Can You Make It Zero?

Here's one quick way of seeing it.

Changing the sign of a given integer doesn't change if its odd or even.

So we have to add five even integers, $\pm 2, \pm 4, \pm 6, \pm 8, \pm 10$ and 5 odd integers, $1, \pm 3, \pm 5, \pm 7, \pm 9$ .

The sum of an odd amount of even numbers is even and the sum of an odd amount of odd numbers is odd.

But even number plus an odd number is odd!

So the total sum of the numbers is always an odd number and zero is not odd.

The total sum of above numbers are 55, since it's odd number, it's impossible to make the above equation true. Because we need -22.5 and +22.5

We just have integers.

Quick logic! If they all are '+' then the S = 55 (odd). Each '+' turns into '-', S will reduce 2 x X => S will always be odd. So there are no solution to make S = 0 (even)

1 + 2 +3 + .... + 10 = 55 ,so inserting either + or - in the boxes , we can only get odd numbers and 0 is an even number...

For example, 1 + 2 - 3 + 4 + 5 +...+ 10 = 55 - 2 $\times$ 3 = 49

Even + odd is odd QED

The sum of 1+2+3+...+10 = 55 that is odd number.
So, it's impossible to + , - to make sum to zero.

sum of 1 to 10 numbers is 55....if we write it as 55 ones then minimum will get a difference of 1