we can model this as a binary tree where each operation is a node, and parentheses define the order of operations.

But maybe that's too abstract. Let me think of possible groupings.

First, note that addition is associative and commutative, but subtraction is neither. So, grouping additions together won't change their result, but grouping subtractions can have different effects.

Wait, but since we can't rearrange terms, the order of operations is fixed except for the parentheses. So, for example, if I have 1 + 2 + 3 - 4 + 5 + 6, the default is left to right:

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

But if I group some operations differently, like 1 + (2 + 3 - 4) + 5 + 6, that would be different. Let me compute that:

First, 2 + 3 - 4 = 1, then 1 + 1 + 5 + 6 = 13. Same result. Wait, so that didn't change anything. Hmm.

But in the example given, they grouped 4 + 5, so that's changing the order. Let me think again.

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

If we group the -4 + 5 as -(4 - 5)? Wait, no. Wait, the example was 1 + 2 + 3 - (4 + 5) + 6. So, that's grouping the 4 + 5, which is subtracted. So, that's changing the order of operations. So, instead of subtracting 4 and then adding 5, it's subtracting (4 + 5). So, that's equivalent to subtracting 9 instead of subtracting 4 and adding 5. Which is a big difference.

So, in that case, the net effect is that instead of (-4 + 5) = +1, it's - (4 + 5) = -9. So, that's a difference of -10. Which is why the total went from 13 to 3. So, that's a significant change.

Therefore, the key is to find groupings where we can have more negative contributions. So, perhaps grouping terms that are being subtracted with other terms, thereby increasing the magnitude of the subtraction.

So, the idea is to find groupings where we can have larger subtractions. Let me think about how to do that.

Let me consider that each subtraction can be thought of as a negative sign. So, if I can group a term with a negative sign, that might help. For example, if I have something like - (a + b), that's equivalent to -a - b. Whereas if it's -a + b, that's different.

So, perhaps if I can create more negative terms by grouping. Let me think of the original expression:

1 + 2 + 3 - 4 + 5 + 6.

So, the operations are:

Start with 1, then add 2, add 3, subtract 4, add 5, add 6.

So, the subtraction is only on the 4. But if we can group the subtraction with other terms, we can make the subtraction affect more numbers.

For example, if we can have - (4 - 5), but that would be -(-1) = +1, which is not helpful. Wait, but maybe grouping in a way that the subtraction applies to more numbers.

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

3 - 4 + 5 = 4, so total is 1 + 2 + 4 + 6 = 13. Same as before.

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

2 + 3 - 4 + 5 = 6, so total is 1 + 6 + 6 = 13. Still same.

Hmm. So, grouping additions and subtractions in different ways doesn't change the result if they are all additions and subtractions. Wait, but in the example given, grouping 4 + 5 changed the result. So, that was a subtraction of (4 + 5) instead of subtracting 4 and adding 5. So, that's different.

So, the key is that when you have a subtraction followed by an addition, grouping them together can change the result. So, in the original expression, the subtraction is followed by an addition. So, if you group them, you can have a larger subtraction.

So, perhaps the strategy is to group as many terms as possible under a subtraction. Let me think.

Let me consider that the original expression has a single subtraction: -4. If I can make that subtraction affect more numbers, that would be better. For example, if I can have - (4 - 5 + 6), but that would be - (5) = -5? Wait, let me check:

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

First, compute inside the parentheses: 4 - 5 + 6 = 5. Then, 1 + 2 + 3 - 5 = 6 - 5 = 1. So, that's 1. Which is lower than 13, but not the minimum.

Alternatively, if I can have more terms subtracted. Let me think. For example, if I can have - (4 + 5 + 6). But how?

Original expression is 1 + 2 + 3 - 4 + 5 + 6. If I can group the -4 + 5 + 6 as -(4 - 5 - 6)? Wait, that would be different. Wait, but how can I group that?

Wait, maybe if I have 1 + 2 + 3 - (4 - 5 - 6). Let me compute that:

Inside the parentheses: 4 - 5 - 6 = -7. Then, 1 + 2 + 3 - (-7) = 6 + 7 = 13. So, same as original. Not helpful.

Alternatively, maybe grouping the entire expression in a way that more terms are subtracted. Let me think.

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

3 - 4 + 5 + 6 = 10. Then, 1 + 2 + 10 = 13. Still same.

Alternatively, 1 + (2 + 3 - 4 + 5 + 6). Let me compute:

2 + 3 - 4 + 5 + 6 = 12. Then, 1 + 12 = 13. Still same.

Hmm. So, grouping additions and subtractions in different ways doesn't change the result if they are all additions and subtractions. But when you have a subtraction followed by an addition, grouping them can change the result. For example, in the original example, they grouped 4 + 5, which was subtracted. So, that's changing the order.

So, perhaps the key is to group terms that are being subtracted with terms that are being added, thereby turning them into a larger subtraction. Let me think.

Let me consider the original expression again:

1 + 2 + 3 - 4 + 5 + 6.

The subtraction is only on 4. But if I can group the -4 with the +5 and +6, so that instead of subtracting 4 and then adding 5 and 6, I subtract (4 - 5 - 6). Wait, but how?

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

Inside the parentheses: 4 - 5 - 6 = -7. Then, 1 + 2 + 3 - (-7) = 6 + 7 = 13. Same as before.

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

Inside the parentheses: 4 + 5 + 6 = 15. Then, 1 + 2 + 3 - 15 = 6 - 15 = -9. Oh! That's a lower value. So, that's -9. Is that allowed?

Wait, can I group the -4 + 5 + 6 as -(4 + 5 + 6)? Let me check the original expression. The original expression is 1 + 2 + 3 - 4 + 5 + 6. So, the subtraction is only on 4. But if I add parentheses around 4 + 5 + 6, then it would be 1 + 2 + 3 - (4 + 5 + 6). Is that a valid placement of parentheses?

Yes, because the original expression is 1 + 2 + 3 - 4 + 5 + 6. So, inserting parentheses around 4 + 5 + 6 would be:

1 + 2 + 3 - (4 + 5 + 6). Which is valid. So, that would be 1 + 2 + 3 - 15 = 6 - 15 = -9. So, that's a valid expression with parentheses, and gives -9. That's lower than the previous example of 3. So, maybe that's a candidate for the minimum.

But can we get even lower?

Let me think. How else can we group the terms?

Another idea: Maybe group more terms to be subtracted. For example, if we can have more subtractions. Let me think.

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

Suppose we try to group the first part as 1 + (2 + 3 - 4 + 5 + 6). Wait, that would be 1 + (2 + 3 - 4 + 5 + 6) = 1 + (12) = 13. Not helpful.

Alternatively, group the first three terms: (1 + 2 + 3) - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 13. Same as before.

Alternatively, group the first two terms: (1 + 2) + 3 - 4 + 5 + 6 = 3 + 3 - 4 + 5 + 6 = 13. Still same.

Alternatively, group the first term with the next operations: 1 + (2 + 3 - 4 + 5 + 6) = same as before.

Alternatively, group the subtraction with the following additions. For example, 1 + 2 + (3 - 4 + 5 + 6) = 1 + 2 + 10 = 13.

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

Alternatively, group 3 - (4 + 5) + 6. Let me compute that:

3 - (4 + 5) + 6 = 3 - 9 + 6 = 0. Then 1 + 2 + 0 = 3. Which is the example given.

So, that gives 3. But earlier, grouping 4 + 5 + 6 as a subtraction gives -9. So, that's lower.

Is there a way to get even lower?

Let me think. What if we group more terms to be subtracted? For example, can we have something like 1 + 2 + 3 - (4 + 5 + 6) = -9 as above. What if we can have more terms subtracted?

Wait, but the original expression only has one subtraction operator. Wait, no. Wait, the original expression is 1 + 2 + 3 - 4 + 5 + 6. So, the operations are +, +, -, +, +. So, there is only one subtraction. So, the rest are additions. Therefore, the only subtraction is on the 4. However, by adding parentheses, we can change the effect of that subtraction. For example, if we can make that subtraction apply to more terms.

But in the case of grouping 4 + 5 + 6, we are effectively making the subtraction apply to 4, 5, and 6. Wait, but how? Because the original expression is 1 + 2 + 3 - 4 + 5 + 6. If we group 4 + 5 + 6, then it's 1 + 2 + 3 - (4 + 5 + 6). So, the subtraction is only on the entire group, which is 4 + 5 + 6. So, that's equivalent to subtracting 15. So, that's a larger subtraction than just subtracting 4. So, that's why it's -9.

But can we make the subtraction apply to even more terms? For example, if we could have something like 1 + 2 + 3 - 4 + 5 + 6, but with parentheses that somehow subtract more terms. Wait, but the original expression only has one subtraction. Unless we can create more subtractions by grouping.

Wait, for example, if we have something like 1 + 2 + (3 - 4 + 5 + 6). Wait, that's 1 + 2 + (3 - 4 + 5 + 6) = 1 + 2 + 10 = 13. Not helpful.

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

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

Alternatively, can we have something like (1 + 2 + 3 - 4) + 5 + 6? Which is (6 - 4) + 5 + 6 = 2 + 5 + 6 = 13. Same.

Alternatively, 1 + (2 + 3 - 4 + 5) + 6. Let me compute that:

2 + 3 - 4 + 5 = 6. Then 1 + 6 + 6 = 13.

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

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

So, all these groupings don't change the result. So, the only way to get a different result is to group terms that are being subtracted with terms that are being added, thereby changing the sign of those added terms. For example, in the example given, grouping 4 + 5 as a subtraction, which turned the +5 into a -5. So, that's a way to make the addition of 5 into a subtraction.

So, perhaps if we can do similar things with other terms. Let me think.

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

If I can group the +5 + 6 as a subtraction. Wait, but how? Because the original expression has a subtraction only on 4. Unless we can create a subtraction on 5 and 6 by grouping.

Wait, for example, if I have 1 + 2 + 3 - 4 + (5 + 6). That's 1 + 2 + 3 - 4 + 11 = 6 - 4 + 11 = 13. Same as before.

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

Alternatively, if I can have 1 + 2 + (3 - 4 + 5 + 6) = 1 + 2 + 10 = 13.

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

Alternatively, if I can have (1 + 2 + 3 - 4 + 5) + 6 = (6 - 4 + 5) + 6 = 7 + 6 = 13.

Alternatively, maybe grouping the first subtraction with the following additions. For example, 1 + 2 + 3 - (4 - 5 - 6). Wait, let me compute that:

Inside the parentheses: 4 - 5 - 6 = -7. Then, 1 + 2 + 3 - (-7) = 6 + 7 = 13. Same as before.

