4. Y. David, On a sequence generated by a sieving process, Riveon Lematematika 11 (1957) 26–31. 5. P. Erd˝os and E. Jabotinsky, On sequences of integers generated by a sieving process I, II, Indag. Math. 20 (1958) 115–128. 6. V. Gautheron, La Tchouka, in Wari et Solo: le Jeu de calculs africain (Les Distracts), A. Deledicq and A. Popova, ed., CEDIC, Paris, 1977, 180–187. 7. B. C. Jones, L. Taalman, and A. Tongen, Solitaire Mancala Games and the Chinese Remainder Theorem (to appear). 8. N. J. A. Sloane, My favorite integer sequences, in Sequences and their applications (Singapore, 1998), Springer, London, 1999, 103–130.

Proof Without Words: Squares Modulo 3 Roger B. Nelsen ([email protected]), Lewis & Clark College, Portland OR 97219 ( 0 (mod 3), n ≡ 0 (mod 3) 2 2 n = 1 + 3 + 5 + · · · + (2n − 1) ⇒ n ≡ 1 (mod 3), n ≡ ±1 (mod 3)

(3k – 1)2 = 1 + 3[(2k – 1)2 – (k – 1)2]

(3k)2 = 3[(2k)2 – k2]

(3k + 1)2 = 1 + 3[(2k + 1)2 – (k + 1)2] Summary. Using the fact that the sum of the first n odd numbers is n 2 , we show visually that n 2 ≡ 0 (mod 3) when n ≡ 0 (mod 3), and n 2 ≡ 1 (mod 3) when n ≡ ±1 (mod 3). http://dx.doi.org/10.4169/college.math.j.44.4.283 MSC: 00A05, 11-01

VOL. 44, NO. 4, SEPTEMBER 2013 THE COLLEGE MATHEMATICS JOURNAL

283

([email protected]), Lewis & Clark College, Portland OR 97219 n2 ...

Y. David, On a sequence generated by a sieving process, Riveon Lematematika 11 (1957) 26–31. 5. P. Erd˝os and E. Jabotinsky, On sequences of integers ... Popova, ed., CEDIC, Paris, 1977, 180–187. 7. B. C. Jones, L. Taalman, and A. Tongen, Solitaire Mancala Games and the Chinese Remainder Theorem (to appear).

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