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
||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
Czepa Knight Tour
||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
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
||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.
The diagram is the answer to problem 340 in Dudeney’
Amusements in Mathematics . 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 did not know if Vandermonde solved the
problem, but this is his solution.
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.
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. I show the first half of each tour in one color, the second half
in a contrasting color.
May 2007. A solution similar to this but with a Magic
Knight Tour on each surface is on my
 H. E. Dudeney, Amusements in
Mathematics, Dover Publ., 1970, 486-20473-1, pp103 and 229. (First published in
| 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.
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
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' the differences are a
variety of values.
NOTE: This cube is not a Group I cube (see
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 and are shown on my
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
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
33911 31393 93199 19973
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
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
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
Les Card had a great idea, and he presented
interesting results, considering that this work was done before the age of
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
 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.
||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’
I Top II
III IV- Bottom
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
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.
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
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
 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
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.
 The Sci.math newsgroup Dec. 2, 1996 and Dec. 10, 1996.
|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
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
 Royal Vale Heath, MatheMagic, Dover Publ. , 1953, 126
pages. ( page 122)
||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
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!
An order 2
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
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
For instructions on how to construct this type of a
cube go to
|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).
However, “G. L.” has directly contributed a number of items to my pages.
Thanks G. L.
Cube of Dice
|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.
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
 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
||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 
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. (immediately above) 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. 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
||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?
 Charles W. Trigg, Second Order Perimeter-magic and
Perimeter Anti-magic Cubes, Mathematics Magazine, 47(3), 1974, pp95-97.
 Charles W. Trigg, Eight Digits on a Cube’s Vertices, JRM, vol.7, no. 1,
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
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
If you look closely at the
drawing, you will recognize cubes and squares. All have numbers at the
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
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 complete this page with 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
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
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.
Danish poet and puzzle inventor Piet Hein developed this puzzle in
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
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
 Berlekamp, Conway, Guy,
Winning Ways II, Academic Press, 1982, 01-12-091102-7, pp 801-814.
 Martin Gardner, Mathematical Games, Scientific American, Sept.
1958 and 1972, July 1969.
||Miscellaneous Cube Puzzles
The three puzzles shown here are from my collection, but I cannot
recall the names of any of them.
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.