Teaser 2410: [Cube numbers]
From The Sunday Times, 30th November 2008 [link]
Ollie was given a cube together with 15 labels, numbered 1 to 15. He put a different label on each edge of the cube so that, if you chose any vertex and added up the numbers on the three edges meeting there, you got the same “vertex sum” for each vertex.
Furthermore, if you chose any face and added up the four numbers on edges around the face, you got the same “face sum” for each face. Ollie was left with three unused numbers; they were not a mixture of odd and even.
What were Ollie’s unused numbers?
This puzzle was originally published with no title.
[teaser2410]
Frits are discussing. 
Ruud van der Ham, and 
Alex.T.Sutherland, 
Hugo, 
S2T2 
Jim Randell 2:44 pm on 18 September 2026 Permalink |
This Python program labels the 12 edges of the cube, and then uses the [[
SubstitutedExpression]] solver from the enigma.py library, to assign different values from 1 to 15 to each edge. We then look at the 3 unused numbers, and check they have the same parity.It runs in 247ms. (Internal runtime is 173ms).
from enigma import ( SubstitutedExpression, irange, subsets, intersect, diff, join, sprintf as f, map2str, printf ) # label the edges A .. L vertices = "ABE BCF CDG DAH ELI FIJ GJK HKL".split() faces = "ABCD BEFI CFGJ DGHK AHEL".split() # collect expressions exprs = list() fsum = lambda xs: join(xs, sep=" + ") # find vertices/faces that share an edge for xs in [vertices, faces]: for (x1, x2) in subsets(xs, size=2): es = intersect([x1, x2]) if not es: continue exprs.append(f("{x1} == {x2}", x1=fsum(diff(x1, es)), x2=fsum(diff(x2, es)))) # remove equivalent rotations/reflections exprs.extend(["A < min(B, C, D, E, F, G, H, I, J, K, L)", "B < min(D, E, H)"]) # construct a solver digits = list(irange(1, 15)) p = SubstitutedExpression(exprs, base=16, digits=digits) for s in p.solve(verbose=0): # find unused digits xs = diff(digits, s.values()) # all unused digits have the same parity if len({x % 2 for x in xs}) > 1: continue # output solution (unused values and labelling of edges) printf("unused values = {xs}") printf("-> {s}", s=map2str(s))Solution: The unused numbers are: 2, 8, 14.
The (incorrect) solution published in the paper was: “2, 8 and 15”.
Here is a diagram showing the labelling of the edges of the cube:
(Rotations/reflections of this arrangement also give viable solutions).
There are also 5 essentially different ways to label the cube where the unused numbers have a mix of even and odd parities. In these labellings the unused numbers are:
These can be seen by removing the condition at line 31 of the program.
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