The die that makes sums uniform

MediumProbability Combinatorics~8m

You have a normal six-sided die (faces $1$ – $6$ ) and a second six-sided die whose faces are unknown (they need not be $1$ – $6$ ). Together they are engineered so the sum of the two dice is equally likely to be any integer from $1$ to $12$ . What is the sum of the six faces on the mystery die?

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Use generating functions: $(x+\cdots+x^6)\cdot M(x)=3(x+\cdots+x^{12})$ .
$M(x)=3(1+x^6)$ , so faces $\{0,0,0,6,6,6\}$ sum to $18$ .

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The die that makes sums uniform

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