So, what did we learn on this exploration? Well, we started by considering the 2d6 case, and went to the 3d6 case; finding a specific formula for 3d6 on the way. Then we considered how it could generalize with other/more dice. To solve this problem we used geometry in higher dimensions, induction for a proof and found the binomial hidden behind almost every part of the formula. Thus we learn that the mystical simplex numbers are, in fact, just the binomial numbers.
In the end we derived the general purpose formula for the amount of events that could roll a particular number on ndk. Thus with our curiosity satisfied we can return to playing board games with a greater knowledge on how the dice decide our fate
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