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Luke_A_Somers comments on Open Thread for February 11 - 17 - Less Wrong Discussion
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You can do better than missing one part in 692693? You can't do it in one roll (not even a chance of one roll) since the dice aren't large enough to ever uniquely identify one result... is there SOME way to get it exactly? No... then it would be a multiple of 1001.
I am presently stumped. I'll think on it a bit more.
ETA: OK, instead of having ONE left over, you leave TWO over. Assuming the new pair is around the same size that nearly doubles your trouble rate, but in the event of trouble, it gives you one bit of information on the outcome. So, you can roll a single 503 sided die instead of retrying the outer procedure?
Depending on the pair of primes that produce the two-left-over, that might be better. 709 is pretty large, though.
The best you can do leaving 2 over is 709*953=675677, coincidentally using the same first die. You can do better.