Puzzle Extension – Consecutive Integers
Given an integer N, N ≠ 0, how many ways can N be written as the sum of consecutive integers?
“How many ways can 1000 be written as the sum of consecutive integers?“ Nice puzzle posted here a few days ago. We had some nice discussion and partial and complete solutions in the comments. Nick linked back to a longer discussion on his blog, Divide by Zero.
But let’s extend and generalize this.
Assume you have a group of kids who brute forced a solution for N = 1000. Come back at them a week or two later with: Given an integer N, N ≠ 0, how many ways can N be written as the sum of consecutive integers?
What will they ask? How are you going to guide them? How are you going to keep them from quitting?
And, oh yeah, what’s the answer, and how did you get it?
I eventually took the same odd-and-even approach that pops up in the comments to the previous puzzle, and then asked myself about the strictly positive solutions. What I didn’t expect, when I actually wrote out the solutions, is that the pairs they appear in (for odd factors) are both “the same” series, one strictly and one semi-trivially adding a run of numbers summing to zero. I knew that answers would pair up in this form, but I didn’t expect the by-highest-term pairings to be the same as the by-corresponding-factor pairings – though once you notice it, it’s trivially proved.
This is the sort of thing we use at school for our competition puzzle of the month, and probably will :).