math in autumn: Nine Men in a Trench

Nine Men in a Trench is a math puzzle invented in 1917 by H. E. Dudeney. Historically, it was a dark time. The battles of World War One were never ending, and casualties were horrendous. At the Battle of the Somme, which lasted over four months, the combined casualties for the Allies and the Germans totalled an unimaginable 1,200,000 people.

WWI battles featured trench warfare, which must have given rise to the setting for Dudeney's puzzle.

Photograph from Canadian government archives showing Canadian soldiers at the Somme. Those recesses dug into the sides of the trench are called funk holes, and they are a factor in Dudeney's Nine Men in a Trench puzzle.

In contrast to all this darkness, we have this wonderful photo of a student from Contentnea-Savannah School in North Carolina. She is displaying her puzzle at a math fair.

Photo posted with permission.

This is the puzzle:

Nine Men in a Trench

Move the red marker from the right end to the left end. You can move markers into the spaces, but cannot double them up. Nor can you jump a marker over another.

(If you look closely at the photo, you can see markers on the puzzle board behind the student.)

The puzzle can be found on our math fair website (link provided below). It is a simplification of Dudeney's puzzle. Here is his original version, complete with funk holes and soldiers:

Nine Men in a Trench

Here are nine men in a trench. Number 1 is the sergeant, who wishes to place himself at the other end of the line––at point 1––all the other men returning to their proper places as at present. There is no room to pass in the trench, and for a man to attempt to climb over another would be a dangerous exposure. But it is not difficult with those three recesses, each of which will hold a man.

How is it to be done with the fewest possible moves? A man may go any distance that is possible in a move.

So where's the math??

The Nine Men in a Trench puzzle does not explicitly involve arithmetic. At SNAP, we promote the use of math based puzzles in the classroom, but puzzles like this sometimes prompt the above question.

The truth is that much of mathematics does not involve arithmetic. Nine Men in a Trench is a sorting problem, and sorting problems are dealt with in both mathematics and computing science courses.

Like all sorting problems, Nine Men deals with a list of objects that have to be put into a specific order, with restrictions on what manoeuvres are allowed when reordering the objects.

To solve the Nine Men puzzle, students have to figure out procedures to use that will allow them accomplish the sort. Although they are dealing with a 100 year old puzzle, when students work with this puzzle they are truly doing up-to-date mathematics. Beyond memorizing and practicing the usual algorithms of their math courses, when they work on this puzzle they are also beginning to invent algorithms of their own making.

A more challenging adaptation of the puzzle makes it very clear that it is a sorting problem:

Five Men in a Trench

Put the tiles in numerical order from left to right. Stay within the playing board. You can move tiles into the spaces, but cannot double them up. Nor can you jump a tile over another.

We can ask for the minimum numbers of moves required to solve this version (although that should probably not be the point of the puzzle for children). I have been able to do it in twenty-five, but I don't know if that is the smallest possible number. As in Dudeney's puzzle, a move is defined as sliding a single tile any number of positions, following the rules for moving a tile of course.

More about connections between math and puzzles

Teresa Sutherland and I have put together a small booklet about the connections between school mathematics and the puzzles on our math fair website. It includes pointers to both the Alberta Math curriculum and the American Common Core State Standards. Teresa is a Designer/Director/Teacher from Maryland for the STEAM-based Odyssey of the Mind Inspired Spontaneous Challenge Program.