Sunday Times Teaser 2966 – On Track
by Howard Williams
Published July 28 2019 (link)
Sarah and Jenny are runners who train together on a circular track, and Sarah can run up to 20 per cent faster than Jenny. They both run at a constant speed, with Sarah running an exact percentage faster than Jenny. To reduce competition they start at the same point, but run round the track in different directions, with Sarah running clockwise.
On one day they passed each other for the seventh time at a point which is an exact number of degrees clockwise from the start. Sarah immediately changed her pace, again an exact percentage faster then Jenny. After a few passes both runners reached the exit, at a point on the track an exact number of degrees clockwise from the start, at the same time.
How fast, relative to Jenny, did Sarah run the final section?
-
Brian Gladman permalink12345678910111213141516171819from math import gcd# With Sarah P% faster than Jenny, Sarah will be at (1 + P/ 100).D degrees# clockwise when Jenny is at D degrees anti-clockwise. They meet each time# the sum of these angles is a multiple of 360 degrees giving for the N'th# meeting:## (200 + P).D = 36000.N## N[min] = (200 + P) / gcd(200 + P, 36000)for P in range(1, 21):g = gcd(36000, 200 + P)Nmin = (200 + P) // gif Nmin < 8:t = divmod(36000 // g, 360)print(f"{P}% faster: meet {Nmin} at position {t} anti-clockwise.")
-
Erling Torkildsen permalink12345678# With Sarah running p % faster than Jenny they meet each time at an# integer degree after n passes when (200 + p) divides 180 * n * pfor p in range(1, 21):for n in range(1, 7):if n * 180 * p % (200 + p) == 0:print(f"{p}% faster after {n} passes.")