Variations On Magic Squares

This page contains:

Reversible Magic Square

Reversible Magic Square Picture Reversible Magic Square made in solid silver by Rex Cooper on behalf of the members of the NMC for services to the Club The first example of something unusual is this Reversible Magic Square. Its "Magic Total" is 264, but what makes this Magic Square different is that, if you turn it upside down, the "Magic Total" is still 264!

Further details of this Magic Square may be found in "Self-Working Number Magic" by Karl Fulves, and in "Greater Magic" by John Northern Hilliard.

The two photographs to the right are of a solid silver version of this Reversible Magic Square. It was commissioned by Richard Stupple and designed and crafted by Rex Cooper in recognition of my services to the Northamptonshire Magicians' Club.

Anti-Magic Square

Anti-Magic Square Picture This Magic Square is called an Anti-Magic Square because it has been constructed to add up to as many different totals as possible. In this case, there are eight different totals - 6, 12, 15, 16, 17, 18, 19 and 21.

Another example may be found in my book on Magic Squares, and further details may be found in "The Magic Numbers Of Dr. Matrix" by Martin Gardner.

Magic Cube

Sorry, your browser doesn't support Java. If it did, you would be able to see a rotating 3D Magic Cube which first appeared in Ripley's "Believe It Or Not" in "The New York American" in October 1932, where each face of the Magic Cube adds up to 194.

Domino Magic Square

Domino 6 x 6 Magic Square Picture This 6 x 6 Magic Square is constructed using dominoes.

If each half of each domino is treated as a single digit in its own right, then the "Magic Total" is 13.

Further examples may be found in my book on Magic Squares, and further details may be found in "Mathematical Recreations" by Maurice Kraitchik and in "Solo Games" by Gyles Brandreth.

Knight's Tour Magic Square

The Knight’s Tour is one of the oldest of knight puzzles on a chess board, and involves moving a knight, using standard rules of chess, so that it occupies each and every square on the board in turn, without moving to any square more than once.

The tour is deemed to be "closed" if the knight returns to its starting square, or "open" if it ends up on a different square to the one on which it started.

Although the Knight’s Tour has no apparent link to magic squares, work was started on "magic" Knight’s Tours by Euler in the eighteenth century, and a "semi-magic" square was published by William Beverley, in "The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science", for August 1848.

If each move is numbered, starting at one and ending at 64, then the pattern formed after an "open" tour creates a "semi-magic" square that adds up to a "magic total" of 260:

1 30 47 52 5 28 43 54
48 51 2 29 44 53 6 27
31 46 49 4 25 8 55 42
50 3 32 45 56 41 26 7
33 62 15 20 9 24 39 58
16 19 34 61 40 57 10 23
63 14 17 36 21 12 59 38
18 35 64 13 60 37 22 11

Note that the two corner diagonals do not add up to 260. It is this fact that prevents it being a full magic square.

Full details may be found in the " Mathematical Magic Show" by Martin Gardner.

Also worth a look is Mario Velucchi's Ultimate Knight's Tour Page Of Links and Dan Thomasson's Knight's Tour pages.

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This page is Copyright © 1998, Mark S. Farrar.
Created: Sunday 15th November, 1998

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