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.
|An order 4 cube that is unconventionally magic because the 64 cells are visited in turn by knight moves.|
|The surface cells of this order 8 cube are linked by 384 knight moves.|
|This closed knight tour visits all 64 cells of this order 4 true magic cube.|
|Each cell in this order 4 cube contains a digit that is part of a prime number.|
|Search for a special cube with NO correct orthogonal lines, and all correct pantriagonals.|
|An impossible figure (cube?) that has all lines sum to the same value.|
|The binary number on each line projects a decimal number on the surface.|
|A 1932 cube with pandiagonal magic squares on all six faces.|
|This Dobnik cube has 6 faces that sum to 50 and many magic lines.|
|Numbering the pips on a die so all faces are magic.|
|An order 3 cube with magical properties constructed from 27 dice.|
|These cubes are magic by the numbers placed on the outline.|
|Numbered corners on a tesseract form order 3 perimeter magic cubes.|
|Rubik and Soma cubes plus 3 others from my collection.|
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. 
 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.
A re-entrant knight tour of the 6 faces of an
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.
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.
 H. E. Dudeney, Amusements in Mathematics, Dover Publ., 1970, 486-20473-1, pp103 and 229. (First published in 1917.)
|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.
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
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.
In 1968, Dr. Lesley E. Card published a paper called Patterns in Primes. 
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:
|“Reversible primes fit conveniently into the pattern of
magic squares. It is possible for example, to construct a 4 x 4 square in
which each row, each column, and each main diagonal is a prime when read
in either direction.”
|This is one of the two examples he presented. It contains 20
distinct 4 digit reversible primes, for a total of 40 prime numbers.
His second example was flawed because the first column and first row contained identical 4 digit primes.
Interestingly, Carlos Rivera rediscovered this same prime magical square 30 years later, and published it as puzzle number 4 in his Prime Puzzles and Problems page in June 1998.
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.
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
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.
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
Addendum: August, 2005 Anurag Sahay has improved greatly on Card's results! See Update-4
 Leslie E. Card, Patterns in Primes, JRM 1:2, 1968, pp 93-99.
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.
Only 8 of 24 orthogonal lines are correct. No diagonals of planes parallel to the cube
faces are correct. All 64 pantriagonals are correct.
I Top II III IV-
Is a cube with no correct
orthogonal lines and all correct pantriagonals lines possible? And also all
correct planar pandiagonals?
Dr. Clifford Pickover, in his excellent book on magic squares , 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.
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.
 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).
This order 4 cube  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 .
 H.D. Heinz and J.R.
Hendricks, Magic Square Lexicon: Illustrated, Self-published, 2000,
0-9687985-0-0, page 25.
|In list form the above cube is:
1 1 1 1 0 0 0 1 1 0 0 0 1 1 0 0
0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 1
0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 0
0 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0
|The decimal digits projected onto the front and right side
08 12 11 09 03 02 04 15
01 07 02 13 13 06 05 01
15 00 03 05 11 10 09 08
10 14 06 04 00 14 07 12
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
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.
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 . 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.
proposed the following parlor trick (assuming this is made into an actual
 Royal Vale Heath, MatheMagic, Dover Publ. , 1953, 126 pages. ( page 122)
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.
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
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  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.
 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).
Imagine an order 3 cube made up of 27 ordinary dice.
Jeramiah Farrell, Indiana, U. S. A.  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.
 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).
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
B. This is figure A. represented as a 2x2x2 array of cubes
Charles W. Trigg  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
 found that there were 12 different solutions that contain only one sum
that is duplicated.
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.
 Charles W. Trigg, Second Order
Perimeter-magic and Perimeter Anti-magic Cubes, Mathematics Magazine, 47(3),
 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?
 A. Pickover, The Zen of Magic Squares, Circles and Stars, Princeton Univ. Pr., 2002, 0-691-07041-5, 404 pages. (pp.102-103).
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.
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
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.
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.
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.
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 .
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.. 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  who claims there are 19,186 solutions.
Conway, Guy, Winning Ways II, Academic Press, 1982, 01-12-091102-7,
|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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This page last updated October 15, 2009
Copyright © 2003 by Harvey D. Heinz