MacMahon's Coloured Cubes
Contents of this Page
What are MacMahon's Coloured Cubes? 
There are 30 MacMahon Cubes
Making of the Cubes
Figures with Cubes 
MacMahon's Problem
The Mayblox Problem
Kowalewski's Problem
Instant Insanity
Coloured Cubes on the Internet
To the Main Page    "Mathematische Basteleien"

What are MacMahon's Coloured Cubes?
MacMahon's cubes are the cubes which develop if you give the six sides six different colours in all possible combinations.
...... The colours are not determined. You can choose any colour. I chose red(1), light blue(2), dark blue(3), dark green(4), light green(5), and  pink(6). 
A cube is drawn on the left as a net and in perspective with three turned square sides.
P.A.MacMahon was an English mathematician and major. He lived from 1854 to 1929. 

There are 30 MacMahon Cubes     top
...... If you give the squares of a cube the numbers 1, 2, 3, 4, 5, 6 and form all permutations of the six numbers, you get 1*2*3*4*5*6=6!=720 cubes. 
Many among the cubes are the same. They can be transferred by turns around one of the 13 axles into one another. There are 24 turns and thus only 720:24 = 30 different cubes.
The following picture illustrates these facts. The numbers below the cubes mean the number of the turns.

Making of the Cubes top
You also get the 30 cubes by systematic colouring.
> All cubes get the colour pink on the reverse side.

> The front sides get one of the six colours in each line.

> The sides underneath get the third suitable colour in each line . 

> The remaining three sides get all permutations of the remaining three colours in a line. 

The names like Ba or Fa come from  J.H.Conway (see below).

If you want to play with the coloured cubes, you must build them yourselves.
...... You can write the numbers 1 to 30 (in place of the colours) on round self-adhesive labels and stick them on the cubes. Each cube should get a name like Ab in order to keep track of things.

It is more beautiful, of course, if you use larger cubes and give them the six colours.

Figures with Cubes  top
Playing with the cubes is looking for right rectangular solids or big cubes with one-coloured sides. 
The following four figures are relatively easy to find. 

The cube on the left is extended twice by a 1x2x2-slice to the left. 3x2x2 and 4x2x2-solids develop. 
The small cube on the left is used for forming the corners of the large cube on the right. You can easily find the 19 remaining cubes.
...... The L-shaped figure on the left is built by symmetrically coloured cubes (see below).

The speciality is that inside the same colours touch as additional condition. This is the so-called domino condition (Gardner).

MacMahon's Problem top
The main problem is MacMahon's problem.
You select a cube from the 30 cubes, e.g. the cube Ab.
... 29 cubes remain. You select eight among them to build a 2x2x2 cube with the same colours as the small cube. The same colours must touch inside, too, to make the puzzle more difficult. 
A description of the way to the solution for the cube Ab follows.

You can't find the solution by accident. You must proceed systematically. 

First you look for the four cubes in the lower layer. Lay all cubes with dark blue underneath.

The cubes Bc, Ca, Df, Ed, and Fe are possible for the cube at the bottom, on the left, in front. Take out Df, Ed, and Fe, because inside and outside would be the same colour. Bc and Ca are left. (Lay Bc aside. This would lead to a second solution.) 

The cubes Bd, Cf, Da, Ec, and Fe are possible for the cube at the bottom, on the right, in front. Bd is left.

The cubes Bd, Cf, Da, Ec, and Fe are possible for the cube at the bottom, on the left, in the back. Bf is left.

The cube Ea is only possible for the cube at the bottom, on the right, in the back

Turn the remaining cubes so that dark green is at the top.

The cubes Be and Cd are possible for the cube at the top, on the left, in the front. Be is left.

The cube Fa is only possible for the cube at the top, on the right, in the front

The cube Da is only possible for the cube at the top, on the left, in the back

The cube Bc is only possible for the cube at the top, on the right, in the back.

Hence the solution is:

There is the second solution on the far right, which I mentioned while looking for the solution. You use the same cubes. They lay symmetrically to the centre and are turned.

J.H.Conway takes  the credit for a complete solution of the problem.
......... He arranged the 30 cubes in a 6x6-field, whereby he kept  the main diagonal free.

