Sunday Times Teaser 3236 – A Fair Start
by Andrew Skidmore
Published Sunday September 29 2024 (link)
We did well selling plants on our charity stall this year: £70 was taken in the first hour. The plants were priced at £1, £2, £3, £4 and £5 but all purchases were made from just three of those categories. All the customers spent different amounts and all bought three items, but no-one bought three items at the same price. Last year the Vicar spent £14 but so far this year no-one has exceeded or even matched that amount. In fact the Vicar was the first customer and spent the average amount in that first hour.
What, in ascending order, were the prices of the items that the Vicar bought?
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BRG permalink123456789101112131415161718192021222324252627282930from itertools import combinations, combinations_with_replacement# when they each spend less than £14, at least# 7 people are needed to spend £70 in totalmin_p = 7# consider the three different costs of the plants soldfor c3 in combinations(range(1, 6), 3):# consider the possible sales involving these three plantsposs_sales = set()for s3 in combinations_with_replacement(c3, 3):# all bought plants costing less than £14, not all the sameif sum(s3) < 14 and len(set(s3)) != 1:poss_sales.add(s3)# now find sets of purchases that result in total sales of £70 ...for ns in range(min_p, len(poss_sales) + 1):for sales in combinations(poss_sales, ns):# ... and different spends with an integer average spend in £sspns = tuple(sum(s) for s in sales)if len(spns) == len(set(spns)) and divmod(sum(spns), ns) == (70 // ns, 0):# find the vicar's purchasesvs = {s for s in sales if sum(s) == 70 // ns}.pop()f, s, l = ', '.join(f"£{v}"for v in vs).rpartition(', ')sls = ' '.join(''.join(f"{s}" for s in ss)for ss in sorted(sales))print(f"The vicar spent {f} and {l} [sales: {sls}].")
