# Enigmatic Code

Programming Enigma Puzzles

## Tantalizer 420: Box and Cox

From New Scientist #970, 9th October 1975 [link]

At noon precisely Box left the Lion at Hogpen and set off on foot for the Plough at Inkwell. At noon precisely Cox left the Plough and set off by bicycle for the Lion. When they met Box had covered 4 miles. After 10 minutes’ chat Cox gave Box the bike and walked on to the Lion, where he sank a pint in 3 minutes and set off again for the Plough. Box meanwhile cycled on to the Plough as soon as Cox left him, lowered a pint in 3 minutes and cycled off again towards the Lion.

They met this time 7 miles from the Plough and discussed wurzels for 10 minutes. then Box relinquished the bike and walked on to the Lion and Cox cycled on to the Plough. Pints were sunk in two minutes this time and off our heroes went again with Cox still mounted and Box afoot. They collided two miles from the Lion.

Let Box walk at p mph and cycle at 9 mph. Let Cox walk at r mph and cycle at s mph. Let the road be ever so wiggly.

How many miles is a single journey along it?

[tantalizer420]

### One response to “Tantalizer 420: Box and Cox”

1. Jim Randell 9 October 2019 at 7:55 am

 Considering the distances travelled from noon to the first meeting:

4 / p = (d – 4) / s
s / p = (d – 4) / 4

 And from the first to second meetings:

(d – 4 + 7) / 9 = (4 + d – 7) / r
r / 9 = (d – 3) / (d + 3)

 And from the second to third meetings:

(d – 7 + 2) / p = (7 + d – 2) / s
s / p = (d + 5) / (d – 5)

From  + :

s / p = (d – 4) / 4 = (d + 5) / (d – 5)
(d – 4)(d – 5) = 4(d + 5)
d² – 9d + 20 = 4d + 20
d² – 13d = 0
d(d – 13) = 0
d = 13 (d ≠ 0)

Solution: The distance between the pubs is 13 miles.

We also get:

r = 5.625
s / p = 2.25

So Cox walked at 5.625 mph (a nice brisk pace), and however fast Box walked Cox cycled 2.25 times faster.

So if Box walked at 4 mph, Cox cycled at 9 mph (the same speed that Box cycles).

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