Write out the positive integers in a row and underline every fifth number. Now ignore the underlined numbers and record the partial sums of the other numbers in a second row, placing each sum directly beneath the last entry that it contains.

Now, in this second row, underline and ignore every fourth number, and record the partial sums in a third row. Keep this up and the entries in the fifth row will turn out to be the perfect fifth powers 1^{5}, 2^{5}, 3^{5}, 4^{5}, 5^{5} …

If we’d started by ignoring every fourth number in the original row, we’d have ended up with perfect fourth powers. In fact,

For every positive integer

k> 1, if everykth number is ignored in row 1, every (k– 1)th number in row 2, and, in general, every (k+ 1 –i)th number in rowi, then thekth row of partial sums will turn out to be just the perfectkth powers 1^{k}, 2^{k}, 3^{k}…

This was discovered in 1951 by Alfred Moessner, a giant of recreational mathematics who published many such curiosa in *Scripta Mathematica* between 1932 and 1957.

(Ross Honsberger, *More Mathematical Morsels*, 1991.)