Prime Magic Square


A prime magic square is a magic square consisting only of prime numbers (although the number 1 is sometimes allowed in such squares). The left square is the 3×3 prime magic square (containing a 1) having the smallest possible magic constant, and was discovered by Dudeney in 1917 (Dudeney 1970; Gardner 1984, p. 86). The second square is the 3×3 magic square consisting of primes only having the smallest possible magic constant (Madachy 1979, p. 95; attributed to R. Ondrejka). The third square is the 3×3 prime magic square consisting of primes in arithmetic progression (199+210n) having the smallest possible magic constant of 3117 (Madachy 1979, p. 95; attributed to R. Ondrejka). The 4×4 prime magic square on the right was found by A. W. Johnson, Jr. (Dewdney 1988).


According to a 1913 proof of J. N. Muncey (cited in Gardner 1984, pp. 86-87), the smallest magic square composed of consecutive odd primes including the number 1 is of order 12, illustrated above.


The 3×3 square whose entries are consecutive primes illustrated above was discovered by Nelson (Guy 1994, p. 18; Rivera) in response to a challenge by Martin Gardner. Nelson collected Gardner's $100 prize, and also found 20 other such squares (Guy 1994, p. 18).


The amazing square above (Madachy 1979, pp. 93-94) is a 13×13 prime magic border square, so that the 11×11, 9×9, ..., and 3×3 subsquares are all also prime magic squares.

See also

Magic Square, Prime Array, Prime Number

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Dewdney, A. K. "Computer Recreations: How to Pan for Primes in Numerical Gravel." Sci. Amer. 259, pp. 120-123, July 1988.Dudeney, E. Problem 408 in Amusements in Mathematics. New York: Dover, 1970.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, 1984.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.Heinz, H. "Prime Magic Squares.", J. S. "Magic and Antimagic Squares." Ch. 4 in Madachy's Mathematical Recreations. New York: Dover, pp. 85-113, 1979.Nelson, H. L. "A Consecutive Prime 3×3 Magic Square." J. Recr. Math. 20, 214-216, 1988.Rivera, C. "Problems & Puzzles: Puzzle 003-Magic Squares with Consecutive Primes.", C. "Problems & Puzzles: Puzzle 004-Prime-Magical Squares.", N. J. A. Sequences A073502 and A073520 in "The On-Line Encyclopedia of Integer Sequences."

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Prime Magic Square

Cite this as:

Weisstein, Eric W. "Prime Magic Square." From MathWorld--A Wolfram Web Resource.

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