Knucklebones: Upper Primary Mathematics Competition Question

Knucklebones were originally made from part of the ankle of a sheep; they were also known by their more anatomically correct name of astragaloi. They survive to this day in the children’s game known as Jacks. However, knucklebones were also used as dice.

Because of their peculiar shape, knucklebones only had four numbered faces, with the values 1, 3, 4 and 6. Ancient Greeks and Romans seem to have enjoyed playing a game that used four such knucklebones as dice. If we add together the values on the four exposed faces, which numbers between 1 and 24 inclusive can never be rolled?


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