Write down the positive integers in row one, cross out every 
th
number, and write the partial sums of the remaining numbers in the row below. Now
cross off every 
th
number and write the partial sums of the remaining numbers in the row below. Continue.
For every positive integer 
, if every 
th number is ignored in row 1, every 
th number in row 2, and every 
th number in row 
, then the 
th row of partial sums will be the 
th powers 
, 
, 
, ....
Moessner's Theorem
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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.Referenced on Wolfram|Alpha
Moessner's TheoremCite this as:
Weisstein, Eric W. "Moessner's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MoessnersTheorem.html