Moessner’s magic

Start with a list of all nonzero natural numbers. Cross out every second one—the even numbers—and take the cumulative sum of the resulting sequence. The resulting list is 1, 3 + 1 = 4, 5 + 4 = 9, 7 + 9 = 16… and these numbers should look familiar: they’re precisely the square numbers!

In 1951, Alfred Moessner discovered the following similar procedure. Start with the same nonzero natural numbers, and cross out every third one. Add the numbers as before, and now cross out every second one. Then you’re left with the third powers of the natural numbers.

Starting with every fourth number results in the fourth powers, and Moessner conjectured (well, said that his not-so-easy proof would follow later) that this holds in general: starting with crossing out every k-th number, summing, crossing out every (k–1)-th number, summing… finally gives you the k-th powers.

What happens if we start with different numbers? Funnily enough, if we start with crossing out the triangular numbers 1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10… (every time incrementing the step size with 1) then we find the factorial numbers 1! = 1, 2! = 2 × 1 = 2, 3! = 3 × 2 × 1 = 6, 4! = 4 × 3 × 2 × 1 = 24…

If we increment the increment in the step size with 1 every time, so that we cross out 1, 1 + 3 = 4, 1 + 3 + 6 = 10, 1 + 3 + 6 + 10 = 20… the resulting sequence may not look that familiar. These numbers are the superfactorials 1!! = 1, 2!! = 2! × 1! = 2, 3!! = 3! × 2! × 1! = 12, 4!! = 4! × 3! × 2! × 1! = 288…

Finally, crossing out the square numbers gives us another unfamiliar sequence. However, notice that the squares are given by 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, 1 + 2 + 3 + 4 + 3 + 2 + 1… and that the resulting numbers are precisely 1, 1 × 2 × 1, 1 × 2 × 3 × 2 × 1, 1 × 2 × 3 × 4 × 3 × 2 × 1…

Moessner’s magic

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