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. 

You can obtain the book from the SNAP math fair website. Or contact me, Teresa, or Sean Graves, respectively at 

(Ted)       tlewis@ualberta.ca

(Teresa)  dsuther@yahoo.com

(Sean)    sgraves@ualberta.ca

The original Dudeney puzzle (and others) can be found in either of the following books:

Who needs algebra anyway?

These five balance and weighing puzzles are intended to be solved using only manipulatives. Some of them are suitable for a puzzle-based math fair. They are fun to solve and require a lot of mathematical thinking.

No algebra is needed, but if you have algebra at your disposal, it is interesting to see how each puzzle can be modelled as a system of linear equations.

#1
There are three different types of boxes. All boxes with the same colour weigh the same.

These boxes balance.

---

How many blue boxes are needed to balance one red box?

So, the solution for puzzle #1 is not difficult: The balance at the top left says that two yellow boxes are the same weight as three blue boxes. So, attending to the balance at the top right, replace the two yellow boxes with three blue boxes to get

By removing three blue boxes from each pan (which  maintains the balance) we see that five blue boxes will balance one red box.


Here's one that is a little more interesting:

#2
There are three different types of boxes. All boxes with the same colour weigh the same.

These boxes balance.

---

How many blue boxes are needed to balance one red box?

And one that is more challenging:

#3
There are three different types of boxes. All boxes with the same colour weigh the same.

These boxes balance.

---

How many blue boxes are needed to balance one red box?

Or this one:

#4
There are three different types of boxes. All boxes with the same colour weigh the same.

 These boxes balance.

---

How many blue boxes are needed to balance one red box?

Here is one that is about inequalities.

#5
There are three different types of boxes. All boxes with the same colour weigh the same.

These boxes balance.

---

These boxes will not balance.
Which side of the balance will tilt down?

Puzzle #5 is from Sam Loyd, where it was not given as a balance and weighing puzzle, but was set up as a "Tug o' war" problem:

In a tug of war contest, four stout boys could tug just as strong as five plump sisters. Two plump sisters and one stout boy could hold their own against two slim twins. If three plump sisters and the slim twins team up against one plump sister and four stout boys, which side will win?

–––––––

There are major differences between using manipulatives and using algebra to solve these particular puzzles.

1. There are no negative numbers when using manipulatives. In the following situation, you cannot isolate the red box by removing the yellow box from the left pan because there is no balancing yellow box that can be removed from the other pan.

Whereas with algebra, we readily subtract y from both sides of the equation

r + y = 5b 

                         to get

r = 5b – y.

2. There are no fractions when using manipulatives. In puzzle #4, I did not ask the question

How many blue boxes are needed to balance one yellow box? 

because there is no answering it using manipulatives alone.

It could be deduced that two and one-half blue boxes are needed to balance one yellow box, but you cannot get this via manipulatives unless your manipulatives are sliced into smaller pieces beforehand. (And you have to use at least a minimal amount of algebra to determine the size of the slices.)

If we use algebra to solve the puzzle, we fairly quickly obtain the fact that

 y =  ⁵⁄₂ b,

even 'though you can argue that this is of dubious meaning in the context of the physical puzzle.

A more complete answer to the question in the post title:

Sometimes you really don't need algebra, but algebra can make thinking a lot easier.