Unusual Magic Cubes

This page is devoted to cubes that are out of the ordinary. While each is magic in it's special way, only one is magic in the sense that rows, columns and triagonals sum correctly.
Because this series of pages is devoted to magic cubes, I thought I would throw this page in for a change of pace, so to speak.

Czepa cube

This cube is not magic in the usual sense. However, the steps between cells with consecutive numbers are equivalent to the moves of a chess knight. That is, 2 moves in 1 direction, then 1 move at right angles. Of course, unlike a chessboard, the moves are in three possible directions instead of two as on a chess board. Because it is also a knight move from cell 64 back to cell 1, this is called a re-entrant knight tour. This cube was published in 1918 in Germany. [1] [1] A. Czepa, Mathematische Spielereien (Mathematical Games), Union Deutsche, 1918,  140 pages, (page 77) (Old German script). There are many magic objects in this small format book but just two magic cubes.

Knight Tour

A re-entrant knight tour of the 6 faces of an order 8 cube.

This illustration is of an order 8 cube that has been flattened out to show all 6 faces. The purple line shows a knight tour that visits all 384 cells of the faces in turn. This solution is the answer to problem 340 in Dudeney’ Amusements in Mathematics [1]. As an introduction, Dudeney says “Some years ago I happened to read somewhere that Abnit Vandermonde, a clever mathematician, who was born in 1736 and died in1793, had devoted a good deal of study to the question of knight’s tours. … he had proposed the question of a tour of the knight over the six surfaces of a cube, each surface being a chessboard.” Dudeney was unaware as to whether Vandermonde solved the problem, but this is his solution (on page 229). It is interesting to note that all 6 faces use the same tour. Two faces are exactly the same, with the others differing only by rotation or reflection. Because these pages are primarily concerned with magic squares and cubes, it would be nice if consecutively numbering the steps of this tour produced a magic square. However, although over 200 different knight tours have now been discovered for the 8 by 8 board, none that form a magic square have ever been discovered. The closest solution discovered to date is only semi-magic. The rows and columns sum correctly to 260, but one diagonal = 256 and the other 264. It is generally believed that it is impossible for an 8×8 knight's tour to be diagonally magic, but no final proof of this has yet been given. May 2007. A solution similar to this but with a Magic Knight Tour on each surface is on my Update-5.page.

[1] H. E. Dudeney, Amusements in Mathematics, Dover Publ., 1970, 486-20473-1, pp103 and 229. (First published in 1917.)

Knight Tour-complete

On October 29, 2003, Guenter Stertenbrink of Germany, sent me an order 4 cube. The 64 consecutive numbers traced out a magic knight tour. Unfortunately, the cube was not quite magic because only two of the four triagonals summed correctly. However, Guenter was not to be denied! On November 9, 2003, I received an email with the cube shown here attached. This cube is pantriagonal magic because all rows, columns pillars AND pantriagonals sum correctly. The 64 integers trace out a magic knight tour. Furthermore, it is a 'reentrant' tour because cells 1 and 64 are also only a knight move apart. For those readers that are not chess players, a knight moves 2 cells in one direction, then turns 90 degrees and moves 1 additional cell. This cube has an unusual feature. For each of the 64 pantriagonals, the difference between the sum of the first and third numbers and the sum of the second and forth numbers equals 2. Normally, if the cube is not 'complete' ( this difference is 0 in all cases), the differences are a variety of values. NOTE: This cube is not a Group I cube (see Cube_Groups. htm).

On November the 15, Guenter sent me an order 15 perfect magic cube. This is not related to knight moves, but was the first cube I have seen of order 15. And being perfect was a great bonus! I put that cube on my Large Cubes page.

Thanks Guenter for sending these cubes.

On April 28, 2007 Awani Kumar announced the first order-8 Magic Knight Tour cube, and
on June 19, 2007 he announced the first order-12 Magic Knight Tour cube.
Both cubes tours were verified by Guenter Stertenbrink.
They are both shown on my Update-5 page.

