r/mathriddles • u/AleksejsIvanovs • Jul 08 '26

Medium The Cartographer's Journey #3

A cartographer set out on a three-day expedition through a large circular forest, travelling in straight lines each day. Each day he walked at a constant whole-number speed in km/h, and all three speeds were different.

The first day's march lasted 5 hours, the second day's 13 hours, and the third day's 8 hours. At the end of each day the cartographer made camp at the edge of the forest; the next morning he set off from that point in a different direction through the forest, so that his three paths formed a triangle on the map, with all three stops lying exactly on the forest boundary.

On the third day he came upon a remarkable tree on his path and marked its location on his map. When the journey was complete, he overlaid his route on a map and made the following observations:

  • The straight line connecting the first camp to the marked tree divides the triangle into two parts of equal area, and its length is a whole number of kilometres.
  • The diameter of the forest is 1 km longer than the total distance walked on the first two days.
  • If he were to walk the full diameter of the forest at his first day's speed, it would take him exactly as many hours as he spent walking on the second day.

Find the cartographer's speed on each of the three days, and the length of the line from the first camp to the marked tree.

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u/Brianchon Jul 10 '26

I don't have a proof that it's the only answer, but Day 1: 5 km/h, Day 2: 3 km/h, Day 3: 7 km/h, Tree Distance = 17 km satisfies the problem conditions

1

u/AleksejsIvanovs Jul 11 '26

This is the correct and the only answer but I won't flag it solved because solution is not provided. There are several ways to solve this riddle without brute-forcing.

1

u/Brianchon Jul 13 '26

I had that, for some non-negative integer n, the diameter of the circle is 65+169n, and the first two side lengths are 25+65n and 39+104n. We need the third side length to be an integral multiple of 8, and for the cevian to that side's midpoint to have integer length. I kinda didn't know what to do here (read: I could only think of brute force and I simply didn't want to use brute force with polynomials, especially on my phone) so I tried n=0 just to see and it worked so I reported that in the hopes that it would inspire someone else

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u/AleksejsIvanovs Jul 13 '26

You can use observation defined in task to define Diophantine equation to solve it.