MacMahon's Coloured Cubes
What are Mac Mahon's Coloured Cubes?
MacMahon's cubes are the cubes which develop if you give the six sides
six different colours in all possible combinations.
P.A.MacMahon was an English mathematician and major. He lived from 1854
||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
A cube is drawn on the left as a net and in perspective with three
turned square sides.
There are 30 Mac Mahon
The following picture illustrates these facts. The numbers below the cubes
mean the number of the turns.
||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.
Making of the Cubes top
You also get the 30 cubes by systematic colouring.
If you want to play with the coloured cubes, you
must build them yourselves.
> 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).
||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
The cube on the left is extended twice by a 1x2x2-slice
to the left. 3x2x2 and 4x2x2-solids develop.
Playing with the cubes is looking for right rectangular
solids or big cubes with one-coloured sides.
The following four figures are relatively easy
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
||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).
Mac Mahon's Problem top
The main problem is Mac Mahon's problem.
||You select a cube from the 30 cubes, e.g. the
A description of the way to the solution for the cube Ab follows.
||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.
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.
The speciality of the table is that it contains all
solutions of MacMahon's problem for all cubes.
||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.
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.
Still another feature:
||Cube and mirror cube have mirror images, for
instance Ab and Ba.
||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
||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
There is one solution on the left. The colours
||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
VIA-Spiele Verlag Elfriede Pauli
Ivars Peterson's MathTrek
Jaap Scherpius (Jaap's Puzzle Page)
John J. O'Connor & Edmund F. Robertson, University
of St Andrews
Kadon Enterprises Inc.
Armbruster invented Instant Insanity
Major Percy Alexander
cubes (Multiple Cubes, Probabilities,
Charles-É. Jean (Dictionnaire
de mathématiques récréatives)
(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
(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
(8) Rüdiger Thiele, Konrad Haase: Teufelsspiele, Leipzig/Jena/Berlin
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.
Feedback: Email address on my main page
page is also available in German.
2002 Jürgen Köller