Prime-magical cube

In 1968, Dr. Lesley E. Card published a paper called Patterns in Primes. [1]

In this paper he gave examples of cubes consisting of a number in each cell such that each row, column, and pillar consisted of an m digit number that was a reversible prime. A reversible prime is a number, that when read in reverse, forms another prime.
After first discussing reversible primes in his paper, he went on to say:

By June of 1999, T.W.A. Baumann had already found a solution for the 11 x 11 matrix. This solution may be seen at the above site, or on my Prime magic squares page.

In this same paper, Card presents cubical arrays based on the same principal. He relaxed the rules by not requiring that the plane diagonals or space diagonal (triagonals) be prime numbers.
He shows a 4 x 4 x 4 array, which I present below in diagram form.
He also provides orders 5, 6 and 7 arrays.
As a point of interest, I provide listings of the order 5 and order 7 prime magical cubes.

Order-5
33911   31393   93199   19973   13933           This cube contains 14 reversible primes (total of 28 unique primes).
31393   17939   39113   93199   39397           The other 122 required primes are duplicates of these 28.
93199   39113   11779   91711   93911           There are NO composite numbers in this cube!
19973   93199   91711   79111   39119
13933   39397   93911   39119   37199

Order-7
9731317  7399391  3913717  1937933  3379391  1913939  7173193
7399391  3131137  9373393  9191311  3137179  9117137  1731971
3913717  9373993  1713319  3999313  7399391  1193771  7393117
1937933  9133171  3333973  7191931  9797371  3737177  3131173
3379391  3191719  7339771  9339791  3133397  9171913  1911733
1913939  9397117  1917731  3113917  9797993  3377119  9717397
7173193  1731971  7393117  3131173  1911733  9717397  3173371

Even a quick look at this order-7 listing, however, reveals some problems. Notice that each plane uses the same prime number for the first row and the first column. In addition, there are many other duplicate numbers used. For example, the last prime in the first plane and the first prime in the last plane!

This cube is Les Card's order 4 example. Consider the digits in each cell as part of a 4-digit number. Ideally; there should be 3m2, or 48 reversible prime numbers in this cube. As per Card's rules, we will not figure on the triagonals being correct. As each number is to be non-palindrome prime, the cube would then consist of 96 distinct four-digit primes. Unfortunately, Mr. Card fell far short of this lofty goal. An inspection reveals that there are only 13 distinct primes. 5 numbers are reversible primes, 3 numbers are prime only in one direction and composite when reversed (5 x 2 + 3 = 13). There are a total of 7 distinct composite numbers. The other 76 numbers are duplicates.

Les Card had a great idea, and he presented interesting results, considering that this work was done before the age of personal computers!

My Challenge… who will be the first to produce a cube that consists of 3m² distinct reversible non-palindromic prime numbers of length m?
Addendum: August, 2005 Anurag Sahay has improved greatly on Card's results! See Update-4

[1] Leslie E. Card,  Patterns in Primes, JRM 1:2, 1968, pp 93-99.

de Winkel's Cube

I was looking for an example of a special (not magic) cube that had all correct pandiagonals and pantriagonals but No correct orthogonal lines. This one from Aale de Winkel’s Encyclopedia comes fairly close!

Rows, columns, and pandiagonals of each horizontal plane sum to a value that is different for each plane. The pillars sum correctly to 130 (too bad), but rows and diagonals of these 8 planes are incorrect.
Rows and columns of the oblique square arrays are incorrect (except for columns of two squares) but pandiagonals of these six squares are correct. (these are the pantriagonals of the cube). All pantriagonals are correct.
 

