Ruffian Ritual

Not everyone is nice. Not everyone even pretends to be nice, but their is still honour and protocol among thieves.  Let's say that you're spying on a den of thieves and ruffians. The first thing you notice when they get together is that they all greet each other with a secret handshake (to be developed by your students.)

Thanks to David H. Lawrence XVII for this photo. I'm not sure if he's really a bad guy 😉

 You need to pair them up so that each minute they are doing the secret handshake with a different member of the gang. They might start by doing the secret handshake with the person across from them.

  They might then shake hands with one of their neighbours...

... and then the other neighbour...

They are finished their ritual and can now get down to their dastardly business. But what happens if a different number of these ruffians gather together?

The ruffians will very quickly figure out that they are doomed to failure if they are an odd number. Odd numbers don't work because the thug that is left out of the first handshake ritual will likely feel thwarted and get violent.

You can see the disaster here...

... and you can see the ruffian who justifiably feels left out.

What about six thugs? Can they meet and get all their ritual hand shaking done so that every minute every thug is shaking the hand of every other thug? Again they could start out with those standing opposite...

Then they could shake their neighbours...

... and the other neighbour...

Each thug has two thugs left to shake hands with...

... does this look good?

No - two of these thugs have already greeted each other. Your fist task as a class is to figure out how six thugs can shake each others hands. Your second task is to figure out how many thugs can come to one of these subversive meetings. Is 8 possible? Is 10 possible? Is there an upper bound that limits the maximum gathering of these nefarious characters?

Spoiler alert - the next slides highlight a solution found by one of my grade 8 students (Matt Slavin) to the 8 ruffian puzzle. Do not show this to your class! Do not even look at it yourself until after your class and you have struggled for a period. Somebody will probably find a different solution and that is far far superior to you entering the class like a god or goddess smugly knowing the answer. It is much more exciting for the students to see their teacher struggling along side them. I did not solve this problem myself. There is no embarrassment in trying and failing. What a fantastic role model you will be to your class. Please stop here.

Matt's idea was to break up the group of eight ruffians into two groups of four. In three minutes each group can handshake with each other. We've already seen how to solve the four ruffian problem. Now what do the ruffians do? The two groups don't know each other. They decide to line up.

Now they do their secret handshake greeting with the person across from them. PS. Notice the hairy guy in the top left. He's part of the red group. He grew hair just so we can track what he does 😉

Now the red ruffians move to their left. The guy at the end comes back to the start.

See where they are going...

Now they repeat...

and move...

and again...

and shake...

and move...

and last time...

and last shake...

Finished. This same technique can be used to tackle many numbers, but alas, it doesn't help with all ;-(

Ruffian Ritual

(unknown authorship)

Use Ruffian Ritual as an icebreaker at the start of the year or on a day where they just need to do something physical. It will get them thinking about patterns and asking tough questions. For how many ruffians is this puzzle solvable? Brute force might help solve some puzzles for ten or less ruffians, but after a while, we really want to encourage students to think up some general solutions. This is not easy which is why I suggest grades 8+.

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

MP1 Toughen up!

Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP2 Think abstractly!

Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.

MP3 Work together!

Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

MP4 Model reality!

Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.

MP5 Use the right tools!

Students should use the right tools: 0-99 wall charts, graph paper, etc.

MP6 Be precise!

Students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist 😉

MP8 Be lazy!?!

Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?


Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉 For a free poster of MathPickle's ideas on elementary math education go here.

Gordon Hamilton

(MMath, PhD)