The six sides of the cube are the numbers a, b, c, d, e and f, ordered by size. If the cube with the largest number f is facing down, then the mean value of the numbers on the other five faces is 1. If instead the second largest number is below e, the mean of the other five numbers increases to 2.

As a result, the difference between the numbers f and e is exactly 5. If the third highest number d is below, the mean of the other five numbers increases to 3, which means that the difference between the numbers e and d is also exactly 5. This also applies to all other four pairs of adjacent numbers: the larger number is always 5 greater than the smaller.

Since the mean of the five smallest numbers on the cube must be 1, 1 = (a + b + c + d + e) / 5 = (a + (a + 5) + (a + 10) + (a + 15) )) + (a + 20))/5 = a + 10, which can be changed to a = -9.

The six digits on the faces of the dice are -9, -4, 1, 6, 11 and 16. Instead of numbers, such dice can also be rendered with eyes using the sums on the bank such as plus and minus signs. Expresses the statement with black and red.

These dice aren’t just a math gimmick, they can actually be bought. The company Trick Dice in Inverness in Scotland manufactures them and sells them worldwide under the name hem dice.

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