Sunday Times Teaser 3171 – What’s Happened To Our Chocolate Bar?
by Michael Fletcher
Published Sunday July 02 2023 (link)
Father Christmas was rewarding his helpers (sprites, elves and fairies). He had 333 helpers, of whom 111 were elves.
He had 28 chocolate bars to give away in proportion to the size of the groups. If the groups’ shares were 9.5, 9.3 and 9.2 then they would actually receive 10, 9 and 9, because the bars can’t be split and the extra bar goes to the group with the highest remainder (for two extra bars, one each would go to the two groups with the highest remainders). In fact the sprites, elves and fairies received 7, 9 and 12 bars respectively. “I’ve got another chocolate bar,” said Father Christmas, “so now the sprites, elves and fairies will receive 6, 10 and 13 bars respectively.” “What’s happened to our chocolate bar?” asked the sprites.
How many sprites are there?
-
BRG permalink123456789101112131415161718192021222324252627282930313233343536# elves and sprites + fairies numberselves, rest = 111, 222# allocations of chocolate ars to sprites,# elves and fairies for 28 and 29 barsa28, a29 = [7, 9, 12], [6, 10, 13]# share <bars> among numbers of sprites, elves and fairies in <nbrs>def share(nbrs, bars):sm = sum(nbrs)# number groups and find (integer, remainder) bars for each groupshares = [(i,) + divmod(n * bars, sm) for i, n in enumerate(helpers)]# find how many bars are left to share out and how they will be sharedleft = sum(s[2] for s in shares)extras = [(0, 0, 0), (0, 0, 1), (0, 1, 1)][left // 333]# sort the shares in order of remainders and add bars where necessaryshares = sorted(shares, key=lambda s: s[2])shares = [(s[0], s[1] + x) for s, x in zip(shares, extras)]# return the numbers of bars in the original group orderreturn [b for a, b in sorted(shares, key=lambda x: x[0])]# fairies: 29.f / 333 >= 12 ; 28.f / 333 < 13 ==> 138 <= f < 155for fairies in range(138, 155):helpers = (rest - fairies, elves, fairies)# share and check results for allocations of 28 and 29 barsif (share(helpers, 28), share(helpers, 29)) != (a28, a29):continueprint(f"{rest - fairies} sprites")
