Sunday Times Teaser 3335 – Trish Tries Triathlon
by John Owen
Published Sunday August 23 2026 (link)
The first two legs of our non-standard Triathlon comprise a run across a field then a swim across a lake. The lake’s parallel shores run from west to east; the starting point for the run is a whole number of metres north of the lake’s northern shore and 1,600 metres north of the lake’s southern shore. The exit from the lake is at a fixed point (a whole number of metres further east than the starting point), but competitors may choose their point of entry to the lake.
Trish chooses her entry point so as to minimise the time she will take for the two legs, knowing that she runs six times as fast as she swims. Interestingly, this makes the two legs the same whole number of metres long.
How long?
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BRG permalink12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152from number_theory import pythagorean_triplesfrom collections import defaultdict# Because the run is six times faster than the swim, the land run# will be a small angle from a West->East line while the sea swim# will be a small angle from a North -> South line as shown below## |---___# | ---___ c# | a ---___# | ---___# .-------------- -------------# | b |\# | | \# | | \# | d | \ c# | | \# | | \# | | e \# -------------------------------------# | g |# with the distances as above and v as the swimming speed# we obtain the total time t as## 6.v.t = (a^2 + b^2)^(1/2) + 6.[d^2 + (g - b)^2]^(1/2)## differentiating:## 6.v.dt/db = b / (a^2 + b^2)^(1/2) - 6.(g - b) / [d^2 + (g - b)^2]## Setting this to zero, rearranging and cross multipying gives:## b.[d^2 + (g - b)^2]^(1/2) == 6.(g - b).(a^2 + b^2)^(1/2)## which simplifies to## b.c = 6.e.c ==> b == 6.e# collect pythagorean triangles indexed on hypotenusec2l_ab = defaultdict(list)for a, b, c in pythagorean_triples(1500):c2l_ab[c] += [(a, b)]for c, l_ab in sorted(c2l_ab.items()):for a, b in l_ab:if (d := 1600 - a) >= c:continueif (e := round(c * c - d * d) ** (1 / 2)) and b == 6 * e:print(f"leg = {c}m ({a=}m, {b=}m {d=}m, e={int(e)}m)")
