Answer: It is 192 metres from Harey’s house to school
The winner is Tim Glanvill of Northallerton, North Yorkshire, UK. There were 180 entries.
Worked answer
Let the distance from Harey’s house to school be H metres and let him win the first race by D = 10A + B metres (A, B non-zero digits).
Advertisement
Then in the second race each runner covers (H + D)/H times the distance they covered in the first race. So Tortus will cover (H + D)(H – D)/H metres and will be this far from school: H – [(H² – D²)/H] = D²/H.
Hence D²/H = (10A + B)²/H = 10B + A.
Therefore (10B + A)/(10A + B) = (10A + B)/H
(and, as H>D, we know that A>B).
What if 10B + A and 10A + B have no factor in common (apart from 1)?
Then the fraction (10A + B)/H “cancels down” to (10B + A)/(10A + B).
So 10A + B = N(10B + A) (and H = N(10A + B)).
But then 10B + A and 10A + B have common factor 10B + A!!
Hence 10A + B and 10B + A have a common factor > 1.
Case 1: A, B both even
In this case, D = 42, 62, 82, 64, 84 or 86.
We need D² divided by its reverse to be H.
Only for 84 is D² divided by its reverse equal to an integer, namely 147.
So the first race would be 147 metres and Harey would win by 84.
But that would mean that Tortus had covered less than half the distance, which is not allowed.
Case II: B odd
The only values of D which have a common factor with their reverse are soon seen to be D=21, 51, 81, 63, 93, 75 and 87.
Clearly none of these makes D² divided by its reverse into an integer
Case III: B even, A odd
The only values of D which have a common factor with their reverse are soon seen to be D=72, 54 and 96.
Only D=72 makes D² divided by its reverse into an integer, namely 192.
So the first race was 192 metres with Harey winning by 72 metres (the ratio of their distances being 192:120 or 8:5).
Then the second race was 264 metres with Harey winning by 27 (the ratio of their distances being 264:165 or 8:5 once again).
