Arrange some dice, stacking them on top of each other, so that the sum of dots on each vertical side of the rectangular parallelepiped is the same.
Of course that using an even number of dice the solution is simple.
What's the smallest odd number of dice that can be used?
What's the smallest sum that can be formed?
This cannot be accomplished with an odd number of regulation dice. Opposite every odd face is an even face and vice versa, as opposite faces add up to 7. So if an odd number of dice are stacked, then for any given pair of faces of the stack, one member of the pair will have an odd number of odd die faces, and the opposite stack face will have an even number of odd die faces; thus, one will have an odd total and the other an even total.
Thus, for the second part we must assume an even number of dice in the stack. The smallest sum on each stack face is 7, formed by a stack of two dice. Both have the same face down--it doesn't matter which--and one is rotated 180° with regard to the other around a vertical axis of rotation. Each face of one is then aligned with the corresponding opposite face and all four sides add up to 7.
Posted by Charlie
on 2017-11-28 09:53:21