An other dicey challenge

Suppose we have the following two dice:

  • a fair, four-sided die with numbers \(D_4 = [1,2,2,3]\)
  • a fair, nine-sided die with numbers \(D_9 = [1,3,3,5,5,5,7,7,9]\).

Rolling these two dice and adding their outcomes together results in a sum between 1 and 12. Remarkably, the probability distribution of this sum is the same as that for two regular, six-sided dice. That is, \((D_4 + D_9) \sim (D_6 + D_6)\).

Now, can you construct another pair of four-sided and nine-sided dice with positive integers different from the above but with the exact same property of \((D_4 + D_9) \sim (D_6 + D_6)?\)

If \(A\) is the highest number on the four-sided die and \(B\) is the highest number on the nine-sided die, post the product \(A\cdot B\).



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