Teaser 2928: Golf balls
From The Sunday Times, 4th November 2018 [link] [link]
Three friends play 18 holes of golf together every week. They have a large supply of golf balls numbered from 1 to 5. At the beginning of a year, they each start playing with a new ball, but all three ball numbers are different. Alf uses the same ball number for exactly 21 holes before changing to the next higher number (or to number 1 if he was using a number 5). He continues to use each number for exactly 21 holes. The same applies to Bert, except that he changes his ball number in the same way after every 22 holes, and to Charlie who changes his ball number after every 23 holes.
Last year, there was just one occasion when they all used the same ball number on a hole.
What was the number of the hole on that occasion?
[teaser2928]
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Jim Randell 8:59 am on 1 April 2019 Permalink |
This Python program finds the solution constructively in 79ms.
Run: [ @replit ]
from enigma import (irange, subsets, singleton, printf) # increment ball count, and ball if necessary def inc(i, x, X): i += 1 if i == X: x = (1 if x == 5 else x + 1) i = 0 return (i, x) # find (week, hole) pairs def solve(A, B, C, a, b, c, n): i = j = k = 0 # consider number of weeks for w in irange(1, n): # and each hole for h in irange(1, 18): if a == b == c: yield (w, h, b) # increment ball counters (i, a) = inc(i, a, A) (j, b) = inc(j, b, B) (k, c) = inc(k, c, C) # choose starting balls for A, B, C # suppose A's ball is 1, we only have to choose B and C a = 1 for (b, c) in subsets((2, 3, 4, 5), size=2, select='P'): # in 2017 there were 52 whole weeks, + 1 extra Sunday rs = singleton(solve(21, 22, 23, a, b, c, 53)) if rs: (w, h, x) = rs printf("a={a} b={b} c={c}: week={w}, hole={h}, ball={x}")Solution: Hole number 10.
It would happen on week number 53.
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