Alternatively, 1 + 2 + 3 - (4 - 5 + 6). Let me compute:

Inside the parentheses: 4 - 5 + 6 = 5. Then, 1 + 2 + 3 - 5 = 6 - 5 = 1. Which is lower than 13, but higher than -9.

Alternatively, 1 + 2 + 3 - (4 + 5 - 6). Let me compute:

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

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

Is there a way to get even lower?

Let me think. What if we can have more subtractions? For example, if we can have two subtractions. But the original expression only has one subtraction. Unless we can create another subtraction by grouping.

Wait, for example, if we have something like 1 + 2 + (3 - 4) + (5 - 6). Wait, but how can we do that? The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses to make it 1 + 2 + (3 - 4) + (5 - 6). Let me check if that's allowed.

Original expression: 1 + 2 + 3 - 4 + 5 + 6. To make it 1 + 2 + (3 - 4) + (5 - 6), we need to add parentheses around (3 - 4) and (5 - 6). Is that allowed? Let me see.

Yes, because the original expression is 1 + 2 + 3 - 4 + 5 + 6. So, inserting parentheses around (3 - 4) and (5 - 6) would be:

1 + 2 + (3 - 4) + (5 - 6). Let me compute that:

First, 3 - 4 = -1, 5 - 6 = -1. Then, 1 + 2 + (-1) + (-1) = 1 + 2 - 1 - 1 = 1. Which is 1. So, that's lower than the previous -9? No, -9 is lower. Wait, 1 is higher than -9. So, -9 is still lower.

Alternatively, can we have more subtractions? For example, if we can have something like 1 + (2 + 3 - 4 - 5 - 6). Let me check if that's possible.

Original expression: 1 + 2 + 3 - 4 + 5 + 6. To make it 1 + (2 + 3 - 4 - 5 - 6). Let me see. How would that be grouped? The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we group 2 + 3 - 4 - 5 - 6, that would require changing the operations. Wait, but the original operations are +, +, -, +, +. So, to have -4 -5 -6, we need to have the subtraction signs. But in the original expression, after the -4, we have +5 and +6. So, unless we can change the +5 and +6 into -5 and -6 by grouping. But how?

Wait, if we have 1 + (2 + 3 - 4 - 5 - 6). That would require that the original expression has -4 -5 -6. But the original expression is -4 + 5 + 6. So, unless we can group the +5 and +6 as -5 -6. How?

For example, if we have 1 + 2 + 3 - 4 - (5 + 6). Let me check if that's allowed. The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around 5 + 6 and put a minus sign in front, that would be 1 + 2 + 3 - 4 - (5 + 6). Let me compute that:

Inside the parentheses: 5 + 6 = 11. Then, 1 + 2 + 3 - 4 - 11 = 6 - 4 - 11 = 2 - 11 = -9. Which is the same as before. So, that's -9. So, same result.

Alternatively, can we have more subtractions? For example, 1 + 2 + 3 - (4 + 5 + 6) = -9. Or 1 + 2 + (3 - 4 - 5 - 6). Let me compute that:

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

So, that's another way to get -9. So, seems like -9 is achievable in multiple ways.

Is there a way to get lower than -9?

Let me think. Let's see. To get lower than -9, we need the total value to be less than -9. Let's think about how that could happen.

The original expression is 1 + 2 + 3 - 4 + 5 + 6. The total is 13. If we can make the expression evaluate to something like -10, -11, etc.

How can we do that? Let me think. For example, if we can have more subtractions. But how?

Wait, the original expression has only one subtraction operator. However, by grouping, we can create more subtractions. For example, if we can have multiple subtractions by grouping.

Wait, for example, if we have something like 1 + 2 + (3 - 4 - 5 - 6). Which we already did, resulting in -9. But is there a way to have even more subtractions?

Wait, let me think. Suppose we have 1 + (2 + 3 - 4 - 5 - 6). Let me compute that:

Inside the parentheses: 2 + 3 - 4 - 5 - 6 = 5 - 4 - 5 - 6 = 1 - 5 - 6 = -10. Then, 1 + (-10) = -9. Same result.

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

1 + 2 + 3 - 4 - 5 - 6 = 6 - 4 - 5 - 6 = 2 - 5 - 6 = -9. Same result.

So, regardless of how we group, if we subtract 4, 5, 6, we get -9. Is there a way to subtract more?

Wait, the original expression is 1 + 2 + 3 - 4 + 5 + 6. The numbers are 1, 2, 3, 4, 5, 6. If we can somehow subtract all numbers except 1 and 2, but how?

Wait, for example, if we have something like (1 + 2) - (3 + 4 + 5 + 6). Let me check if that's possible.

Original expression: 1 + 2 + 3 - 4 + 5 + 6. To get (1 + 2) - (3 + 4 + 5 + 6), we need to group the first two terms as a sum and then subtract the rest. Let me see:

Original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around (1 + 2) and then subtract (3 + 4 + 5 + 6). But how?

Wait, the original expression is 1 + 2 + 3 - 4 + 5 + 6. To make it (1 + 2) - (3 + 4 + 5 + 6), we need to have the subtraction apply to the entire rest of the expression. But the original expression has a subtraction only on 4. So, unless we can group the rest of the terms as a subtraction. Let me think.

Wait, if we have 1 + 2 + 3 - 4 + 5 + 6. If we want to make it (1 + 2) - (3 + 4 + 5 + 6), that would require changing the operations. Because originally, after the subtraction of 4, we have +5 and +6. To make them -5 and -6, we need to add parentheses around them and put a minus sign. But how?

For example, if we have 1 + 2 + 3 - 4 - (5 + 6). Which we already did, resulting in -9. But that's only subtracting 5 and 6. If we could subtract 3, 4, 5, 6, that would be better. How?

Wait, if we have something like 1 + 2 - (3 + 4 + 5 + 6). Let me check if that's possible. Original expression is 1 + 2 + 3 - 4 + 5 + 6. To get 1 + 2 - (3 + 4 + 5 + 6), we need to have the subtraction apply to 3, 4, 5, 6. But in the original expression, the subtraction is only on 4. So, unless we can group the 3 with the rest. Let me think.

If we have 1 + 2 + (3 - 4 - 5 - 6). Which is 1 + 2 + (3 - 4 - 5 - 6) = 1 + 2 + (-12) = -9. Same as before.

Alternatively, if we have (1 + 2 + 3) - (4 + 5 + 6) = 6 - 15 = -9. Same result.

So, regardless of grouping, the maximum subtraction we can get is 15 (4 + 5 + 6), leading to -9. Is there a way to get a larger subtraction?

Wait, what if we can subtract more numbers? For example, subtract 1, 2, 3, 4, 5, 6? But how?

Original expression is 1 + 2 + 3 - 4 + 5 + 6. If we could somehow subtract all numbers, but the original expression starts with additions. Unless we can create a subtraction on the first terms.

For example, if we have -(1 + 2 + 3 + 4 + 5 + 6). But how can we do that? The original expression is 1 + 2 + 3 - 4 + 5 + 6. To make it -(1 + 2 + 3 + 4 + 5 + 6), we need to have a negative sign in front. But the original expression doesn't have that. Unless we can group the entire expression with a negative sign. But the original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around the entire expression and put a negative sign? Wait, but the problem says we can't change the operations. So, we can't add a negative sign unless it's part of the original expression.

Wait, the original expression is 1 + 2 + 3 - 4 + 5 + 6. There is no negative sign at the beginning. So, we can't just put a negative sign in front. Unless we can group the entire expression as a subtraction. But how?

Wait, for example, if we have -(1 + 2 + 3 - 4 + 5 + 6). But that would require adding a negative sign, which isn't part of the original expression. The problem says we can't change the operations, so I think we can't add a negative sign. So, that's not allowed.

Therefore, the only way to get a negative value is by grouping terms that are being subtracted. So, the maximum subtraction we can get is 4 + 5 + 6 = 15, leading to -9. Is that the minimum?

Wait, let me think again. What if we can have more subtractions by grouping in a different way?

For example, suppose we have 1 + (2 + 3 - 4) - (5 + 6). Let me check if that's allowed. Original expression is 1 + 2 + 3 - 4 + 5 + 6. To make it 1 + (2 + 3 - 4) - (5 + 6), we need to add parentheses around (2 + 3 - 4) and (5 + 6). Let me compute that:

Inside the first parentheses: 2 + 3 - 4 = 1. Inside the second parentheses: 5 + 6 = 11. Then, 1 + 1 - 11 = -9. Same result.

Alternatively, 1 + 2 + (3 - 4 - 5 - 6) = 1 + 2 + (-12) = -9.

Alternatively, 1 + (2 + 3 - 4 - 5 - 6) = 1 + (-10) = -9.

So, all these ways give -9. Is there a way to get lower?

Wait, let me think. What if we can have more subtractions? For example, if we can have something like 1 - (2 + 3 + 4 + 5 + 6). But how?

Original expression is 1 + 2 + 3 - 4 + 5 + 6. To get 1 - (2 + 3 + 4 + 5 + 6), we need to have the subtraction apply to the rest of the terms. But the original expression starts with a +, so unless we can group the first term with a negative sign. But again, we can't add a negative sign unless it's part of the original expression.

Wait, unless we can group the entire expression as follows: (1 + 2 + 3 - 4 + 5 + 6) with some parentheses that somehow negate parts. But I don't think that's possible without changing the operations.

Alternatively, if we can have something like (1 + 2 + 3) - (4 + 5 + 6) = -9. Which is the same as before.

Alternatively, if we can have (1 + 2) - (3 + 4 + 5 + 6) = 3 - 18 = -15. Wait, is that possible?

Wait, original expression is 1 + 2 + 3 - 4 + 5 + 6. To make it (1 + 2) - (3 + 4 + 5 + 6), we need to have the subtraction apply to 3, 4, 5, 6. But in the original expression, after the subtraction of 4, we have +5 and +6. So, how can we make them subtracted?

If we have 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around 3 - 4 + 5 + 6 as -(3 - 4 + 5 + 6)? Wait, no. Wait, if we want to have (1 + 2) - (3 + 4 + 5 + 6), that would require that the original expression has a subtraction between 1 + 2 and the rest. But the original expression is 1 + 2 + 3 - 4 + 5 + 6. So, the subtraction is only on 4. To make it (1 + 2) - (3 + 4 + 5 + 6), we need to have the subtraction between 1 + 2 and the rest. But how?

Wait, if we have 1 + 2 - (3 + 4 + 5 + 6). Let me check if that's possible. The original expression is 1 + 2 + 3 - 4 + 5 + 6. To make it 1 + 2 - (3 + 4 + 5 + 6), we need to change the +3 - 4 + 5 + 6 into - (3 + 4 + 5 + 6). How?

