A normal P-multimagic square is a normal magic square that is also magic when its numbers are replaced by their kth power for 1 ≤ k ≤ P:
Notes on squares presented here:
. The number range is 1 to n^{2}; some originals had range
0 to n^{2}-1.
. The aspect is Frénicle standard form; this may differ from
the original aspect.
. The ½ and the minus sign are dropped from the
bones
numbers.
The first bimagic square is this order 8 made by G.Pfeffermann in 1890:
Here is an order 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, and 25.
Orders 8 to 24 are from Christian Boyer's tremendous site www.multimagie.com where history, details, and full credits are given.
Orders 8 and 9 are by G.Pfeffermann, 1890: Bimagic.htm, and 1891: Bimagic9.htm. The order 9 is associative.
Orders 10 and 11 are by Fredrik Jansson, 2004: Bimagic10&11.htm.
The first order 12 was made by Walter Trump in 2002. It is also
trimagic!
The order 12 here is by Pan Fengchu, 2007. It is self-complement, bent diagonal 2-way,
U zigzag_{2,3} 2-way.
Order 13 is by Chen Qinwu and Chen Mutian, 2006.
Order 14 is by Chen Qinwu and Jacques Guéron, 2006.
Order 15 is by Chen Qinwu, 2006. It is associative.
Order 16 is by Gaston Tarry, 1903. See
Bitrimagic12_16.htm. It is
V zigzag_{2,3}, U zigzag, U zigzag_{4} 2-way.
Orders 17, 18, 19, 21, 22, and 23 are by Jacques Guéron, 2006: Bitrimagic17_64.htm#Gueron.
Order 20 is by Su Maoting, 2006: Bitrimagic17_64.htm#Maoting. It is associative.
Order 24 is by Chen Qinwu, 2005: Bitrimagic17_64.htm#Qinwu. It is self-complement, bent diagonal 2-way, U zigzag_{2,3,4,6} 2-way.
The first order 25 was published by Édouard Barbette in 1912.
The order 25 here is by Harm Derksen, Christian Eggermont,
and Arno van den Essen, 2005.
It is ultramagic.
Boyer points out on his
Highly multimagic squares page
that this square can be made by the Tarry-Cazalas method.
In 2006, Su Maoting made the first normal bimagic square with all the broken diagonals also bimagic: Panbimagic.htm! It is V zigzag_{2,3,5}, V zigzag_{9} 2-way, U zigzag_{2,4}, U zigzag_{8} 2-way. It is an order 32.
In 2014, Walter Trump and Francis Gaspalou completed calculation of all order 8 bimagic and bisemimagic squares. Here is the bimagic type report from program GetType:
29376 squares are pandiagonal, complete, and compact_{4}, 96 are associative and V zigzag, 96 are pandiagonal and bentdiagonal, and 64 are pandiagonal and adjacent corner paired.
The first trimagic square was discovered by Gaston Tarry in 1905: Trimagic128.htm. It is an order 128 normal trimagic square. It is V zigzag_{2,3}, U zigzag.
The first order 64 normal trimagic square was found in 1933 by Eutrope Cazalas: Trimagic64.htm. It is associative, V zigzag_{2,3,5}, U zigzag_{2,4}, U zigzag_{8} 2-way.
The first known order 32 normal trimagic square was made by William H. Benson in 1949, (published in 1976): Trimagic32.htm. It is pandiagonal, compact_{16}, V zigzag, V zigzag_{3,5} 2-way, U zigzag_{2,4} 2-way.
In 2002, Walter Trump set a record with the first order 12 normal trimagic square: Trimagic12.htm. It is U zigzag_{2,3} 2-way. It is also self-complement, (top-to-bottom symmetric) and, so is bent diagonal 2-way, (left, right):
See Records.htm.
The first tetramagic square is an order 256 made by Charles Devimeux, announced in 1983: Tetramagic256.htm.
The first pentaamagic square is an order 1024 made by Christian Boyer and André Viricel in 2001: Tetra-penta.htm.
The first hexamagic square is an order 4096 made by Pan Fengchu in 2003: Hexamagic4096.htm.
In their 2005 paper Multimagic Squares, Harm Derksen, Christian Eggermont, and Arno van den Essen prove that it is always possible to construct a n-multimagic square for any n.
The n-multimagic sums of rows, columns, and main diagonals are preserved by Transform 1 and Transform 2. So, from each multimagic square, the total number of resulting squares is:
Order Transforms1_2All -------- ------------------ 8, 9 192 10, 11 1,920 12, 13 23,040 14, 15 322,560 16, 17 5,160,960 24, 25 9.8 x 10^{11} 32, 33 6.9 x 10^{17} 64, 65 5.7 x 10^{44}
Boyer, Christian "multimagic squares site" www.multimagie.com
Derksen, Harm, Christian Eggermont, and Arno van den Essen
"Multimagic Squares"
https://arxiv.org/abs/math/0504083
de Winkel, Aale "The Magic Encyclopedia" magichypercubes.com/Encyclopedia/
Gaspalou, Francis "GaspalouTarry" GaspalouTarry.pdf
Heinz, Harvey "Multimagic Squares" www.magic-squares.net/multimagic.htm
Trump, Walter "Bimagic Squares of Order 8" www.trump.de/magic-squares/bimagic-8/
Trump, Walter "Estimates of the number of magic squares, ..."
www.trump.de/magic-squares/estimates/multi.htm
Weisstein, Eric W. "Multimagic Square."
From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/MultimagicSquare.html