Summary: Only 8 of 24 orthogonal lines are correct. No diagonals of planes parallel to the cube faces are correct. All 64 pantriagonals are correct. Corners of all 2x2 squares parallel to a cube face , sum to one of eight values (if the value was 130, this would be ‘compact’). There are two complement pairs spaced m/2 apart, on each pantriagonal (the ‘complete’ feature). I Top                        II                             III                            IV- Bottom   1   27   10   20        32     6   23   13       37   63   46   56       60   34   51   41 26    4    17   11          7   29   16   22       62   40   53   47       35   57   44   50 19    9    28     2        14   24     5   31       55   45   64   38       42   52   33   59 12   18     3   25        21   15   30     8       48   54   39   61       49   43   58   36 Is a cube with no correct orthogonal lines and all correct pantriagonals lines possible? And also all correct planar pandiagonals? Yes! See my Update-1 page.

The Impossible Cube

Dr. Clifford Pickover, in his excellent book on magic squares [1], illustrates two rather whimsical cubes that are unconventionally magic. These illustrations are from the Dover Pictorial Archive and in each case, Arlin Anderson, Alabama, U. S. A. has managed to assign numbers to the small cubes such that the figure is magic.

The impossible cube uses the numbers 1 to 43 in 8 lines of 5, 2 lines of 7 and 1 line of 3. The sum is 108. Note that one cube (number 34) is hidden. I had fun drawing this figure. My visual perception of it would suddenly change from one orientation to the other, alternately looking up at it, and then changing to a downward view. Once the numbers were put in, this optical illusion disappeared.
The cryonic cube uses consecutive numbers from 1 to 27, but in this case the numbers are assigned to the faces of the cubes. All six lines of 6 numbers sum to 84.

[1] C. A. Pickover, The Zen of Magic Squares, Circles and Stars, Princeton Univ. Pr., 2002, 0-691-07041-5, 404 pages (pp 356 and 359).

The projection cube

This order 4 cube [1] consists of zeros or ones in each of the 64 cells. The 4-digit binary number that each line represents, ‘projects’ a decimal integer from 0 to 15 on the appropriate cube face, depending on which of the two directions the line is being read. For example: The second from back row in the top plane, binary 0100,  'projects' a decimal 4 on the left side and a decimal 2 on the right side. Another example; front right corner pillar contains binary 1110. It 'projects' a decimal 14 on the top surface and a decimal 7 on the bottom surface! The original idea was proposed by K. S. Brown and answered by Dan Cass [2]. [1] H.D. Heinz and J.R. Hendricks, Magic Square Lexicon: Illustrated, Self-published, 2000, 0-9687985-0-0,  page 25. [2] The Sci.math newsgroup Dec. 2, 1996 and Dec. 10, 1996.

With an order-4 cube, each of the 16 four digit binary numbers appear once in each direction of each orientation, of the 4 planes.
Put another way, each decimal number from 0 to 15 is 'projected' onto each of the six faces of the cube.

The principal may be extended to higher orders. However, the quantity of binary numbers of the order length is greater then the number of spaces available on the cube face. The best we can hope for is to use all the available binary numbers between the two directions of each orientation of the planes.
Order-5 requires 25 different numbers, but there are 32 5-digit binary numbers.
Order-6 requires 36 different numbers, but there are 64 6-digit binary numbers.

Addendum:  Feb. 22, 2004
On Feb. 10, 2004, Peter  Manyakhin (who was obviously unaware of this page) proposed an order 6 cube of this type. A spirited discussion subsequently took place between him, Walter Trump, Guenter Stertenbrink and especially Aale de Winkel. However, mostly concerned with the order 4 cube and how many basic cubes of that order there are. See both order 4 and order 6 cubes of this type  in the de Winkel encyclopedia.
May 6, 2004. I show an order 6 projection cube, and more details on my Update-2 page.

