The Adventures of Pinocchio were written by Carlo Collodi in 1883. This puzzle was inspired by imagining Pinocchio’s classroom – full of little people who always lied or always told the truth.

In one of his classroom the truth teller students and liar students sat in sixteen desks. The teacher asked “Are you sitting beside exactly two children like yourself?” The answer was unanimous – “YES!”

(Students sitting diagonally adjacent are not “beside.”)

If we color the truth tellers green – and the liars yellow…

Does this work?

No – The truth tellers (green) are happy. But the liars (yellow) would have answered “No!” to the teachers question… so this cannot be right. Try to find an answer.

Here is one answer. The truth tellers are all sitting beside exactly two other truth tellers. The liars say “YES!” they are sitting beside two liars. This of course is a lie.


Here is the only other answer.


Moving up a grade… Pinocchio’s class size increases. The teacher asks the same question. The students unanimously say “YES!” Can you find which students are liars?


There are 9 yellow liars. This is the only solution so we know exactly where they are.


Now solve for 6×6 and 7×7…

Start asking other questions…



Here is the solution for 6×6.


Here is a solution for 7×7.

Is this a solution for 6×8- or did I fail?

Yes – that was a fail. I don’t think 6×8 has a solution.

What other questions can you ask? What about different grids? What about 3D objects? What about infinite tiling patterns? What about have two clans – both truth tellers, and two clans – both liars.

Here is a solution if you ask how to tile an infinite plane under the constraint that everyone answers “Are you beside exactly one truth teller” with a “YES!”

There are other solutions.


Here is a solution if you ask how to tile an infinite plane under the constraint that everyone answers “Are you beside exactly three truth tellers” with a “YES!”

There are other solutions.


Here is a failure if you ask “Are you beside exactly three truth tellers” and want to get a “YES!”

Sadly – the three corners of the triangle give a “No!”



Well two of them can be fixed by duplicating the triangle…

Sadly that still leaves three “No!” answers from the corner.


But if we perform this an infinite number of times… always duplicating our last result… do we fix the problem?

This construction is called Sierpinski’s triangle. It is a fractal that you’ll re-discover all over the place 😉

Yoshiyuki Kotani and I independently explored this problem. I came across Yoshiyuki at the 2016 Gathering for Gardner in Atlanta.

Pinocchio’s Playmates

(MathPickle, 2013)

This is one of the most extensive patterning puzzles. It can be explored deeply in many directions. Here are some puzzle-sheets to get your students started.


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 Know the tools.

Students master the tools at their fingertips - whether it's a pencil or an online app. 

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)