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Let \(a\), \(b\), \(c\) and \(d\) be the numbers on the four faces. The following simultaneous equations must hold:

These can be solved to find that the numbers on the faces must be \(-\frac{2}{3}\), \(\frac{1}{3}\), \(\frac{4}{3}\) and \(\frac{7}{3}\).

Extension

Is it possible to make a six-sided die with one number on each face such that the value of the roll can be calculated by adding up the five visible numbers?

Is it possible to make an \(n\)-sided die with one number on each face such that the value of the roll can be calculated by adding up the \((n-1)\) visible numbers?

Is it possible to make a die with one integer on each face such that the value of the roll can be calculated by adding up the visible numbers?

$$a+b+c=1$$ $$a+b+d=2$$ $$a+c+d=3$$ $$b+c+d=4$$