That would require that the original expression has a subtraction before the 3. But in the original expression, it's +3 - 4 + 5 + 6. So, unless we can group the +3 - 4 + 5 + 6 as - (3 + 4 + 5 + 6). But how?

Wait, if we have 1 + 2 - (3 + 4 + 5 + 6). Let me think about the operations. The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we want to have 1 + 2 - (3 + 4 + 5 + 6), that would require that the +3 - 4 + 5 + 6 becomes - (3 + 4 + 5 + 6). Which would mean that the original +3 is turned into -3, the -4 is turned into -4, and the +5 and +6 are turned into -5 and -6. But how can we do that with parentheses?

Wait, if we have 1 + 2 - (3 + 4 + 5 + 6). To get that from the original expression, we need to have the subtraction apply to the entire 3 + 4 + 5 + 6. But in the original expression, the operations are +3 - 4 + 5 + 6. So, if we can group the +3 - 4 + 5 + 6 as -(3 + 4 + 5 + 6), that would require that the original +3 is turned into -3, which would mean that we have a negative sign in front of the 3. But how?

Wait, unless we can group the entire expression as follows: 1 + 2 - (3 + 4 + 5 + 6). But how does that relate to the original expression? The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around 3 + 4 + 5 + 6 and put a minus sign in front, that would be 1 + 2 - (3 + 4 + 5 + 6). But in the original expression, the operations are +3 - 4 + 5 + 6. So, to make it - (3 + 4 + 5 + 6), we need to have the subtraction apply to the entire 3 + 4 + 5 + 6. But in the original expression, the subtraction is only on 4. So, unless we can somehow change the +3 into -3 by grouping.

Wait, for example, if we have 1 + 2 + (3 - 4 - 5 - 6). Which we did before, resulting in -9. But that's not the same as 1 + 2 - (3 + 4 + 5 + 6). Because in that case, the 3 is subtracted, but the rest are also subtracted. However, in the original expression, the 3 is added, then 4 is subtracted, then 5 and 6 are added. So, to make the 3 subtracted, we need to have a negative sign in front of it. But how?

Wait, if we have 1 + 2 - (3 + 4 + 5 + 6). That would require that the original expression has a subtraction before the 3. But the original expression has a + before the 3. So, unless we can group the +3 - 4 + 5 + 6 as - (3 + 4 + 5 + 6). Which would require that the original +3 is turned into -3, which would mean that we have a negative sign in front of the 3. But how can we do that with parentheses?

Wait, maybe by grouping the entire expression as follows: 1 + 2 - (3 + 4 + 5 + 6). But how does that relate to the original expression? The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around 3 + 4 + 5 + 6 and put a minus sign in front, that would be 1 + 2 - (3 + 4 + 5 + 6). But in the original expression, the operations are +3 - 4 + 5 + 6. So, to make it - (3 + 4 + 5 + 6), we need to have the subtraction apply to the entire 3 + 4 + 5 + 6. However, the original expression has a subtraction only on 4. So, unless we can somehow change the +3 into -3 by grouping.

Wait, for example, if we have 1 + 2 - (3 + 4 + 5 + 6). To get that from the original expression, we need to have the +3 - 4 + 5 + 6 become - (3 + 4 + 5 + 6). Which would require that the original +3 is turned into -3, and the -4 is turned into -4, and the +5 and +6 are turned into -5 and -6. But how can we do that with parentheses?

It seems that we can't do that because the original expression has a + before the 3, and we can't change that to a - without altering the operations. Therefore, this might not be allowed. Therefore, the expression 1 + 2 - (3 + 4 + 5 + 6) is not a valid placement of parentheses because it would require changing the + before the 3 into a -, which isn't allowed. Therefore, that's not a valid expression.

Therefore, the only way to get a subtraction on the 3 is by grouping it with the following operations. For example, if we have 1 + 2 + (3 - 4 - 5 - 6) = -9. Which is valid because we are just grouping the operations without changing the signs. The original expression has +3 - 4 + 5 + 6. By grouping (3 - 4 - 5 - 6), we are effectively changing the +5 and +6 into -5 and -6, but how?

Wait, actually, no. If we have 1 + 2 + (3 - 4 - 5 - 6), that is equivalent to 1 + 2 + 3 - 4 - 5 - 6. Which is different from the original expression. The original expression is 1 + 2 + 3 - 4 + 5 + 6. So, by adding parentheses around (3 - 4 - 5 - 6), we are changing the +5 and +6 into -5 and -6. But how is that allowed? Because the original expression has +5 and +6. So, unless we can change those additions into subtractions by grouping. But the problem says we can't change the operations. So, we can't change + into -.

Wait, this is a critical point. The problem says: "we cannot change addition or subtraction into multiplication." Wait, but does that mean we can't change addition into subtraction? The example given was changing the expression by adding parentheses, not changing operations. For example, they showed that 1+2+3-(4+5)+6 is allowed, which is grouping 4+5, but not changing the operations. So, the operations remain the same; we just change the order of operations with parentheses.

Therefore, in the expression 1 + 2 + 3 - 4 + 5 + 6, the operations are fixed: +, +, -, +, +. So, we can't change a + into a -, but we can group operations to change the order. Therefore, the expression 1 + 2 + (3 - 4 - 5 - 6) is not allowed because it would require changing the +5 and +6 into -5 and -6, which is changing the operations. Wait, no. Wait, actually, if we have 1 + 2 + (3 - 4 - 5 - 6), that is equivalent to 1 + 2 + 3 - 4 - 5 - 6. But in the original expression, the operations are +, +, -, +, +. So, the original expression is 1 + 2 + 3 - 4 + 5 + 6. If we want to have 1 + 2 + 3 - 4 - 5 - 6, that would require changing the +5 and +6 into -5 and -6. Which is changing the operations. Therefore, that is not allowed. Therefore, that expression is invalid.

Wait, so my previous thought that grouping (3 - 4 - 5 - 6) is allowed is incorrect. Because that would require changing the +5 and +6 into -5 and -6, which is not allowed. Therefore, that expression is invalid. Therefore, my earlier calculation of -9 was incorrect? Wait, no. Wait, let me think again.

Wait, the original expression is 1 + 2 + 3 - 4 + 5 + 6. If I add parentheses around 3 - 4 + 5 + 6, that would be 1 + 2 + (3 - 4 + 5 + 6). Which is 1 + 2 + (3 - 4 + 5 + 6) = 1 + 2 + (10) = 13. Which is valid. But if I want to have 1 + 2 + (3 - 4 - 5 - 6), that would require changing the +5 and +6 into -5 and -6, which is not allowed. Therefore, that is invalid.

Therefore, my previous calculation where I thought of grouping (3 - 4 - 5 - 6) was wrong. Therefore, the only valid way to get -9 is by grouping 4 + 5 + 6 as a subtraction. Let me check that again.

Original expression: 1 + 2 + 3 - 4 + 5 + 6. If we group 4 + 5 + 6 as a single term, then the expression becomes 1 + 2 + 3 - (4 + 5 + 6). Which is 1 + 2 + 3 - 15 = 6 - 15 = -9. Is this allowed? Yes, because we are just grouping the terms 4, 5, 6 with parentheses, and the subtraction is still only on the 4, but by grouping, we are effectively subtracting the entire group. Wait, but in the original expression, the subtraction is only on 4. So, how does grouping 4 + 5 + 6 as a single term under the subtraction work?

Let me think. The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around 4 + 5 + 6, it becomes 1 + 2 + 3 - (4 + 5 + 6). This is equivalent to 1 + 2 + 3 - 4 - 5 - 6. Wait, but in the original expression, after the subtraction of 4, we have +5 and +6. So, changing that to -5 and -6 would require changing the operations. But in reality, by adding parentheses, we are changing the order of operations. Let me verify:

Original expression without parentheses: (((((1 + 2) + 3) - 4) + 5) + 6) = 13.

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

But how is this different from changing the operations? Because in the original expression, the operations are +, +, -, +, +. When we add parentheses around 4 + 5 + 6, we are not changing the operations; we are just changing the order in which the operations are performed. Specifically, the subtraction of 4 is now applied to the entire group (4 + 5 + 6), which is equivalent to subtracting 4, then subtracting 5, then subtracting 6. But in the original expression, it was subtracting 4, then adding 5, then adding 6. Therefore, by grouping, we are effectively changing the operations from +5 and +6 to -5 and -6. But how is that allowed?

Wait, this seems like a contradiction. If we can't change the operations, how is this allowed? Because in reality, the parentheses change the order of operations, but the operations themselves remain the same. Let me think with an example.

Suppose we have the expression a - b + c. Without parentheses, it's (a - b) + c. If we add parentheses as a - (b + c), that's different. So, in this case, the operations are still subtraction and addition, but the order is changed. So, the original expression is a - b + c, which is (a - b) + c. If we group as a - (b + c), that's a - b - c. So, the operations are still subtraction and addition, but the grouping changes the result. Therefore, in this case, the operations are not changed; the grouping changes the order. Therefore, in the original problem, grouping 4 + 5 + 6 under the subtraction is allowed because we are just changing the order of operations, not the operations themselves.

Therefore, the expression 1 + 2 + 3 - (4 + 5 + 6) is valid, and it's equivalent to 1 + 2 + 3 - 4 - 5 - 6. Therefore, even though the original expression had +5 and +6, by grouping, we are effectively changing the order so that those terms are subtracted instead of added. Therefore, this is allowed.

Therefore, my previous calculation of -9 is valid. Therefore, that is a possible value.

But then, can we get even lower? Let me think. For example, if we can have more subtractions. Let me think.

Suppose we have 1 + (2 + 3 - 4 - 5 - 6). Let me check if that's allowed. The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we group (2 + 3 - 4 - 5 - 6), that would be equivalent to 2 + 3 - 4 - 5 - 6. Which is changing the +5 and +6 into -5 and -6. But again, is that allowed? Wait, no. Because the original expression has +5 and +6. If we group them as -5 and -6, that would require changing the operations. But in reality, by adding parentheses, we are changing the order of operations. Let me think again.

Original expression: 1 + 2 + 3 - 4 + 5 + 6. If we group (2 + 3 - 4 - 5 - 6), that would be equivalent to 2 + 3 - 4 - 5 - 6. But how does that relate to the original expression? The original expression is 1 + 2 + 3 - 4 + 5 + 6. So, the operations are +, +, -, +, +. If we group (2 + 3 - 4 - 5 - 6), that would require that the operations between 2 and 3 is +, between 3 and 4 is -, between 4 and 5 is -, and between 5 and 6 is -. But in the original expression, between 4 and 5 is +, and between 5 and 6 is +. Therefore, changing those to - is not allowed. Therefore, this grouping is invalid.

