Some Logical or Mathematical Puzzles

Several months ago, a number of puzzles were posted in the School's Coffee Room as a challenge to coffee-drinkers. Some were trivial (how to obtain 4 quarts of water, given a large supply, and jugs holding 3 and 5 quarts); others were standard logic puzzles (given five professors, five PhD students, five topics, five countries of origin, five ages and a dozen statements, match professors, students, topics, ...)

Three were more interesting, and though the puzzles are well-known, their solutions seem not to be. These are reproduced here along with my own solutions. To these I have added an old and very interesting cross-number puzzle which I first encountered when I was a mathematics undergraduate over 50 years ago.

I posted the weighings puzzle in response to a much easier one posted earlier. The puzzle is well-known, but several people asked for the solution and some claimed that it could not be done in less than four weighings.

The pirates' gold problem is discussed in Wikipedia:

and a solution by Ian Stewart appeared in Scientific American, May 1999. The solution which I wrote without being aware of his is essentially the same.

The bean-division problem is discussed in:

The suggested solutions appearing there (as of July 2007), however, are simplistic and lack any proper analysis of the problem.

The cross-number puzzle is quite different from any other I have seen. As I have said, I first saw it over 50 years ago, though I didn't remember where, or who created it. Google ("dogsmead") lists a few dozen webpages. The one with the most comprehensive history reveals that the puzzle was created by William Thomas Williams (1913-1995), and was first published as: The Little Pigley Farm, in "The Strand Problems Book, a modern anthology of perplexities and tantalizers", by W.T. Williams and G.H. Savage. I won't reveal the date here as it can help in the solution. A paper: On Cross- Number Puzzles and the Lucas-Bonaccio Farm 1998, by William Y Sit, can be accessed in PDF via:

Contrary to the opinions expressed on some of the webpages, the puzzle is completely logical, and requires no guesswork and no genius. Some knowledge of the properties of numbers is helpful. (My puzzle box is layed out using simple ASCII encoding and must be displayed/printed in a fixed-width font to be seen correctly.) Solution.

It is hoped that these puzzles will exert a calming influence on any reader who has become over-excited by the pages of the VINCI manual.

Michael Levison