Nothing funny about feldspar

From the February 28 New Scientist:

(paraphrased) Harey and Tortus race to school, which is a whole number of meters away from Harey’s house. They each travel at their own steady paces. On the first day Harey wins, but Tortus has traveled more than half way (and loses by a whole 2-digit number of meters).

The next day they race again, but this time Harey starts further away from school — by the distance Tortus lost. He wins again. Tortus has lost again by a 2-digit whole number of meters, though this time the digits are reversed from the first race.

How far is Harey’s house from school?

Plagioclase and I spent a fun half-hour revisiting our simple algebra substitutions, as well as brute-forcing quadratic equations in a spreadsheet. There is probably a more elegant (easier?) way to figure this, but we didn’t look for it.

As a hint, let me remind you that ratios are useful things.

The distance from Harey’s house to schools is 192 meters.

In first race, Tortus travels 120 meters (a margin of 72 meters), and the second race he travels 165 meters (a margin of 27 meters).