A kid is adding consecutive integers on a calculator, one at a time, starting with 1 + 2 + 3 + … and so on. At one point you notice the sum is now 100, but that shouldn’t be possible if the kid was adding this way. The kid tells you that he made an error and *subtracted *exactly one of the numbers he was supposed to add.

What is the number he subtracted?

#### Solution

10

If the kid subtracted one of the numbers he should have added, the correct sum would have been greater than 100. So we take a look at the sums that are greater than 100:

- Adding 1 through 14 would have given him 105. Subtracting a number instead of adding it decreases the sum by twice that number. He would have had to subtract 2.5 to get to 100, and that’s not an integer, so this is not a solution.
- Adding 1 through 15 would have given him 120. Subtracting 10 instead of adding 10 would have gotten him to 100, so this is a correct solution.
- We still need to rule out other possible solutions. Adding 1 through 16 would have given him 136. He would have needed to subtract 18 to get to 100, but the last number added was only 16, so this is not a solution.
- This continues to be true for any sum for numbers greater than 16, since the sums increase more than twice as fast as the numbers, so no other solution is possible.

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