Teaser 2739: Funny dice
From The Sunday Times, 22nd March 2015 [link] [link]
I have two cube-shaped dice, one red and one blue, with a positive whole number on each face. When I throw the dice and add up the two numbers, the most likely total is 7. The next most likely totals are 6 and 8, the next are 5 and 9, the next are 4 and 10, the next are 3 and 11, and the least likely are 2 and 12. However, my dice are not standard: indeed, the total of the six numbers on the red dice is higher than the total of those on the blue dice.
What are the six numbers on the red dice?
[teaser2739]
Hugh+Casement and 
S2T2 
Jim Randell 10:05 pm on 13 February 2022 Permalink |
See also: Teaser 3098, Enigma 382, Enigma 646b.
We assume that the 6-sided dice are “fair” (i.e. when one is thrown each face has a 1/6 chance of showing), and also that each of the mentioned values (2-12) is an achievable throw with the pair.
The probability of each combination must be expressible as n/36 and the sum of the probabilities must be 1.
If we consider the lowest possible probabilities:
Then we find this accounts for 36/36, so must be the required distribution, and is the same distribution as a pair of standard dice.
So the dice we are looking for are a non-standard pair with the same throw distribution as a standard pair.
There is only one such value for 6-sided dice, known as the Sicherman dice.
We can use the [[
sicherman()]] function I wrote for Teaser 3098 to solve this problem.This Python program runs in 46ms (internal run time is 841µs).
Run: [ @replit ]
from enigma import printf from sicherman import sicherman for (d1, d2) in sicherman(6): if sum(d1) != sum(d2): printf("{d1} + {d2}")Solution: The numbers on the red die are: 1, 3, 4, 5, 6, 8.
And the numbers on the blue die are: 1, 2, 2, 3, 3, 4.
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Hugh+Casement 7:12 am on 14 February 2022 Permalink |
We are told ” the least likely are 2 and 12″ but there is no stipulation that they have to occur at all.
I found eight sets where only sums from 3 to 11 can occur, with frequencies in the order given.
So for me the puzzle is ill-defined — that’s apart from the misuse of the plural ‘dice’.
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