The columns are called a, b, c, d, e, f, 
the lines A, B, C, D, E, F.

Thus each cube gets a pair of a large and a small letter as a name depending on its position.

The speciality of the table is that it contains all solutions of MacMahon's problem for all cubes.
If you want to build i.e. the cube Ab as a 2x2x2-cube, you can easily find the eight cubes:
You go from Ab to the mirror cube Ba and choose the remaining eight cubes in the line and the column with cube Ba.
...... Cube and mirror cube have mirror images, for instance Ab and Ba.
Still another feature:
....... If you choose the five cubes of a line or of a column and build a 1x1x5-bar and make sure that any colour lies down, the remaining five colours lie above.

The Mayblox Problem top
Eight cubes are given. Form a 2x2x2 cube with six colours at every side and the same colours touching inside. You have no small cube as a model. This is an additional difficulty.

Kowalewski's Problem top
The German mathematician G. Kowalewski varied the 8-cube-problem.
...... It required the same colours on the right and on the left and in front and on the reverse side. A third and a fourth colour are at the top and underneath. Inside equal colours should meet. 

This problem has only two solutions. One solution is shown. You need eight more cubes for the second solution. 

Instant Insanity  top
"Parker Brothers" introduced the "color matching box" with the name "Instant Insanity" in the year 1967. The puzzle was called  "Vier verrückt" or "Katzenjammer-Puzzle" in Germany. Twelve millions were sold world wide (!?).
...... This puzzle has four coloured cubes. In contrast to Mac Mahon's cubes only four different colours are used.

The problem is to arrange the cubes to a 1x1x4-bar, so that four different colours appear on all four sides.

Here is one solution.

...... It is also possible to build a bar from Mac Mahon's cubes, so that six different colours appear on all four sides. In addition the same colours touch inside. Even the ends of the bars have the same colour. 

There is one solution on the left. The colours lie diagonally. 

...... You can divide the 30 cubes into five groups with six cubes each, so that you can form five bars just described  [ Zoltan Perjés, book 6].
You can assemble them in the Conway scheme and name them with Roman letters. The drawn bar has the number I.
The speciality is, that the cubes lay together and that you can partly assemble them not only side by side, but also before one another or underneath the other without changing their properties of the different coulors in one row and the domino condition (noticed by Torsten Sillke).

Coloured Cubes on the Internet   top


Percy Alexander MacMahon


Jaap Scherpius  (Jaap's Puzzle Page)
Octacube, Instant Insanity

John J. O'Connor & Edmund F. Robertson, University of St Andrews
Percy Alexander MacMahon

Kadon Enterprises Inc.
Frank Armbruster invented Instant Insanity

Torsten Sillke
MacMahon's cubes (Multiple Cubes, Probabilities, References)

Percy MacMahon, Instant Insanity, MacMahon Squares


Charles-É. Jean   (Dictionnaire de mathématiques récréatives)
Cube coloré

References   top
(1) Gerhard Kowalewski: Alte und neue mathematische Spiele, Leipzig 1930 (Reprint bei Teubner, Stuttgart 1978)
(2) Bruno Kerst: Mathematische Spiele, Berlin 1933 (Nachdruck: Martin Sändig, Wiesbaden 1968)
(3) Gardner, Martin: Mathematische Knobeleien, Braunschweig 1980 (Vieweg)
(4) Rüdiger Thiele: Das große Spielvergnügen, Leipzig 1984 (Urania-Verlag)
(5) Gardner, Martin: Mathematische Hexereien, Berlin 1988 (Ullstein)
(6) Gardner, Martin:  Fractal Music, Hypercards and More Math. Recreations from SA Magazin,   New York 1991 (Freeman)
(7) Rüdiger Thiele, Konrad Haase: Der verzauberte Raum, Leipzig/Jena/Berlin 1991 (Urania-Verlag)
(8) Rüdiger Thiele, Konrad Haase: Teufelsspiele, Leipzig/Jena/Berlin 1991 (Urania-Verlag)

Thanks to Sabine Sprankel for the idea of this topic and Torsten Sillke for supporting me.
Thanks to Gail from Oregon for supporting me in my translation.

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©  2002 Jürgen Köller