Circus Tent Puzzle The best tents are the ones that have the smallest perimeter. Both of these tents have a perimeter of 7. That's the best you can do for $17. Each triangle is worth $3. Each square is worth $4.
Given a tent price find the smallest perimeter. Let's start by finding the smallest perimeter possible for a $6 tent. Try it.
There is only one way to stitch together $3 triangles and $4 squares to make a $6 tent. This is it and the perimeter is 4. What about a $7 tent?
Again, there is only one way to stitch it. The perimeter is 5.
These simple initial ones may be skipped through quickly if the class is advanced - otherwise, this is a great place for students to link geometry and arithmetic.
Let's fast forward to a $16 tent...
These are both good designs giving the minimal perimeter of 8.
These are both good designs giving the minimal perimeter of 8.
For $19 the lowest perimeter I could find was 9. I'm choosing a unitless measure for perimeter, but in your classroom you should use yards or meters - whatever is standard in your country.
For $20 the lowest perimeter I could find was 8. By no means should you trust that I have found the lowest perimeter. As the price gets higher and higher my guesses get less and less certain.
If some of your students would find it inspiring to be asked to jump ahead and work on $30 to $40 tents, let them. It is often children in the middle or bottom of a class that will feel most inspired by being ask to go forth and blaze new territory.
This just looks like an ugly solution for the $22 tent. Let's keep looking for something with a smaller perimeter. Let's remove 3 squares and add 4 triangles. That will keep the cost fixed.
That's better. If your students do end up beating my answers (which is inevitable) I'd love you to brag to me about it 😉
It is surprising that all shapes with minimal perimeter are not convex. Some are concave. The way I would describe "concave" to a class is to say that two wasps can play hide and seek in a concave tent like the ones above. They can find positions where they cannot see each other. This is not true of the next few convex tents...
This looks great, but...
This is better.
Let's fast forward to $30. My favourite part of this puzzle was from $30 to $40 where things were just difficult enough to make me feel uneasy...
There are actually lots of equally good solutions for 27...
I'm starting to really enjoy the minimalist beauty of some of these creations.
This probably has the least perimeter. I'm sad because it looks a lot less cool than the fox mask on the previous slide.
This can't be optimal, but I have not found anything with a smaller perimeter.
This escape pod is my favourite shape yet. I don't mind if students start to diverge from the tent theme into a star wars theme. That shows that they understand the malleability of the superficial story whilst preserving the mathematics underneath.

... but before I get excited about a score of 10... Glen Whitney asked (July 3rd, 2021) if the shape created has to be solid. If a hollow shape is allowed he can do better than the space pod on the previous slide. He can get a perimeter of only nine. What do you think? The great thing about playful mathematics is that you get to make the rules!

Your students should be expected to ask big questions for which they are ill-equipped to find the answers. That natural curiosity to wonder beyond their ability is great to behold...
I'm in the same place. I ask questions - most of which are beyond my ability or perseverance to answer: I wonder if triangles will eventually take over nearly all of the lowest-perimeter answers. Of course some squares are needed for those prices which are not a multiple of three.
...but perhaps from here on, multiples of three only contain triangles...
If I find a particularly pleasing shape like the one above, I'd like to know if it is the lowest perimeter solution for other square and triangle prices. For example, if triangles are $5 and squares are $7 - is this shape a minimal perimeter shape for $68. Let's jump ahead...
Now I'm just enjoying jumping ahead to beautiful solutions... forgetting the base ones in-between. Flip this on your students - and for a homework assignment - ask them to find "the most beautiful tent" under $100. This is a great homework assignment where the arithmetic and simple geometry is consumed by a loftier aesthetic.
We have enough experience now that we can guess that this won't work...
...and we would be right. But this doesn't look that great either... but I'm too lazy to figure out right now if this is as low a perimeter as we can achieve.
I'll wager a lot of money that this is the best solution for $72.
But I'd wager only a little that 18 is the smallest perimeter for $132... especially since it can be made with all triangles. I'm off to make another puzzle. Enjoy exploring this one.

Glen Whitney asks - what is the smallest perimeter that can contain a set of polygons costing $n. An n-sided polygon costs $n. This problem gets tougher a lot faster than our puzzle.

Circus Tent Puzzle

(MathPickle, 2015)
This puzzle links geometry and arithmetic. For younger students, manipulatives are needed or the teacher must retain control. This is because young students will make too many mistakes doing free-hand drawings. Junior high students can learn to sketch equilateral triangles and insure that the sum of angles meeting at a point adds to 360 degrees.

 Circus Tent Puzzle II

(Glen Whitney, 2021)

Glen Whitney suggested a generalized problem that you pay $n for a regular n-sided polygon. I recommend his version for older children because there are aha! moments regarding pentagons, hexagons and bigger polygons. For younger children working with manipulatives, the original Circus Tent Puzzle that uses only $3 triangles and $4 squares is going to be both more cost-effective and makes talking about the sums of angles around a point easier. Both puzzles turn out to be equally challenging.

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)