There is one great minaret at Samarra, but what happened if Caliph al-Mutawakkil decided to build twelve!?!


The Caliph loved beautiful geometry, so if he had built twelve minarets, he might have chosen to build them in an array with 90 degree rotational symmetry. Let’s see how that might work…


He builds one minaret.


Because the first minaret was built… and because we have 90 degree rotational symmetry… we know another must be built here…

and another here…

and another here…

This looks like its going to be easy, but there is one other rule that the Caliph decides to follow.

He wants that the singers on all 12 minarets can see each other.

All these four can see each other, but we must make sure that we don’t have three minarets in a line.

Let’s build another…




Because of rotational symmetry, we must build three more…


Everything is still looking good. No three minarets are in a line.

We need to build again (remember we need to build 12.)


Because we have 90 degree rotational symmetry we must build three more…


Here they are… and it looks that we have no three minarets on the same line. Is that right? The green line certainly looks good.

No! We have failed. This line has three minarets. Not all the singers can see each other.

This is your first task – build twelve minarets in Samarra.

At the end of this slide show there will be downloadable puzzle-sheets.


For the next challenge we are travelling to the court of Suleiman the Magnificent in Istanbul in the 1500s. You are an apprentice city planner working under Mimar Sinan, the chief architect of Suleiman the Magnificent. Again, your job is to build minarets so that no three lie in a line…


Your task is more difficult – you have 20 minarets to construct, but at least you have been given three to start… and told that the final plan must have mirror symmetry as shown.


…because of this line of mirror symmetry – which new minarets must be built?

One of the minarets sits right on the line of symmetry, so we only needed to build two more minarets.

Now let’s consider this line of symmetry. Where do we build new minarets because of this line?

We have ten of our twenty minarets already built. We KNOW these are right since the initial three minarets were correct, but now we must venture forth into the unknown…

Does this work or not?

This line is good…

…but again we have an error.

Instead of always getting bigger and bigger as you move through these puzzle-sheets, you can try to find ALL the solutions for cities 4×4 and smaller.


There is nothing wrong to flip the tables and give your top students the task of finding ALL small solutions while giving middling students the thrill of tackling the toughest problems BEFORE the top kids tackle them. The video below actually shows grade one students tackling the 4×4 solutions. I have never had a grade one student find all solutions.

Download puzzle-sheets here.


No three minarets in a line

(Dudeney, 1917)

The puzzle presented in the slide show above is for students working on symmetry. The video on the right shows the related unsolved problem posed by Henry Dudeney in 1917 used in a younger grade.

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)