Given d dice each with n sides, find the probability that when they are rolled at once, there are no two consecutive numbers.
This task may be quite difficult. For a warm-up, try finding the numerators for fixed d such as {1,2,3} or for fixed n such as {2,3,4}.
Note 1: I don't have a formula so much as an algorithm.
Note 2: This problem arose as an attempt to solve http://perplexus.info/show.php?pid=12342 by Larry which uses non-independent cards instead of dice.
The numerators (successes):
sides #dice, count of successes
2 2 2 3 2 4 2 5 2 6 2
3 2 5 3 9 4 17 5 33 6 65
4 2 10 3 22 4 46 5 94 6 190
5 2 17 3 47 4 125 5 335 6 917
6 2 26 3 90 4 290 5 906 6 2786
7 2 37 3 157 4 601 5 2197 6 7897
8 2 50 3 254 4 1142 5 4838 6 19910
9 2 65 3 387 4 2021 5 9819 6 45845
10 2 82 3 562 4 3370 5 18610 6 97882
The denominators (as expected, sides^dice):
2 2 4 3 8 4 16 5 32 6 64
3 2 9 3 27 4 81 5 243 6 729
4 2 16 3 64 4 256 5 1024 6 4096
5 2 25 3 125 4 625 5 3125 6 15625
6 2 36 3 216 4 1296 5 7776 6 46656
7 2 49 3 343 4 2401 5 16807 6 117649
8 2 64 3 512 4 4096 5 32768 6 262144
9 2 81 3 729 4 6561 5 59049 6 531441
10 2 100 3 1000 4 10000 5 100000 6 1000000
Remember, the number of dice precedes each count in any row.
Edited on January 26, 2021, 1:19 pm
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Posted by Charlie
on 2021-01-26 13:16:09 |
numerators and denominators separately
