The local craft shop stocks an ornament made of a piece of carved wood. Its surface consists of a number of faces, all being different colours but the same-sized regular polygon, and with the same number of faces meeting at each vertex. Abel bought a number of these ornaments, all absolutely identical, and placed them on his mantelpiece.
Abel's nephew saw them and noted that three of them had the same colour face resting on the mantelpiece and that in all other cases the colours were different. “That’s an unnatural set-up,” he commented, and he arranged them so that all the colours resting on the surface were different.
However, he was embarrassed when Abel proved to him that the previous situation was more natural because, if the ornaments were placed at random, then there was 50 per cent more chance of having three the same (and the rest different) than having them all different.
How many ornaments did Abel have, and what shape were they?
Note: Adapted from Enigma numnber: 1662 which appeared in New Scientist on 2011.
The first paragraph implies a Platonic solid so the number of sides s must be one of 4,6,8,12,20 (this also equals the number of colors)
Call the number purchased x.
The probability of 3 the same face down, but no others is
s!*x(x-1)(x-2)
p1= -----------------
6(s-x+2)!*s^x
The probability of all faces different
s!
p2=------------
(s-x)!*s^x
From here I just searched a table of p1(n)/p2(n) and found a single solution for 1.5 when s=20, n=11 (and no other s)
Abel had 11 icosahedra.
https://www.desmos.com/calculator/p7pojm4ams
The ratio p1(x)/p2(x)=1.5
yields an equation that simplifies to the cubic
x^3 - 12x^2 + (18s+29)x - 9(s+2)(s+1) = 0
but this doesn't make things much easier.
Only s=20 has an integer solution x=11.
If you ask wolframalpha for any integer solutions it also finds s=15, x=9
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Posted by Jer
on 2024-06-27 13:30:33 |
Solution
