68 GEOMETRICAL RECREATIONS [CH. IV 



thirty such cubes, no two of which are coloured alike. Take 

 any one of these cubes, K , then it is desired to select eight out 

 of the remaining twenty-nine cubes, such that they can be 

 arranged in the form -of a cube (whose linear dimensions are 

 double those of any of the separate cubes) coloured like the 

 cube K, and placed so that where any two cubes touch each 

 other the faces in contact are coloured alike. 



Only one collection of eight cubes can be found to satisfy 

 these conditions. These eight cubes can be determined by the 

 following rule. Take any face of the cube K: it has four 

 angles, and at each angle three colours meet. By permuting the 

 colours cyclically we can obtain from each angle two other 

 cubes, and the eight cubes so obtained are those required. A 

 little consideration will show that these are the required cubes, 

 and that the solution is unique. 



For instance suppose that the six colours are indicated 

 by the letters a, b, c, d, e, f. Let the cube K be put on a 

 table, and to fix our ideas suppose that the face coloured f 

 is at the bottom, the face coloured a is at the top, and the 

 faces coloured b, c, d, and e front respectively the east, north, 

 west, and south points of the compass. I may denote such an 

 arrangement by (/; a; b, c, d, e). One cyclical permutation 

 of the colours which meet at the north-east corner of the top 

 face gives the cube (/; c; a, b, d, e), and a second cyclical 

 permutation gives the cube (/; 6; c, a, d, e). Similarly 

 cyclical permutations of the colours which meet at the north- 

 west corner of the top face of K give the cubes (/; d; b,a, c, e) 

 and (/; c ; b, d, a, e). Similarly from the top south-west corner 

 of K we get the cubes (/; e ; b, c, a, d) and (/; d ; b, c, e, a) : 

 and from the top south-east corner we get the cubes 

 (/; e; a, c, d, b) and (/; b; e, c, d, a). 



The eight cubes being thus determined it is not difficult to 

 arrange them in the form of a cube coloured similarly to K, 

 and subject to the condition that faces in contact are coloured 

 alike; in fact they can be arranged in two ways to satisfy 

 these conditions. One such way, taking the cubes in the 

 numerical order given above, is to put the cubes 3, 6, 8, and 2 



