Sunday Times Teaser 3308 – New Towns
by Peter Good
Published Sunday February 15 2026 (link)
Three new towns (A, B and C) were being built, with straight roads joining them. Each road was a whole number of miles in length, with AB the same as BC and a total length of the three roads of 108 miles.
A new power station was needed for the three towns and two plans were considered. The Plan 1 site at point P was to be the same distance from each town (a perfect square number of miles). The Plan 2 site was to be a whole number of miles along PB such that the total length of cable connecting the power station to each town was a whole number of miles. Given the above, no shorter total cable length of a whole number of miles was possible, so plan 2 was adopted.
What was the total cable length?
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BRG permalink1234567891011121314151617181920# 2.AB + AC == 108 and AC < 2.ABfor AB in range(28, 54):AC = 108 - 2 * ABsin_abc = (1 - (0.5 * AC / AB) ** 2) ** 0.5# the (perfect square) circumradius R = AB / (2.sin(ABC))r = 0.5 * AB / sin_abcif abs(r - (R := round(r))) < 1e-14 and round(R ** 0.5) ** 2 == R:# clen = cable length, x = height of lower triangleclen = 0for x in range(1, int(AB * sin_abc) + 1):# y = half lower cable lengthy2 = (AC / 2) ** 2 + x * xif (y := round(y2 ** 0.5)) ** 2 == y2:ln = AB * sin_abc - x + 2 * y# store minimum total cable lengthif not clen or ln < clen:clen = lnprint(f"Cable length = {clen:.0f} [{AB = }, {AC = }, {R = }].")
