Sunday Times Teaser 2542 – Two Routes
by Graham Smithers
Published: 12 June 2011 (link)
A line of equally spaced poles is marked in order with odd numbers, 1, 3, 5, etc. Directly opposite is a parallel line of an equal number of poles with even numbers, 2, 4, 6, etc. There are fewer than 100 poles. The distance in metres between adjacent poles is a twodigit prime; the distance between opposite poles is another twodigit prime. Jan walks the route 12435, etc to reach the final pole. John walks the odds 135, etc to the last oddnumbered pole; then walks diagonally to pole 2; then walks the evens 246, etc, also finishing at the final pole. Jan’s and John’s routes are of equal length.
How many poles are there?
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Brian Gladman permalink1234567891011121314151617181920212223242526272829303132# Let the number of poles be 2.N (since the routes end on the# even poles there must be an even number. The layout is:## p# 1 3___5 ___2.n1 2.n1#     # q    ...  OR ... # ___ ___  ___# 2 4 6 2.n 2.n## We have:## N.q + (N  1).p == 2.(N  1).p + sqrt((N  1)^2.p^2 + q^2)## [N.q  (N  1).p]^2 = (N  1)^2.p^2 + q^2## (N^2  1).q^2  2.N.(N  1).p.q = 0## (N  1).q.[(N + 1).q  2.N.p]## which gives (N + 1).q = 2.N.p and hence N = q / (2.p  q),# which means that 2.p  q = 1 and the number of poles is 2.q# so we are looking for a two digit prime 20 < q < 50 such that# (q + 1) / 2 is also primefrom number_theory import Primes, is_primefor q in Primes().range(20, 50):if is_prime((q + 1) // 2):print('There are {} poles.'.format(2 * q))