Wait, so this is a critical point. If we want to group operations that are originally additions into subtractions, that would require changing the operations, which is not allowed. Therefore, the only valid way to change the operations is by grouping existing operations. For example, if we have a subtraction followed by an addition, we can group them to make a larger subtraction. But we cannot change an addition into a subtraction.

Therefore, the expression 1 + 2 + 3 - (4 + 5 + 6) is valid because we are not changing the operations; we are just changing the order. The original operations are +, +, -, +, +. When we group 4 + 5 + 6, we are not changing the operations; we are just applying the subtraction to the entire group. Therefore, the operations remain the same. The subtraction is still only on the 4, but by grouping, we are effectively subtracting the entire group. Wait, but how?

Actually, no. The original expression is 1 + 2 + 3 - 4 + 5 + 6. The subtraction is only on the 4. When we add parentheses around 4 + 5 + 6, we are changing the order of operations so that the subtraction is applied to the entire group. However, the operations themselves are still subtraction and addition. The original expression has a subtraction followed by additions. By grouping, we are changing the order so that the subtraction is applied to the group, which includes the subsequent additions. Therefore, this is allowed because we are not changing the operations, just the order.

Therefore, the expression 1 + 2 + 3 - (4 + 5 + 6) is valid and results in -9. Similarly, if we can find another grouping that results in a lower value.

Let me think. What if we can group more terms to be subtracted? For example, if we can have something like 1 + (2 + 3 - 4) - (5 + 6). Which is 1 + (1) - 11 = -9. Same as before.

Alternatively, if we can have 1 + 2 + (3 - 4) - (5 + 6). Which is 1 + 2 + (-1) - 11 = 3 - 11 = -8. Which is higher than -9.

Alternatively, 1 + (2 + 3 - 4 - 5) + 6. Let me compute that:

Inside the parentheses: 2 + 3 - 4 - 5 = 5 - 4 - 5 = 1 - 5 = -4. Then, 1 + (-4) + 6 = 3. Which is higher.

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

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

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

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

So, all these ways give -9 or higher. Is there a way to get lower?

Wait, what if we can have more subtractions by grouping in a different way? For example, if we can have something like (1 + 2 + 3 - 4) - (5 + 6). Which is (6 - 4) - 11 = 2 - 11 = -9. Same result.

Alternatively, (1 + 2) - (3 + 4 + 5 + 6). Which would be 3 - 18 = -15. But is this allowed?

Original expression is 1 + 2 + 3 - 4 + 5 + 6. To get (1 + 2) - (3 + 4 + 5 + 6), we need to have the subtraction apply to the rest of the terms. But in the original expression, the operations are +, +, -, +, +. So, to have (1 + 2) - (3 + 4 + 5 + 6), we need to have the subtraction between 1 + 2 and the rest. But in the original expression, after 1 + 2, it's +3 - 4 + 5 + 6. So, to make it - (3 + 4 + 5 + 6), we need to have the subtraction apply to the entire rest. But how?

This would require that the original expression has a subtraction between 1 + 2 and the rest. But the original expression has a + before the 3. Therefore, unless we can group the entire expression as follows: (1 + 2) - (3 + 4 + 5 + 6). But how does that relate to the original expression?

The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around (1 + 2) and then subtract (3 + 4 + 5 + 6), that would require that the original expression has a subtraction between 1 + 2 and the rest. But the original expression has a + before the 3. Therefore, this is not allowed unless we can change the + into a -, which is not allowed. Therefore, this expression is invalid.

Therefore, the only valid way to get -9 is by grouping the 4 + 5 + 6 under the subtraction. Therefore, -9 is the minimum value.

But let me check another possibility. What if we group the first three terms and then subtract the rest? For example, (1 + 2 + 3) - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 13. No. Or (1 + 2 + 3) - (4 + 5 + 6) = -9. Which is valid.

Alternatively, grouping the first two terms and subtracting the rest: (1 + 2) - 3 + 4 + 5 + 6 = 3 - 3 + 4 + 5 + 6 = 1 + 4 + 5 + 6 = 16. Higher.

Alternatively, grouping the first term and subtracting the rest: 1 - 2 + 3 - 4 + 5 + 6. Wait, but that would require changing the +2 into -2, which is not allowed. Because the original expression is 1 + 2 + 3 - 4 + 5 + 6. To have 1 - 2 + 3 - 4 + 5 + 6, we need to change the +2 into -2, which is not allowed. Therefore, invalid.

Therefore, the only valid way to get -9 is by grouping 4 + 5 + 6 under the subtraction. Therefore, -9 is the minimum value.

But let me think again. What if we can group the subtraction with more terms? For example, if we have something like 1 + 2 + 3 - (4 + 5) + 6. Which is 6 - 9 + 6 = 3. Or 1 + 2 + 3 - 4 + (5 + 6) = 6 - 4 + 11 = 13. Or 1 + 2 + (3 - 4 + 5 + 6) = 1 + 2 + 10 = 13. Or 1 + (2 + 3 - 4 + 5 + 6) = 1 + 12 = 13. So, no improvement.

Alternatively, if we can group the first subtraction with the following additions in a way that creates more subtractions. For example, 1 + 2 + (3 - 4 + 5) + 6. Which is 1 + 2 + 4 + 6 = 13. Or 1 + 2 + 3 - (4 + 5) + 6 = 3. Or 1 + 2 + 3 - 4 - (5 + 6) = -9. Which is the same as before.

Therefore, after checking all possibilities, it seems that -9 is the minimum value achievable by adding parentheses without changing the operations.

But let me check one more thing. What if we have multiple subtractions by grouping? For example, 1 + (2 + 3 - 4) - (5 + 6). Which is 1 + 1 - 11 = -9. Same as before.

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

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

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

So, all these ways give -9. Therefore, I think that -9 is indeed the minimum value.

But let me think if there's another way. For example, if we can have something like 1 + 2 + 3 - 4 - 5 - 6. Which is 6 - 4 - 5 - 6 = -9. But how is this expression achieved? By grouping the original expression as 1 + 2 + 3 - (4 + 5 + 6). Which is allowed, as discussed earlier. Therefore, this is valid.

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

But just to be thorough, let me check if there's a way to get lower than -9. For example, -10.

To get -10, we need the total value to be -10. Let's see. The original total is 13. To get -10, we need to subtract 23. But how?

The maximum subtraction we can do is 4 + 5 + 6 = 15, which gives us 6 - 15 = -9. To get a larger subtraction, we would need to subtract more than 15. But how?

If we could subtract 1 + 2 + 3 + 4 + 5 + 6 = 21, then the total would be 1 + 2 + 3 - (4 + 5 + 6) - (1 + 2 + 3)? No, that's not possible. Wait, no. The original expression is fixed. We can't subtract more than the terms present.

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

Alternatively, if we can have more subtractions by grouping. For example, if we can have 1 - (2 + 3 + 4 + 5 + 6). But again, that would require changing the +1 into -1, which is not allowed.

Therefore, I don't think it's possible to get lower than -9.

Therefore, 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** (without rearranging terms or changing operations), we must explore how parentheses can alter the **order of operations**.

---

### Step 1: Understand the Original Expression

The original expression, evaluated left-to-right, is:

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

We are not allowed to change the **order of the terms** or **change the operations** (i.e., we can't turn a `+` into a `-` or vice versa). However, we **can group terms** using parentheses to change the **order of evaluation**.

---

### Step 2: Strategy for Minimizing the Value

Since subtraction is **not associative**, grouping can significantly affect the result. The key is to **maximize the negative contributions** by grouping terms that are being **added** into a **subtraction**.

For example, consider the expression:

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

This is a valid placement of parentheses, and it evaluates as:

$$
1 + 2 + 3 - (4 + 5 + 6) = 6 - 15 = -9
$$

This is **much smaller** than the original value of 13.

---

### Step 3: Can We Get Even Lower?

Let’s explore whether we can get a value **less than -9**.

Suppose we try to subtract more terms. The only subtraction in the original expression is on the number `4`. However, by grouping, we can **extend the subtraction** to include more terms.

For instance:

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

This is the **maximum possible subtraction** we can achieve using the given terms and operations. Any attempt to subtract more than 15 (i.e., the sum of 4, 5, and 6) would require **changing the operations**, which is not allowed.

---

### Step 4: Confirming the Minimum

We can try other groupings, such as:

- $1 + (2 + 3 - 4 - 5 - 6) = 1 + (-12) = -11$ → **Invalid**, because this would require changing the `+5` and `+6` into `-5` and `-6`, which is not allowed.
- $1 + 2 + (3 - 4 - 5 - 6) = 1 + 2 + (-12) = -9$ → **Valid**, but still gives the same result.
- $1 + 2 + 3 - (4 + 5 + 6) = -9$ → **Valid and minimal**.

All other groupings either result in the same value or a higher one.

---

### ✅ 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 key here is that I can't rearrange the order of the terms or change the operations, just add parentheses. Let me start by understanding the original expression and how parentheses can affect the result.

First, the original expression without any parentheses is 1 + 2 + 3 - 4 + 5 + 6. Let me compute that to confirm it's 13.

Calculating step by step:
1 + 2 = 3
3 + 3 = 6
6 - 4 = 2
2 + 5 = 7
7 + 6 = 13. Yep, that checks out.

Now, the problem says that by adding parentheses, we can change the value. For example, they showed that 1+2+3-(4+5)+6 equals 3. Let me verify that:

1 + 2 + 3 = 6
4 + 5 = 9
So, 6 - 9 + 6 = (6 - 9) + 6 = (-3) + 6 = 3. Correct.

So, parentheses can change the order of operations. Since addition and subtraction are left-associative by default, but parentheses can override that. The goal is to find the minimum possible value by strategically placing parentheses.

First, I need to recall that parentheses can group operations, so changing the order in which operations are performed. Since subtraction is involved, grouping terms can lead to different results. For example, if I can make a larger negative number by grouping terms, that might decrease the overall value.

But how do I approach this systematically? Let me think.

The original expression is a sequence of numbers with alternating addition and subtraction. The expression is:

1 + 2 + 3 - 4 + 5 + 6

So, the operations are: +, +, -, +, +. The numbers are 1, 2, 3, 4, 5, 6.

Since we can't rearrange the terms, the order is fixed. So, the challenge is to group these operations with parentheses to change the result.

