The Moscow Puzzles by Boris A Kordemsky
English version translated by Albert Parry, edited by Martin Gardner, published 1972
Puzzle 9: Down and Up
A boy presses a side of a blue pencil to a side of a yellow pencil, holding both pencils vertically. One inch of the pressed side of the blue pencil, measuring from its lower end, is smeared with paint. The yellow pencil is held steady while the boy slides the blue pencil down 1 inch, continuing to press it against the yellow one. He returns the blue pencil to its former position, then again slides it down 1 inch. He continues until he has lowered the blue pencil 5 times and raised it 5 times – 10 moves in all. How many inches of each pencil will be smeared with paint after the tenth move?
This problem was thought up by the mathematician Leonid Mikhailovich Rybakov while on his way home after a successful duck hunt. What led him to make up this puzzle is explained in the answer, but don’t read it until you have solved the problem.
Answer: [for the actual answer see end of entry] … Looking at his boots, Leonid Mikhailovich noticed that their entire lengths were muddied where they usually rub each other while he walks. ‘How puzzling’ he thought. ‘I didn’t walk in any deep mud, yet my boots are muddied up to the knees.’
Now you understand the origin of the puzzle.
observations: Today is the first day of the duck-hunting season in the UK (both inland and foreshore, and also geese), so naturally our minds immediately flew to a very memorable incident of duck-hunting in a book. The Moscow Puzzles is the best-selling mathematical puzzlebook of all time, selling a million copies in the former Soviet Union alone, and it is full of Russian charm, and splendid illustrations. For those of us who like our mathematical puzzles, it is a classic. Apparently when it was first produced in English, the hardest questions were too difficult for Westerners, and were omitted.
Martin Gardner, who had a big role in bringing the book to an English-speaking audience, was the doyen of American puzzling, and wrote many books of recreational mathematics, as well as the marvellous Annotated Alice, an unusually-footnoted version of Lewis Carroll’s book. He died in 2010 and is still remembered at the biennial Gathering 4 Gardner.
The answer to the question is: six inches (presumably it was centimetres in the original Russian).
Links up with: Duck shooting is a very popular Russian sport, and features in Anna Karenina. Mrs Zuckerman wore boots to scrub the pig in Charlotte’s Web, while Rosie took her boots off…. The three sisters wanted to get to Moscow.
The photograph of duck boots was taken by Emma Weatherill.