The Believe-it-or-not cube

In Oct.1932, an order 4 cube was printed in Ripley’s Believe It Or Not column in the New York American. It was designed and submitted by Royal V. Heath [1]. Each of the six faces of this cube consists of an order 4 pandiagonal magic square, all with the magic sum of 194. Because each square is magic , all 4 rows, 4 columns and 2 main diagonals sum to the constant. Because it is pandiagonal magic, there are many more combinations of 4 cells that sum correctly. All broken diagonal pairs, a feature of all pandiagonal magic squares. All 2x2 square arrays, including wrap-around (order 4 only). Corners of all 3x3 and 4x4 square arrays, including wrap-around (order 4 only). Total combinations for each square is 10 + 6 + 16 + 16. Royal Heath proposed the following parlor trick (assuming this is made into an actual cube. Before your guests arrive, pick up any book that is handy. Turn to the first line of the ninth page, write down the fourth word on a piece of paper, which you then put in your pocket. To do the trick, have a guest sum the four numbers in any row, column or diagonal (or other combination mentioned above). Then present him with the book and ask him to tell you the word found by using the 3 digits of the number (as you originally did). When he tells you the word, show him the word you pull out of your pocket!

[1] Royal Vale Heath, MatheMagic, Dover Publ. , 1953, 126 pages. ( page 122)

An order 2 Magic Cube

In 1995 Mirko Dobnik of Slovenia designed an order 2 and an order 4 cube that had unusual magic features.

The six faces of this 2x2x2 cube are numbered so that each face of four numbers sums to 50. In addition, each of the six lines of 8 numbers as you go around the cube, sums to 100. Examples: 8 + 17 + 19 + 6  = 50,    1 + 22 + 8 + 17 + 3 + 24 + 20 + 20 + 5 = 100. 11 + 16 + 8 + 19 + 9 + 14 + 18 + 5 = 100 The third pair of lines is not too easy to visualize. This is one of them. 1 + 4 + 14 + 15 + 21 + 24 + 10 + 11. David Singmaster pointed out that there are at least another 12 zigzag lines that also sum to 100. For example: 1 + 23 + 8 + 6 + 3 + 21 + 20 + 18 = 100. The order 4 cube above has the same feature. Because each face is a magic square, any continuous line on four faces sums to 776. Mirko Dobnik drew this to my attention (and provided this cube).

The following pictures show another kind of magic cube. The cube shown is the above Dobnik cube with each number on an individual cubelet. These 8 cubelets are hinged so that the cube can be unfolded to show the 3 different arrangements of the blank (inside) faces of the cubelets.
On these faces, I have written 3 sets of 8 numbers that sum to 100 (see above examples).

For instructions on how to construct this type of a cube go to http://www.mathematische-basteleien.de/magiccube.htm

The Magic Dice

In 1999, G. L. Honaker invented an original puzzle for his high school students. It consists of numbering the pips (dots) on a regular die so as to obtain the same minimum sum on each of the six faces. Dr. Clifford Pickover [1] provided a simple proof that the sum in the solution shown here is the smallest possible. He also proposed distributing distinct numbers over two (or more) dice so as to obtain the smallest possible different sum for each of them. [1] I obtained the above from C. A. Pickover, The Zen of Magic Squares, Circles and Stars, Princeton Univ. Pr., 2002, 0-691-07041-5, 404 pages. (pp.289-292). However, “G. L.” has directly contributed a number of items to my pages. Thanks G. L.

The Farrell Cube of Dice

Imagine an order 3 cube made up of 27 ordinary dice. Jeramiah Farrell, Indiana, U. S. A. [1] designed such a cube in 1999 in which the six faces have magical powers.
To start, all the pips in any of the three rows and 3 columns of a face add to the same value. But there is more...

Place the cube in any orientation, but disregard the top and bottom faces for the following.  Pick any row, column or main diagonal and sum the pips in it . Then add the corresponding pips in each of the other three lateral faces. The sum will always be 42! Some examples with face 1, 6, 2; 6, 2, 1; 2, 1, 6 up and 6, 4, 3; 3, 6, 4; 4, 3, 6 facing you. Start with the top row on this front face and add the four top rows as you turn the cube. Then 6, 4, 3; 2, 5, 3; 1, 4, 3; and 5, 2 4 sum to 42. Another example, but a little harder to visualize from the diagram (easy with a made up cube). Start from the same face as above but this time use the pips in the main diagonal. 6, 6, 6; 2, 3, 5; 1, 1, 1; 5, 4, 2 sum to 42.

