Class Openers

Class Opener – Day 4 – the 36 Officers Problem

As students entered today, each group found on their tables the mystery envelope…{cue Law and Order music}.

DiceInside each evenlope were 36 cut out pieces of paper.  Yesterday we had discussed sample spaces, theoretical and experimental probability involving 2 dice, using applets and online resources to think about similarities / differences, so these pieces were not unfamiliar.  Instructions for the pieces was provided on the board:

Arrange your pieces into a 6×6 grid, so that no duplicate red die appears in any row or column, nor any duplicate white die.

TryingGroups immediately gravitatied towards this task, and after some initial misunderstandings over the directions, got down to business looking for patterns. One student who was a Sudoku fan became quite obsessed. I let this problem hang through the period, just having them stop when we completed notes or had class problems to do.  Many students felt they were close, but couldn’t quite solve the puzzle. Half-way through the period I reminded students that it is often helpful to think about a smaller problem before tackling a larger one. What if you just looked at the dice numbered 1 through 4 – can you complete that problem?  Learn from smaller steps – then tackle the big ones.

If you want to try this task on your own, cut out the pieces from the picture above. Then come back here and visit for the solution.

OR if you just want to know how it end….you unadventurous soul…then keep reading…

We’ll wait for you to come back.


The problem I gave today is an interpretation of the 36 Officers Problem, a problem often credited to Euler.  Instead of dice, the Officers challenge is to arrange 6 ranks and 6 regiments into rows and columns.  My alma mater, Muhlenberg College, hosts a high school math competition every February – with a free t-shirt to all who attend. For the 36th year of the contest, the shirt features the 36 Officers Problem, and gave me inspiration for today’s activity.


As for the solution….here’s the thing….the 36 Officers Problem is impossible.  It’s do-able for all other numbers of rows and columns (except 2), but is impossible for a 6×6.  Some examples of solvable squares appear on this untamed blog, and I’d love to have others contribute their thoughts and or/resources.


By Bob Lochel

HS Math Teacher. Hatboro-Horsham School District, Horsham, PA.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s