Taxi Cab Squares 

(inspired by the Inscribed square problem of Otto Toeplitz, 1911)

Give your students practice with Cartesian coordinates as they explore a new variant of a famous, unsolved problem of Otto Toeplitz (1911).

This problem has a very wide spectrum of curricular engagement: Elementary students will be able to find imbedded squares and create Taxi Cab puzzles of their own; Middle school students will find right angle triangles the key to searching for solutions; High school students will find Cartesian connections with parallel and perpendicular line segments.

Printable puzzle-sheet here

It’s clearly a budget. It’s got a lot of numbers in it
George W. Bush

MathPickle’s original theme from 2011 was Amoeba Squares.

William Boeck (upper left) wrote: “I would also like to add I did this in less than 45 minutes!” Daniel Mascadri (lower left) wrote: “I finished this problem!!!!!!!!!!!!!!!! Is this worth $50 or 1,000,000?” Teachers – please remind students that the reward is NOT for finding a square inside one of the loops that my students came up with. The reward is for finding a loop for which no such square exists. This is hard. Everyone (including me) has failed to find such a loop.
What are the positives and negatives of using prize money. Does it take away the joy of learning for learnings sake? Does it inspire a subset of students who would otherwise be ambivalent? Does it distract the classroom? Are huge monetary rewards ($1,000,000) the best and cheapest way of disseminating awesome problems worldwide? If prizes are to be used – what is the best way to do it? Email me your opinions 😉

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)