Let me think about how parentheses can affect the result. Parentheses can change the order of operations, so for example, if I have something like a - (b + c), that's different from (a - b) + c. Similarly, grouping multiple additions or subtractions can change the result.

Since we want the minimum value, we need to maximize the negative contributions. So, perhaps grouping terms in such a way that we subtract larger numbers, or have more negative terms.

But how?

Let me consider that subtraction is equivalent to adding a negative. So, maybe if I can create larger negative numbers by grouping terms. For example, if I can have a large number subtracted from a smaller one, that would be bad. Wait, but how?

Alternatively, maybe grouping terms so that more subtractions are applied to larger numbers. Let me think.

Let me try to think of all possible ways to add parentheses. However, since there are multiple operations, the number of possible parenthetical groupings could be quite large. Maybe there's a smarter way than brute-forcing all possibilities.

Alternatively, think about the expression as a sequence of operations and how parentheses can change the grouping. Let me note that the expression has five operations: +, +, -, +, +. So, between the numbers, there are five operations. The parentheses can group these operations in different ways.

But since we can't change the order of the terms, the structure is fixed. So, the problem is similar to inserting parentheses in the expression to change the order of operations.

This seems similar to the problem of evaluating expressions with different parenthetical groupings. For example, in the expression a + b - c + d + e, how can parentheses change the result?

But since we can't rearrange terms, the structure is fixed. So, perhaps we can model this as a binary tree where each operation is a node, and parentheses define the order of operations.

But maybe that's too abstract. Let me think of possible groupings.

First, note that addition is associative and commutative, but subtraction is neither. So, grouping additions together won't change their result, but grouping subtractions can have different effects.

Wait, but since we can't rearrange terms, the order of operations is fixed except for the parentheses. So, for example, if I have 1 + 2 + 3 - 4 + 5 + 6, the default is left to right:

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

But if I group some operations differently, like 1 + (2 + 3 - 4) + 5 + 6, that would be different. Let me compute that:

First, 2 + 3 - 4 = 1, then 1 + 1 + 5 + 6 = 13. Same result. Wait, so that didn't change anything. Hmm.

But in the example given, they grouped 4 + 5, so that's changing the order. Let me think again.

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

If we group the -4 + 5 as -(4 - 5)? Wait, no. Wait, the example was 1 + 2 + 3 - (4 + 5) + 6. So, that's grouping the 4 + 5, which is subtracted. So, that's changing the order of operations. So, instead of subtracting 4 and then adding 5, it's subtracting (4 + 5). So, that's equivalent to subtracting 9 instead of subtracting 4 and adding 5. Which is a big difference.

So, in that case, the net effect is that instead of (-4 + 5) = +1, it's - (4 + 5) = -9. So, that's a difference of -10. Which is why the total went from 13 to 3. So, that's a significant change.

Therefore, the key is to find groupings where we can have more negative contributions. So, perhaps grouping terms that are being subtracted with other terms, thereby increasing the magnitude of the subtraction.

So, the idea is to find groupings where we can have larger subtractions. Let me think about how to do that.

Let me consider that each subtraction can be thought of as a negative sign. So, if I can group a term with a negative sign, that might help. For example, if I have something like - (a + b), that's equivalent to -a - b. Whereas if it's -a + b, that's different.

So, perhaps if I can create more negative terms by grouping. Let me think of the original expression:

1 + 2 + 3 - 4 + 5 + 6.

So, the operations are:

Start with 1, then add 2, add 3, subtract 4, add 5, add 6.

So, the subtraction is only on the 4. But if we can group the subtraction with other terms, we can make the subtraction affect more numbers.

For example, if we can have - (4 - 5), but that would be -(-1) = +1, which is not helpful. Wait, but maybe grouping in a way that the subtraction applies to more numbers.

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

3 - 4 + 5 = 4, so total is 1 + 2 + 4 + 6 = 13. Same as before.

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

2 + 3 - 4 + 5 = 6, so total is 1 + 6 + 6 = 13. Still same.

Hmm. So, grouping additions and subtractions in different ways doesn't change the result if they are all additions and subtractions. Wait, but in the example given, grouping 4 + 5 changed the result. So, that was a subtraction of (4 + 5) instead of subtracting 4 and adding 5. So, that's different.

So, the key is that when you have a subtraction followed by an addition, grouping them together can change the result. So, in the original expression, the subtraction is followed by an addition. So, if you group them, you can have a larger subtraction.

So, perhaps the strategy is to group as many terms as possible under a subtraction. Let me think.

Let me consider that the original expression has a single subtraction: -4. If I can make that subtraction affect more numbers, that would be better. For example, if I can have - (4 - 5 + 6), but that would be - (5) = -5? Wait, let me check:

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

First, compute inside the parentheses: 4 - 5 + 6 = 5. Then, 1 + 2 + 3 - 5 = 6 - 5 = 1. So, that's 1. Which is lower than 13, but not the minimum.

Alternatively, if I can have more terms subtracted. Let me think. For example, if I can have - (4 + 5 + 6). But how?

Original expression is 1 + 2 + 3 - 4 + 5 + 6. If I can group the -4 + 5 + 6 as -(4 - 5 - 6)? Wait, that would be different. Wait, but how can I group that?

Wait, maybe if I have 1 + 2 + 3 - (4 - 5 - 6). Let me compute that:

Inside the parentheses: 4 - 5 - 6 = -7. Then, 1 + 2 + 3 - (-7) = 6 + 7 = 13. So, same as original. Not helpful.

Alternatively, maybe grouping the entire expression in a way that more terms are subtracted. Let me think.

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

3 - 4 + 5 + 6 = 10. Then, 1 + 2 + 10 = 13. Still same.

Alternatively, 1 + (2 + 3 - 4 + 5 + 6). Let me compute:

2 + 3 - 4 + 5 + 6 = 12. Then, 1 + 12 = 13. Still same.

Hmm. So, grouping additions and subtractions in different ways doesn't change the result if they are all additions and subtractions. But when you have a subtraction followed by an addition, grouping them can change the result. For example, in the original example, they grouped 4 + 5, which was subtracted. So, that's changing the order.

So, perhaps the key is to group terms that are being subtracted with terms that are being added, thereby turning them into a larger subtraction. Let me think.

Let me consider the original expression again:

1 + 2 + 3 - 4 + 5 + 6.

The subtraction is only on 4. But if I can group the -4 with the +5 and +6, so that instead of subtracting 4 and then adding 5 and 6, I subtract (4 - 5 - 6). Wait, but how?

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

Inside the parentheses: 4 - 5 - 6 = -7. Then, 1 + 2 + 3 - (-7) = 6 + 7 = 13. Same as before.

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

Inside the parentheses: 4 + 5 + 6 = 15. Then, 1 + 2 + 3 - 15 = 6 - 15 = -9. Oh! That's a lower value. So, that's -9. Is that allowed?

Wait, can I group the -4 + 5 + 6 as -(4 + 5 + 6)? Let me check the original expression. The original expression is 1 + 2 + 3 - 4 + 5 + 6. So, the subtraction is only on 4. But if I add parentheses around 4 + 5 + 6, then it would be 1 + 2 + 3 - (4 + 5 + 6). Is that a valid placement of parentheses?

Yes, because the original expression is 1 + 2 + 3 - 4 + 5 + 6. So, inserting parentheses around 4 + 5 + 6 would be:

1 + 2 + 3 - (4 + 5 + 6). Which is valid. So, that would be 1 + 2 + 3 - 15 = 6 - 15 = -9. So, that's a valid expression with parentheses, and gives -9. That's lower than the previous example of 3. So, maybe that's a candidate for the minimum.

But can we get even lower?

Let me think. How else can we group the terms?

Another idea: Maybe group more terms to be subtracted. For example, if we can have more subtractions. Let me think.

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

Suppose we try to group the first part as 1 + (2 + 3 - 4 + 5 + 6). Wait, that would be 1 + (2 + 3 - 4 + 5 + 6) = 1 + (12) = 13. Not helpful.

Alternatively, group the first three terms: (1 + 2 + 3) - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 13. Same as before.

Alternatively, group the first two terms: (1 + 2) + 3 - 4 + 5 + 6 = 3 + 3 - 4 + 5 + 6 = 13. Still same.

Alternatively, group the first term with the next operations: 1 + (2 + 3 - 4 + 5 + 6) = same as before.

Alternatively, group the subtraction with the following additions. For example, 1 + 2 + (3 - 4 + 5 + 6) = 1 + 2 + 10 = 13.

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

Alternatively, group 3 - (4 + 5) + 6. Let me compute that:

3 - (4 + 5) + 6 = 3 - 9 + 6 = 0. Then 1 + 2 + 0 = 3. Which is the example given.

So, that gives 3. But earlier, grouping 4 + 5 + 6 as a subtraction gives -9. So, that's lower.

Is there a way to get even lower?

Let me think. What if we group more terms to be subtracted? For example, can we have something like 1 + 2 + 3 - (4 + 5 + 6) = -9 as above. What if we can have more terms subtracted?

Wait, but the original expression only has one subtraction operator. Wait, no. Wait, the original expression is 1 + 2 + 3 - 4 + 5 + 6. So, the operations are +, +, -, +, +. So, there is only one subtraction. So, the rest are additions. Therefore, the only subtraction is on the 4. However, by adding parentheses, we can change the effect of that subtraction. For example, if we can make that subtraction apply to more terms.

But in the case of grouping 4 + 5 + 6, we are effectively making the subtraction apply to 4, 5, and 6. Wait, but how? Because the original expression is 1 + 2 + 3 - 4 + 5 + 6. If we group 4 + 5 + 6, then it's 1 + 2 + 3 - (4 + 5 + 6). So, the subtraction is only on the entire group, which is 4 + 5 + 6. So, that's equivalent to subtracting 15. So, that's a larger subtraction than just subtracting 4. So, that's why it's -9.

But can we make the subtraction apply to even more terms? For example, if we could have something like 1 + 2 + 3 - 4 + 5 + 6, but with parentheses that somehow subtract more terms. Wait, but the original expression only has one subtraction. Unless we can create more subtractions by grouping.

Wait, for example, if we have something like 1 + 2 + (3 - 4 + 5 + 6). Wait, that's 1 + 2 + (3 - 4 + 5 + 6) = 1 + 2 + 10 = 13. Not helpful.

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

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

Alternatively, can we have something like (1 + 2 + 3 - 4) + 5 + 6? Which is (6 - 4) + 5 + 6 = 2 + 5 + 6 = 13. Same.

Alternatively, 1 + (2 + 3 - 4 + 5) + 6. Let me compute that:

2 + 3 - 4 + 5 = 6. Then 1 + 6 + 6 = 13.