[1] The Mathemagician and Pied Puzzler: a collection in tribute to Martin Gardner, edited by E. Berlekamp and T. Rodgers, A. K. Peters Ltd, 1999, 1-56881-075-X, 266 pages, (pp.148-149).

Perimeter-magic cubes

The cubes shown in this section represent another branch of magic objects. Here the objective is to number the outline (perimeter) of the object in such a way that all lines or surfaces sum to a constant.
Just as magic squares, cubes, etc are classified into orders, so are perimeter magic objects. The order is determined by how many numbers are placed on each line.

	Figure A. The numbers 1 to 12 are assigned to the edges of the cube. The four edges of each face sums to 26. B. The numbers on the cube graph mapped to a magic star graph. The four numbers on each of the six lines of the hexagon sum to 26. This cube would be classed as face-magic order 1. There are no perimeter magic order 1 cubes.
	Figure A. is wire frame of a cube with corners assigned numbers 1 to 8. The four edges of each face sums to 18 (but the individual lines do not all sum the same) so this is face-magic, a subclass of perimeter magic. B. This is figure A. represented as a 2x2x2 array of cubes Charles W. Trigg [1] refers to these cubes with numbers on the corners as v-type or second-order perimeter-magic cubes. There are only 3 fundamentally different cubes of this type.
	These two figures are true perimeter-magic (almost), so we are concerned with the total of the two numbers for each of the 12 lines that make up the cube. Figure A. It is impossible to position the numbers from 1 to 8 in such a way so as to obtain 12 identical sums of two numbers . So there are no order 2 perimeter (line) magic cubes. Are there any order 3 cubes of this type? Figure A. is one of only 3 configurations that have like sums for opposite parallel lines.
Figure B. is an almost anti-perimeter-magic cube. Again, it is impossible to form a second order cube that has 12 different sums in consecutive order. But we can come close. Charles Trigg [2] found that there were 12 different solutions that contain only one sum that is duplicated. Illustration B. is one of 8 of these that have duplicate sums of 9’s. This solution gives consecutive totals from 4 to 14 (with number 9 duplicated).
	This is an order 4 perimeter magic cube, using consecutive numbers from 1 to 32. Each line of four numbers adds up to 66. There are no order 1 or order 2 normal perimeter magic cubes. Are there any order 3 cubes like this?

[1] Charles W. Trigg, Second Order Perimeter-magic and Perimeter Anti-magic Cubes, Mathematics Magazine, 47(3), 1974, pp95-97.
[2] Charles W. Trigg, Eight Digits on a Cube’s Vertices, JRM, vol.7, no. 1, !974, pp49-55

Before 1900, Pao Chhi-shou published a perimeter-magic cube in which the two numbers in each line between the vertices sum to 41. Because the 4 corners of each face sum to 18 (i.e. the same value), the four edges of each face of this cube sums to 182.
Clifford Pickover improved on this design by rearranging the numbers between the vertices so that all four numbers in each line sum the same. Because the 4 corners of each face still sum to 18, the four edges of each face of this cube also sums to 182. Not that in the previously shown order 4 perimeter-magic cube the faces were not magic.

Each line (edge) in Pickover's cube sums to 50. This is the smallest possible value because the vertices use the 8 smallest numbers in the series. Each line in the previous example of the order 4 summed to 66. However, this is not the largest possible. What is the largest possible sum for an order 4 perimeter-magic cube? How many different sums are possible?

[1] A. Pickover, The Zen of Magic Squares, Circles and Stars, Princeton Univ. Pr., 2002, 0-691-07041-5, 404 pages. (pp.102-103).

