Moessner's Theorem

Write down the Positive Integers in row one, cross out every $k_1$th number, and write the partial sums of the remaining numbers in the row below. Now cross off every $k_2$th number and write the partial sums of the remaining numbers in the row below. Continue. For every Positive Integer $k>1$, if every $k$th number is ignored in row 1, every $(k-1)$th number in row 2, and every $(k+1-i)$th number in row $i$, then the $k$th row of partial sums will be the $k$th Powers $1^k$, $2^k$, $3^k$, ....

References

Conway, J. H. and Guy, R. K. ``Moessner's Magic.'' In The Book of Numbers. New York: Springer-Verlag, pp. 63-65, 1996.

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 268-277, 1991.

Long, C. T. ``On the Moessner Theorem on Integral Powers.'' Amer. Math. Monthly 73, 846-851, 1966.

Long, C. T. ``Strike it Out--Add it Up.'' Math. Mag. 66, 273-277, 1982.

Moessner, A. ``Eine Bemerkung über die Potenzen der natürlichen Zahlen.'' S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 29, 1952.

Paasche, I. ``Ein neuer Beweis des moessnerischen Satzes.'' S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952, 1-5, 1953.

Paasche, I. ``Ein zahlentheoretische-logarithmischer `Rechenstab'.'' Math. Naturwiss. Unterr. 6, 26-28, 1953-54.

Paasche, I. ``Eine Verallgemeinerung des moessnerschen Satzes.'' Compositio Math. 12, 263-270, 1956.