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

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

So, all these groupings don't change the result. So, the only way to get a different result is to group terms that are being subtracted with terms that are being added, thereby changing the sign of those added terms. For example, in the example given, grouping 4 + 5 as a subtraction, which turned the +5 into a -5. So, that's a way to make the addition of 5 into a subtraction.

So, perhaps if we can do similar things with other terms. Let me think.

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

If I can group the +5 + 6 as a subtraction. Wait, but how? Because the original expression has a subtraction only on 4. Unless we can create a subtraction on 5 and 6 by grouping.

Wait, for example, if I have 1 + 2 + 3 - 4 + (5 + 6). That's 1 + 2 + 3 - 4 + 11 = 6 - 4 + 11 = 13. Same as before.

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

Alternatively, if I can have 1 + 2 + (3 - 4 + 5 + 6) = 1 + 2 + 10 = 13.

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

Alternatively, if I can have (1 + 2 + 3 - 4 + 5) + 6 = (6 - 4 + 5) + 6 = 7 + 6 = 13.

Alternatively, maybe grouping the first subtraction with the following additions. For example, 1 + 2 + 3 - (4 - 5 - 6). Wait, let me compute that:

Inside the parentheses: 4 - 5 - 6 = -7. Then, 1 + 2 + 3 - (-7) = 6 + 7 = 13. Same as before.

Alternatively, 1 + 2 + 3 - (4 - 5 + 6). Let me compute:

Inside the parentheses: 4 - 5 + 6 = 5. Then, 1 + 2 + 3 - 5 = 6 - 5 = 1. Which is lower than 13, but higher than -9.

Alternatively, 1 + 2 + 3 - (4 + 5 - 6). Let me compute:

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

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

Is there a way to get even lower?

Let me think. What if we can have more subtractions? For example, if we can have two subtractions. But the original expression only has one subtraction. Unless we can create another subtraction by grouping.

Wait, for example, if we have something like 1 + 2 + (3 - 4) + (5 - 6). Wait, but how can we do that? The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses to make it 1 + 2 + (3 - 4) + (5 - 6). Let me check if that's allowed.

Original expression: 1 + 2 + 3 - 4 + 5 + 6. To make it 1 + 2 + (3 - 4) + (5 - 6), we need to add parentheses around (3 - 4) and (5 - 6). Is that allowed? Let me see.

Yes, because the original expression is 1 + 2 + 3 - 4 + 5 + 6. So, inserting parentheses around (3 - 4) and (5 - 6) would be:

1 + 2 + (3 - 4) + (5 - 6). Let me compute that:

First, 3 - 4 = -1, 5 - 6 = -1. Then, 1 + 2 + (-1) + (-1) = 1 + 2 - 1 - 1 = 1. Which is 1. So, that's lower than the previous -9? No, -9 is lower. Wait, 1 is higher than -9. So, -9 is still lower.

Alternatively, can we have more subtractions? For example, if we can have something like 1 + (2 + 3 - 4 - 5 - 6). Let me check if that's possible.

Original expression: 1 + 2 + 3 - 4 + 5 + 6. To make it 1 + (2 + 3 - 4 - 5 - 6). Let me see. How would that be grouped? The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we group 2 + 3 - 4 - 5 - 6, that would require changing the operations. Wait, but the original operations are +, +, -, +, +. So, to have -4 -5 -6, we need to have the subtraction signs. But in the original expression, after the -4, we have +5 and +6. So, unless we can change the +5 and +6 into -5 and -6 by grouping. But how?

Wait, if we have 1 + (2 + 3 - 4 - 5 - 6). That would require that the original expression has -4 -5 -6. But the original expression is -4 + 5 + 6. So, unless we can group the +5 and +6 as -5 -6. How?

For example, if we have 1 + 2 + 3 - 4 - (5 + 6). Let me check if that's allowed. The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around 5 + 6 and put a minus sign in front, that would be 1 + 2 + 3 - 4 - (5 + 6). Let me compute that:

Inside the parentheses: 5 + 6 = 11. Then, 1 + 2 + 3 - 4 - 11 = 6 - 4 - 11 = 2 - 11 = -9. Which is the same as before. So, that's -9. So, same result.

Alternatively, can we have more subtractions? For example, 1 + 2 + 3 - (4 + 5 + 6) = -9. Or 1 + 2 + (3 - 4 - 5 - 6). Let me compute that:

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

So, that's another way to get -9. So, seems like -9 is achievable in multiple ways.

Is there a way to get lower than -9?

Let me think. Let's see. To get lower than -9, we need the total value to be less than -9. Let's think about how that could happen.

The original expression is 1 + 2 + 3 - 4 + 5 + 6. The total is 13. If we can make the expression evaluate to something like -10, -11, etc.

How can we do that? Let me think. For example, if we can have more subtractions. But how?

Wait, the original expression has only one subtraction operator. However, by grouping, we can create more subtractions. For example, if we can have multiple subtractions by grouping.

Wait, for example, if we have something like 1 + 2 + (3 - 4 - 5 - 6). Which we already did, resulting in -9. But is there a way to have even more subtractions?

Wait, let me think. Suppose we have 1 + (2 + 3 - 4 - 5 - 6). Let me compute that:

Inside the parentheses: 2 + 3 - 4 - 5 - 6 = 5 - 4 - 5 - 6 = 1 - 5 - 6 = -10. Then, 1 + (-10) = -9. Same result.

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

1 + 2 + 3 - 4 - 5 - 6 = 6 - 4 - 5 - 6 = 2 - 5 - 6 = -9. Same result.

So, regardless of how we group, if we subtract 4, 5, 6, we get -9. Is there a way to subtract more?

Wait, the original expression is 1 + 2 + 3 - 4 + 5 + 6. The numbers are 1, 2, 3, 4, 5, 6. If we can somehow subtract all numbers except 1 and 2, but how?

Wait, for example, if we have something like (1 + 2) - (3 + 4 + 5 + 6). Let me check if that's possible.

Original expression: 1 + 2 + 3 - 4 + 5 + 6. To get (1 + 2) - (3 + 4 + 5 + 6), we need to group the first two terms as a sum and then subtract the rest. Let me see:

Original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around (1 + 2) and then subtract (3 + 4 + 5 + 6). But how?

Wait, the original expression is 1 + 2 + 3 - 4 + 5 + 6. To make it (1 + 2) - (3 + 4 + 5 + 6), we need to have the subtraction apply to the entire rest of the expression. But the original expression has a subtraction only on 4. So, unless we can group the rest of the terms as a subtraction. Let me think.

Wait, if we have 1 + 2 + 3 - 4 + 5 + 6. If we want to make it (1 + 2) - (3 + 4 + 5 + 6), that would require changing the operations. Because originally, after the subtraction of 4, we have +5 and +6. To make them -5 and -6, we need to add parentheses around them and put a minus sign. But how?

For example, if we have 1 + 2 + 3 - 4 - (5 + 6). Which we already did, resulting in -9. But that's only subtracting 5 and 6. If we could subtract 3, 4, 5, 6, that would be better. How?

Wait, if we have something like 1 + 2 - (3 + 4 + 5 + 6). Let me check if that's possible. Original expression is 1 + 2 + 3 - 4 + 5 + 6. To get 1 + 2 - (3 + 4 + 5 + 6), we need to have the subtraction apply to 3, 4, 5, 6. But in the original expression, the subtraction is only on 4. So, unless we can group the 3 with the rest. Let me think.

If we have 1 + 2 + (3 - 4 - 5 - 6). Which is 1 + 2 + (3 - 4 - 5 - 6) = 1 + 2 + (-12) = -9. Same as before.

Alternatively, if we have (1 + 2 + 3) - (4 + 5 + 6) = 6 - 15 = -9. Same result.

So, regardless of grouping, the maximum subtraction we can get is 15 (4 + 5 + 6), leading to -9. Is there a way to get a larger subtraction?

Wait, what if we can subtract more numbers? For example, subtract 1, 2, 3, 4, 5, 6? But how?

Original expression is 1 + 2 + 3 - 4 + 5 + 6. If we could somehow subtract all numbers, but the original expression starts with additions. Unless we can create a subtraction on the first terms.

For example, if we have -(1 + 2 + 3 + 4 + 5 + 6). But how can we do that? The original expression is 1 + 2 + 3 - 4 + 5 + 6. To make it -(1 + 2 + 3 + 4 + 5 + 6), we need to have a negative sign in front. But the original expression doesn't have that. Unless we can group the entire expression with a negative sign. But the original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around the entire expression and put a negative sign? Wait, but the problem says we can't change the operations. So, we can't add a negative sign unless it's part of the original expression.

Wait, the original expression is 1 + 2 + 3 - 4 + 5 + 6. There is no negative sign at the beginning. So, we can't just put a negative sign in front. Unless we can group the entire expression as a subtraction. But how?

Wait, for example, if we have -(1 + 2 + 3 - 4 + 5 + 6). But that would require adding a negative sign, which isn't part of the original expression. The problem says we can't change the operations, so I think we can't add a negative sign. So, that's not allowed.

Therefore, the only way to get a negative value is by grouping terms that are being subtracted. So, the maximum subtraction we can get is 4 + 5 + 6 = 15, leading to -9. Is that the minimum?

Wait, let me think again. What if we can have more subtractions by grouping in a different way?

For example, suppose we have 1 + (2 + 3 - 4) - (5 + 6). Let me check if that's allowed. Original expression is 1 + 2 + 3 - 4 + 5 + 6. To make it 1 + (2 + 3 - 4) - (5 + 6), we need to add parentheses around (2 + 3 - 4) and (5 + 6). Let me compute that:

Inside the first parentheses: 2 + 3 - 4 = 1. Inside the second parentheses: 5 + 6 = 11. Then, 1 + 1 - 11 = -9. Same result.

Alternatively, 1 + 2 + (3 - 4 - 5 - 6) = 1 + 2 + (-12) = -9.

Alternatively, 1 + (2 + 3 - 4 - 5 - 6) = 1 + (-10) = -9.

So, all these ways give -9. Is there a way to get lower?

Wait, let me think. What if we can have more subtractions? For example, if we can have something like 1 - (2 + 3 + 4 + 5 + 6). But how?

Original expression is 1 + 2 + 3 - 4 + 5 + 6. To get 1 - (2 + 3 + 4 + 5 + 6), we need to have the subtraction apply to the rest of the terms. But the original expression starts with a +, so unless we can group the first term with a negative sign. But again, we can't add a negative sign unless it's part of the original expression.

