(MathPickle 2021)

This geometric square-gluing puzzle is one of my best from 2021. The rules are simple (but probably best seen on the video rather than read):

1) Start with two 1×1 squares labelled “1.”

2) Glue a new square labelled with one more than the last square’s label.

3) Adding the labels of all squares that the new square shares a portion of an edge equals the new square’s label. 

4) The edge length of the new square must be smaller or equal to its label.

With these simple rules try to get as many squares glued together as possible using a blend of geometry and arithmetic.

Elementary school students will revel in creating something beautiful. I’ll add their beautiful solutions here. Please email me the photos you would like me to post to:

Older students may wonder why I have constrained the size of the squares. If I start with two 1×1 squares labelled 1, but let my squares labelled 2 and 3 be large, you can show that you can get an arbitrarily large number of squares glued together.

Computer Scientists may enjoy the extra challenge of finding optimal solutions when starting with an additional 1×1 square labelled 1.


Teaching is an experimental science. Don’t expect to be a great teacher your first year standing in front of a class. 


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)