More Dice

I can't remember found I came by these dice. (They may possibly have come from Christmas crackers.) Inside each of them is a heavy ball which helps them come to rest. As you can see from the die on the right, it is not entirely clear which of two numbers are uppermost. So if they were used competitively, no doubt there would be no end of argument between the players.

More subtly, the dots are not painted on in a consistent way. I just checked and found that if the 1, 2, 5 and 6 of both dice are lined up, the 3 and 4 are in reversed positions by comparison.

In fact there are 30 different ways in which the dots can be arranged on each die. The following (from Mathematische Basteleien) shows all 30:

(click)

The red die (number 3, in the diagram above) is the one used as standard. The person who painted the dots on the die on the right (top picture) has been clumsy and produced a number 9 instead of a number 3.

In addition to there being 30 different ways in which the dots can be arranged on each die, as Mathematische Basteleien goes on to point out, each set of dots can be can be arranged in four ways (since each face can be rotated four quarter turns). Because of rotational symmetry, this doesn't affect the appearance of the 1, 4 and 5 - they look the same at each turn. However, for 2, 3 and 6 there will be two different appearances (since the 2 and 3 are on a diagonal and the dots of the 6 are arranged 2x3). That, by my reckoning, would make a total of 240 different dice, if the orientation of the dots was also taken into consideration.