Wait, unless we can group the entire expression as follows: (1 + 2 + 3 - 4 + 5 + 6) with some parentheses that somehow negate parts. But I don't think that's possible without changing the operations.

Alternatively, if we can have something like (1 + 2 + 3) - (4 + 5 + 6) = -9. Which is the same as before.

Alternatively, if we can have (1 + 2) - (3 + 4 + 5 + 6) = 3 - 18 = -15. Wait, is that possible?

Wait, original expression is 1 + 2 + 3 - 4 + 5 + 6. To make it (1 + 2) - (3 + 4 + 5 + 6), we need to have the subtraction apply to 3, 4, 5, 6. But in the original expression, after the subtraction of 4, we have +5 and +6. So, how can we make them subtracted?

If we have 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around 3 - 4 + 5 + 6 as -(3 - 4 + 5 + 6)? Wait, no. Wait, if we want to have (1 + 2) - (3 + 4 + 5 + 6), that would require that the original expression has a subtraction between 1 + 2 and the rest. But the original expression is 1 + 2 + 3 - 4 + 5 + 6. So, the subtraction is only on 4. To make it (1 + 2) - (3 + 4 + 5 + 6), we need to have the subtraction between 1 + 2 and the rest. But how?

Wait, if we have 1 + 2 - (3 + 4 + 5 + 6). Let me check if that's possible. The original expression is 1 + 2 + 3 - 4 + 5 + 6. To make it 1 + 2 - (3 + 4 + 5 + 6), we need to change the +3 - 4 + 5 + 6 into - (3 + 4 + 5 + 6). How?

That would require that the original expression has a subtraction before the 3. But in the original expression, it's +3 - 4 + 5 + 6. So, unless we can group the +3 - 4 + 5 + 6 as - (3 + 4 + 5 + 6). But how?

Wait, if we have 1 + 2 - (3 + 4 + 5 + 6). Let me think about the operations. The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we want to have 1 + 2 - (3 + 4 + 5 + 6), that would require that the +3 - 4 + 5 + 6 becomes - (3 + 4 + 5 + 6). Which would mean that the original +3 is turned into -3, the -4 is turned into -4, and the +5 and +6 are turned into -5 and -6. But how can we do that with parentheses?

Wait, if we have 1 + 2 - (3 + 4 + 5 + 6). To get that from the original expression, we need to have the subtraction apply to the entire 3 + 4 + 5 + 6. But in the original expression, the operations are +3 - 4 + 5 + 6. So, if we can group the +3 - 4 + 5 + 6 as -(3 + 4 + 5 + 6), that would require that the original +3 is turned into -3, which would mean that we have a negative sign in front of the 3. But how?

Wait, unless we can group the entire expression as follows: 1 + 2 - (3 + 4 + 5 + 6). But how does that relate to the original expression? The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around 3 + 4 + 5 + 6 and put a minus sign in front, that would be 1 + 2 - (3 + 4 + 5 + 6). But in the original expression, the operations are +3 - 4 + 5 + 6. So, to make it - (3 + 4 + 5 + 6), we need to have the subtraction apply to the entire 3 + 4 + 5 + 6. But in the original expression, the subtraction is only on 4. So, unless we can somehow change the +3 into -3 by grouping.

Wait, for example, if we have 1 + 2 + (3 - 4 - 5 - 6). Which we did before, resulting in -9. But that's not the same as 1 + 2 - (3 + 4 + 5 + 6). Because in that case, the 3 is subtracted, but the rest are also subtracted. However, in the original expression, the 3 is added, then 4 is subtracted, then 5 and 6 are added. So, to make the 3 subtracted, we need to have a negative sign in front of it. But how?

Wait, if we have 1 + 2 - (3 + 4 + 5 + 6). That would require that the original expression has a subtraction before the 3. But the original expression has a + before the 3. So, unless we can group the +3 - 4 + 5 + 6 as - (3 + 4 + 5 + 6). Which would require that the original +3 is turned into -3, which would mean that we have a negative sign in front of the 3. But how can we do that with parentheses?

Wait, maybe by grouping the entire expression as follows: 1 + 2 - (3 + 4 + 5 + 6). But how does that relate to the original expression? The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around 3 + 4 + 5 + 6 and put a minus sign in front, that would be 1 + 2 - (3 + 4 + 5 + 6). But in the original expression, the operations are +3 - 4 + 5 + 6. So, to make it - (3 + 4 + 5 + 6), we need to have the subtraction apply to the entire 3 + 4 + 5 + 6. However, the original expression has a subtraction only on 4. So, unless we can somehow change the +3 into -3 by grouping.

Wait, for example, if we have 1 + 2 - (3 + 4 + 5 + 6). To get that from the original expression, we need to have the +3 - 4 + 5 + 6 become - (3 + 4 + 5 + 6). Which would require that the original +3 is turned into -3, and the -4 is turned into -4, and the +5 and +6 are turned into -5 and -6. But how can we do that with parentheses?

It seems that we can't do that because the original expression has a + before the 3, and we can't change that to a - without altering the operations. Therefore, this might not be allowed. Therefore, the expression 1 + 2 - (3 + 4 + 5 + 6) is not a valid placement of parentheses because it would require changing the + before the 3 into a -, which isn't allowed. Therefore, that's not a valid expression.

Therefore, the only way to get a subtraction on the 3 is by grouping it with the following operations. For example, if we have 1 + 2 + (3 - 4 - 5 - 6) = -9. Which is valid because we are just grouping the operations without changing the signs. The original expression has +3 - 4 + 5 + 6. By grouping (3 - 4 - 5 - 6), we are effectively changing the +5 and +6 into -5 and -6, but how?

Wait, actually, no. If we have 1 + 2 + (3 - 4 - 5 - 6), that is equivalent to 1 + 2 + 3 - 4 - 5 - 6. Which is different from the original expression. The original expression is 1 + 2 + 3 - 4 + 5 + 6. So, by adding parentheses around (3 - 4 - 5 - 6), we are changing the +5 and +6 into -5 and -6. But how is that allowed? Because the original expression has +5 and +6. So, unless we can change those additions into subtractions by grouping. But the problem says we can't change the operations. So, we can't change + into -.

Wait, this is a critical point. The problem says: "we cannot change addition or subtraction into multiplication." Wait, but does that mean we can't change addition into subtraction? The example given was changing the expression by adding parentheses, not changing operations. For example, they showed that 1+2+3-(4+5)+6 is allowed, which is grouping 4+5, but not changing the operations. So, the operations remain the same; we just change the order of operations with parentheses.

Therefore, in the expression 1 + 2 + 3 - 4 + 5 + 6, the operations are fixed: +, +, -, +, +. So, we can't change a + into a -, but we can group operations to change the order. Therefore, the expression 1 + 2 + (3 - 4 - 5 - 6) is not allowed because it would require changing the +5 and +6 into -5 and -6, which is changing the operations. Wait, no. Wait, actually, if we have 1 + 2 + (3 - 4 - 5 - 6), that is equivalent to 1 + 2 + 3 - 4 - 5 - 6. But in the original expression, the operations are +, +, -, +, +. So, the original expression is 1 + 2 + 3 - 4 + 5 + 6. If we want to have 1 + 2 + 3 - 4 - 5 - 6, that would require changing the +5 and +6 into -5 and -6. Which is changing the operations. Therefore, that is not allowed. Therefore, that expression is invalid.

Wait, so my previous thought that grouping (3 - 4 - 5 - 6) is allowed is incorrect. Because that would require changing the +5 and +6 into -5 and -6, which is not allowed. Therefore, that expression is invalid. Therefore, my earlier calculation of -9 was incorrect? Wait, no. Wait, let me think again.

Wait, the original expression is 1 + 2 + 3 - 4 + 5 + 6. If I add parentheses around 3 - 4 + 5 + 6, that would be 1 + 2 + (3 - 4 + 5 + 6). Which is 1 + 2 + (3 - 4 + 5 + 6) = 1 + 2 + (10) = 13. Which is valid. But if I want to have 1 + 2 + (3 - 4 - 5 - 6), that would require changing the +5 and +6 into -5 and -6, which is not allowed. Therefore, that is invalid.

Therefore, my previous calculation where I thought of grouping (3 - 4 - 5 - 6) was wrong. Therefore, the only valid way to get -9 is by grouping 4 + 5 + 6 as a subtraction. Let me check that again.

Original expression: 1 + 2 + 3 - 4 + 5 + 6. If we group 4 + 5 + 6 as a single term, then the expression becomes 1 + 2 + 3 - (4 + 5 + 6). Which is 1 + 2 + 3 - 15 = 6 - 15 = -9. Is this allowed? Yes, because we are just grouping the terms 4, 5, 6 with parentheses, and the subtraction is still only on the 4, but by grouping, we are effectively subtracting the entire group. Wait, but in the original expression, the subtraction is only on 4. So, how does grouping 4 + 5 + 6 as a single term under the subtraction work?

Let me think. The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around 4 + 5 + 6, it becomes 1 + 2 + 3 - (4 + 5 + 6). This is equivalent to 1 + 2 + 3 - 4 - 5 - 6. Wait, but in the original expression, after the subtraction of 4, we have +5 and +6. So, changing that to -5 and -6 would require changing the operations. But in reality, by adding parentheses, we are changing the order of operations. Let me verify:

Original expression without parentheses: (((((1 + 2) + 3) - 4) + 5) + 6) = 13.

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

But how is this different from changing the operations? Because in the original expression, the operations are +, +, -, +, +. When we add parentheses around 4 + 5 + 6, we are not changing the operations; we are just changing the order in which the operations are performed. Specifically, the subtraction of 4 is now applied to the entire group (4 + 5 + 6), which is equivalent to subtracting 4, then subtracting 5, then subtracting 6. But in the original expression, it was subtracting 4, then adding 5, then adding 6. Therefore, by grouping, we are effectively changing the operations from +5 and +6 to -5 and -6. But how is that allowed?

Wait, this seems like a contradiction. If we can't change the operations, how is this allowed? Because in reality, the parentheses change the order of operations, but the operations themselves remain the same. Let me think with an example.

Suppose we have the expression a - b + c. Without parentheses, it's (a - b) + c. If we add parentheses as a - (b + c), that's different. So, in this case, the operations are still subtraction and addition, but the order is changed. So, the original expression is a - b + c, which is (a - b) + c. If we group as a - (b + c), that's a - b - c. So, the operations are still subtraction and addition, but the grouping changes the result. Therefore, in this case, the operations are not changed; the grouping changes the order. Therefore, in the original problem, grouping 4 + 5 + 6 under the subtraction is allowed because we are just changing the order of operations, not the operations themselves.

