Teaser 3032: Darts display
From The Sunday Times, 1st November 2020 [link]
I noticed a dartboard in a sports shop window recently. Three sets of darts were positioned on the board. Each set was grouped as if the darts had been thrown into adjacent numbers (e.g., 5, 20, 1) with one dart from each set in a treble. There were no darts in any of the doubles or bulls.
The darts were in nine different numbers but the score for the three sets was the same. If I told you whether the score was odd or even you should be able to work out the score. The clockwise order of numbers on a dartboard is:
20, 1, 18, 4, 13, 6, 10, 15, 2, 17, 3, 19, 7, 16, 8, 11, 14, 9, 12, 5
What was the score that all three sets of darts made?
[teaser3032]
S2T2 
Jim Randell 10:05 am on 30 October 2020 Permalink |
This Python program runs in 49ms.
Run: [ @repl.it ]
from collections import defaultdict from enigma import tuples, unpack, subsets, union, filter_unique, printf # the scores on the dartboard board = (20, 1, 18, 4, 13, 6, 10, 15, 2, 17, 3, 19, 7, 16, 8, 11, 14, 9, 12, 5) # group possible scores together d = defaultdict(list) for (a, b, c) in tuples(board, 3, circular=1): for k in (3 * a + b + c, a + 3 * b + c, a + b + 3 * c): d[k].append((a, b, c)) # find scores with three disjoint sets of sectors rs = list() for (k, vs) in d.items(): for sss in subsets(vs, size=3): if len(union(sss)) == 9: rs.append((k, sss)) # find scores unique by parity rs = filter_unique(rs, unpack(lambda k, sss: k % 2), unpack(lambda k, sss: k)).unique # output solution for (k, sss) in rs: printf("score = {k} {sss}")Solution: The score was 56.
The three groups of darts are:
This is the only disjoint collection with an even score.
It is possible to make odd scores of 43, 47, 61.
And the score of 61 can be made in 4 different ways, so is the answer to a variation on the puzzle where: “if I told you the score, you still wouldn’t be able to work out the set of sectors where the darts were placed”.
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