Tesseract-Cube-Square

A tesseract is a 4 dimensional cube and if it is suitably numbered is a magic 4 dimensional cube. Here I show a tesseract with just the 16 corners numbered. I apologize for using the old fashioned method of illustrating this 4-D object. However, in this case I think it better serves the purpose of showing the cubes and squares it is composed of. If you look closely at the drawing, you will recognize cubes and squares. All have numbers at the corners. The four numbers of all the squares (or parallelograms) sum to 34, and by extension all the cubes are therefore face perimeter-magic. They are order 2, but not normal because they do not consist of consecutive numbers. This drawing of the tesseract may be used to quickly compose order 4 pandiagonal magic squares.

To compose such a square, start at any number in the tesseract. Moving in either direction around the quadrilateral, write down the four numbers to form the first row of the order 4 pandiagonal magic square. Fill in the other three rows of the square by visiting the other parallelograms of the same shape and orientation, starting at the same corner, and moving in the same direction.
Here are 3 examples:

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Volume 2, Academic Press, Inc 1982, 01-12-091102-7, p. 783.
Dominic Olivastro, Ancient Puzzles, Bantam Books, 1993,  280 pages, pp110-113.

Cube Puzzles

I’ve included pictures of some of my cube puzzles. You could say they are magic because of the large number of possible solutions. Also, the hours magically fly by while you're trying  to find solutions.

	Rubik's Cube The most famous of recent puzzles is Rubik’s cube invented by the Hungarian Ernö Rubik. Invented in 1974, patented in 1975 it was put on the market in Hungary in 1977. However it did not really begin as a craze until 1981. By 1982 10 million cubes had been sold in Hungary, more than the population of the country. It is estimated that 100 million were sold world-wide. It is really a group theory puzzle, although not many people realize this. The cube consists of 27 smaller cubes which, in the initial configuration, are colored so that the 6 faces of the large cube are colored in 6 distinct colors. The 9 cubes forming one face can be rotated through 45 degrees. There are 43,252,003,274,489,856,000 different arrangements of the small cubes, only one of these arrangements being the initial position. http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Mathematical_games.html#66
	Soma Cube Danish poet and puzzle inventor Piet Hein developed this puzzle in 1936. It consists of 6 shapes made from 4 small cubes each and 1 shape of 3 small cubes. Together they may be formed into a large 3x3x3 cube in 240 different ways [1]. The Soma puzzle was marketed by Parker Brothers, Co. around 1970 after it gained popularity among math hobbyists as a result of Martin Gardner’s Scientific American Column.[2]. It is one of the best known cube puzzles in the world. I have two copies of this puzzle, so numbered the cubes of one of them to form a magic cube on completion. There is a now a 4x4x4 version of this puzzle. It consists of 12 pieces consisting of 5 small cubes and 1 piece of 4 small cubes. It is named after the inventor, Bruce Bedlam [3] who claims there are 19,186 solutions. [1] Berlekamp, Conway, Guy, Winning Ways II, Academic Press, 1982, 01-12-091102-7, pp 801-814. [2] Martin Gardner, Mathematical Games, Scientific American, Sept. 1958 and 1972, July 1969. [3] http://www.bedlamcube.com/
	Miscellaneous Cube Puzzles The three puzzles shown here are from my collection, but I cannot recall the names of any of them. The back two puzzles are similar types. Both consist of 4 cubes. One has numbers 1 to 4 on the six faces. The other has 4 colors on the six faces. In both cases, the object is to arrange the cubes side by side so each face of the group is the same number (or color). I believe the color version was called Instant Insanity. The puzzle in the front has three cubes that are joined together, but free to rotate independently. Also, each of the four exposed faces is a sliding panel. Because one of the cube faces has no panel, it is possible to move a panel to this adjacent cube. The object is to manipulate the puzzle so that all chains are the same color.

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Harvey Heinz   harveyheinz@shaw.ca
This page last updated October 15, 2009
Copyright © 2003 by Harvey D. Heinz