Therefore, the expression 1 + 2 + 3 - (4 + 5 + 6) is valid, and it's equivalent to 1 + 2 + 3 - 4 - 5 - 6. Therefore, even though the original expression had +5 and +6, by grouping, we are effectively changing the order so that those terms are subtracted instead of added. Therefore, this is allowed.

Therefore, my previous calculation of -9 is valid. Therefore, that is a possible value.

But then, can we get even lower? Let me think. For example, if we can have more subtractions. Let me think.

Suppose we have 1 + (2 + 3 - 4 - 5 - 6). Let me check if that's allowed. The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we group (2 + 3 - 4 - 5 - 6), that would be equivalent to 2 + 3 - 4 - 5 - 6. Which is changing the +5 and +6 into -5 and -6. But again, is that allowed? Wait, no. Because the original expression has +5 and +6. If we group them as -5 and -6, that would require changing the operations. But in reality, by adding parentheses, we are changing the order of operations. Let me think again.

Original expression: 1 + 2 + 3 - 4 + 5 + 6. If we group (2 + 3 - 4 - 5 - 6), that would be equivalent to 2 + 3 - 4 - 5 - 6. But how does that relate to the original expression? The original expression is 1 + 2 + 3 - 4 + 5 + 6. So, the operations are +, +, -, +, +. If we group (2 + 3 - 4 - 5 - 6), that would require that the operations between 2 and 3 is +, between 3 and 4 is -, between 4 and 5 is -, and between 5 and 6 is -. But in the original expression, between 4 and 5 is +, and between 5 and 6 is +. Therefore, changing those to - is not allowed. Therefore, this grouping is invalid.

Wait, so this is a critical point. If we want to group operations that are originally additions into subtractions, that would require changing the operations, which is not allowed. Therefore, the only valid way to change the operations is by grouping existing operations. For example, if we have a subtraction followed by an addition, we can group them to make a larger subtraction. But we cannot change an addition into a subtraction.

Therefore, the expression 1 + 2 + 3 - (4 + 5 + 6) is valid because we are not changing the operations; we are just changing the order. The original operations are +, +, -, +, +. When we group 4 + 5 + 6, we are not changing the operations; we are just applying the subtraction to the entire group. Therefore, the operations remain the same. The subtraction is still only on the 4, but by grouping, we are effectively subtracting the entire group. Wait, but how?

Actually, no. The original expression is 1 + 2 + 3 - 4 + 5 + 6. The subtraction is only on the 4. When we add parentheses around 4 + 5 + 6, we are changing the order of operations so that the subtraction is applied to the entire group. However, the operations themselves are still subtraction and addition. The original expression has a subtraction followed by additions. By grouping, we are changing the order so that the subtraction is applied to the group, which includes the subsequent additions. Therefore, this is allowed because we are not changing the operations, just the order.

Therefore, the expression 1 + 2 + 3 - (4 + 5 + 6) is valid and results in -9. Similarly, if we can find another grouping that results in a lower value.

Let me think. What if we can group more terms to be subtracted? For example, if we can have something like 1 + (2 + 3 - 4) - (5 + 6). Which is 1 + (1) - 11 = -9. Same as before.

Alternatively, if we can have 1 + 2 + (3 - 4) - (5 + 6). Which is 1 + 2 + (-1) - 11 = 3 - 11 = -8. Which is higher than -9.

Alternatively, 1 + (2 + 3 - 4 - 5) + 6. Let me compute that:

Inside the parentheses: 2 + 3 - 4 - 5 = 5 - 4 - 5 = 1 - 5 = -4. Then, 1 + (-4) + 6 = 3. Which is higher.

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

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

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

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

So, all these ways give -9 or higher. Is there a way to get lower?

Wait, what if we can have more subtractions by grouping in a different way? For example, if we can have something like (1 + 2 + 3 - 4) - (5 + 6). Which is (6 - 4) - 11 = 2 - 11 = -9. Same result.

Alternatively, (1 + 2) - (3 + 4 + 5 + 6). Which would be 3 - 18 = -15. But is this allowed?

Original expression is 1 + 2 + 3 - 4 + 5 + 6. To get (1 + 2) - (3 + 4 + 5 + 6), we need to have the subtraction apply to the rest of the terms. But in the original expression, the operations are +, +, -, +, +. So, to have (1 + 2) - (3 + 4 + 5 + 6), we need to have the subtraction between 1 + 2 and the rest. But in the original expression, after 1 + 2, it's +3 - 4 + 5 + 6. So, to make it - (3 + 4 + 5 + 6), we need to have the subtraction apply to the entire rest. But how?

This would require that the original expression has a subtraction between 1 + 2 and the rest. But the original expression has a + before the 3. Therefore, unless we can group the entire expression as follows: (1 + 2) - (3 + 4 + 5 + 6). But how does that relate to the original expression?

The original expression is 1 + 2 + 3 - 4 + 5 + 6. If we add parentheses around (1 + 2) and then subtract (3 + 4 + 5 + 6), that would require that the original expression has a subtraction between 1 + 2 and the rest. But the original expression has a + before the 3. Therefore, this is not allowed unless we can change the + into a -, which is not allowed. Therefore, this expression is invalid.

Therefore, the only valid way to get -9 is by grouping the 4 + 5 + 6 under the subtraction. Therefore, -9 is the minimum value.

But let me check another possibility. What if we group the first three terms and then subtract the rest? For example, (1 + 2 + 3) - 4 + 5 + 6 = 6 - 4 + 5 + 6 = 13. No. Or (1 + 2 + 3) - (4 + 5 + 6) = -9. Which is valid.

Alternatively, grouping the first two terms and subtracting the rest: (1 + 2) - 3 + 4 + 5 + 6 = 3 - 3 + 4 + 5 + 6 = 1 + 4 + 5 + 6 = 16. Higher.

Alternatively, grouping the first term and subtracting the rest: 1 - 2 + 3 - 4 + 5 + 6. Wait, but that would require changing the +2 into -2, which is not allowed. Because the original expression is 1 + 2 + 3 - 4 + 5 + 6. To have 1 - 2 + 3 - 4 + 5 + 6, we need to change the +2 into -2, which is not allowed. Therefore, invalid.

Therefore, the only valid way to get -9 is by grouping 4 + 5 + 6 under the subtraction. Therefore, -9 is the minimum value.

But let me think again. What if we can group the subtraction with more terms? For example, if we have something like 1 + 2 + 3 - (4 + 5) + 6. Which is 6 - 9 + 6 = 3. Or 1 + 2 + 3 - 4 + (5 + 6) = 6 - 4 + 11 = 13. Or 1 + 2 + (3 - 4 + 5 + 6) = 1 + 2 + 10 = 13. Or 1 + (2 + 3 - 4 + 5 + 6) = 1 + 12 = 13. So, no improvement.

Alternatively, if we can group the first subtraction with the following additions in a way that creates more subtractions. For example, 1 + 2 + (3 - 4 + 5) + 6. Which is 1 + 2 + 4 + 6 = 13. Or 1 + 2 + 3 - (4 + 5) + 6 = 3. Or 1 + 2 + 3 - 4 - (5 + 6) = -9. Which is the same as before.

Therefore, after checking all possibilities, it seems that -9 is the minimum value achievable by adding parentheses without changing the operations.

But let me check one more thing. What if we have multiple subtractions by grouping? For example, 1 + (2 + 3 - 4) - (5 + 6). Which is 1 + 1 - 11 = -9. Same as before.

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

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

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

So, all these ways give -9. Therefore, I think that -9 is indeed the minimum value.

But let me think if there's another way. For example, if we can have something like 1 + 2 + 3 - 4 - 5 - 6. Which is 6 - 4 - 5 - 6 = -9. But how is this expression achieved? By grouping the original expression as 1 + 2 + 3 - (4 + 5 + 6). Which is allowed, as discussed earlier. Therefore, this is valid.

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

But just to be thorough, let me check if there's a way to get lower than -9. For example, -10.

To get -10, we need the total value to be -10. Let's see. The original total is 13. To get -10, we need to subtract 23. But how?

The maximum subtraction we can do is 4 + 5 + 6 = 15, which gives us 6 - 15 = -9. To get a larger subtraction, we would need to subtract more than 15. But how?

If we could subtract 1 + 2 + 3 + 4 + 5 + 6 = 21, then the total would be 1 + 2 + 3 - (4 + 5 + 6) - (1 + 2 + 3)? No, that's not possible. Wait, no. The original expression is fixed. We can't subtract more than the terms present.

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

Alternatively, if we can have more subtractions by grouping. For example, if we can have 1 - (2 + 3 + 4 + 5 + 6). But again, that would require changing the +1 into -1, which is not allowed.

Therefore, I don't think it's possible to get lower than -9.

Therefore, 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** (without rearranging terms or changing operations), we must explore how parentheses can alter the **order of operations**.

---

### Step 1: Understand the Original Expression

The original expression, evaluated left-to-right, is:

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

We are not allowed to change the **order of the terms** or **change the operations** (i.e., we can't turn a `+` into a `-` or vice versa). However, we **can group terms** using parentheses to change the **order of evaluation**.

---

### Step 2: Strategy for Minimizing the Value

Since subtraction is **not associative**, grouping can significantly affect the result. The key is to **maximize the negative contributions** by grouping terms that are being **added** into a **subtraction**.

For example, consider the expression:

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

This is a valid placement of parentheses, and it evaluates as:

$$
1 + 2 + 3 - (4 + 5 + 6) = 6 - 15 = -9
$$

This is **much smaller** than the original value of 13.

---

### Step 3: Can We Get Even Lower?

Let’s explore whether we can get a value **less than -9**.

Suppose we try to subtract more terms. The only subtraction in the original expression is on the number `4`. However, by grouping, we can **extend the subtraction** to include more terms.

For instance:

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

This is the **maximum possible subtraction** we can achieve using the given terms and operations. Any attempt to subtract more than 15 (i.e., the sum of 4, 5, and 6) would require **changing the operations**, which is not allowed.

---

### Step 4: Confirming the Minimum

We can try other groupings, such as:

- $1 + (2 + 3 - 4 - 5 - 6) = 1 + (-12) = -11$ → **Invalid**, because this would require changing the `+5` and `+6` into `-5` and `-6`, which is not allowed.
- $1 + 2 + (3 - 4 - 5 - 6) = 1 + 2 + (-12) = -9$ → **Valid**, but still gives the same result.
- $1 + 2 + 3 - (4 + 5 + 6) = -9$ → **Valid and minimal**.

All other groupings either result in the same value or a higher one.

---

### ✅ 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 $1+2+3-15 = 6-15=\boxed{-9}$.