This puzzle should be explored after Square Sardine Packing.

Choose some shapes – order them from first at the top (green rectangle) to the last at the bottom (purple rectangle). Now use the same algorithm that you used in Square Sardine Packing to pack the square. In case you forget that algorithm – I’ll show it to you on the upcoming slides.

Challenge #1: You win if you can get the percentages covered by all shapes to be equal. There are ways to “cheat” – find them.


So we start by placing the first shape. We always try to cover the highest square possible. In the case of a tie – we try to cover the leftmost square possible. Unlike Tetris you cannot rotate the shapes.

Now get ready to place the second shape…

Placing the second shape. Where does the third shape go?

It has to go as high as possible. Since there is a tie – we look for the leftmost placement.

Time to place the next shape… the T

Finally it is time to place the last shape… the purple rectangle.

This is one complete pass. Now we start again at the top. Can you visualize how the packing will look like after the second pass? After you’ve predicted an answer – check it out on the next slide.

Imagine what it will look like after the third complete pass…

Imagine what it will look like after the fourth complete pass…

Imagine what it will look like after the fifth complete pass…

Imagine what it will look like after the sixth complete pass…

For the first time we will fail to get all five shapes fitted in.

The T failed to find a spot during that last pass. That’s ok. Keep going…


After many complete passes we get the solution. Did we win challenge #1? Did all shapes cover the same percentage of the square?




No – the T covered 25%, while the green triangle covered only 14%.

The difference 25% – 14% = 11%.

That’s pretty good, but can you make it zero? Of course there are ways to cheat. Find them. I’ll share some ways to cheat on the following slides. After we find some ways to cheat we’ll look for some more exciting solutions. Cheats will look boring.


Will these squares produce a solution where all colors cover the same percentage?


No – the orange covers 28%.

The others colors cover 24% each.

So that cheat didn’t work. Remember – we want all to cover the same percentage.


Will this work?


Sure – this is a trivial kind of solution. It is kind of a cheat. There are lots of rectangles that will do this.


Here is another uninteresting solution. How do we add a rule to make these not happen? One rule we could add is that the shapes should all be different – or at least in different orientations. Will that get rid of all the silly cheats? No…


The big shape has area 50. It doesn’t need to be a rectangle – there are many boring shapes that it could be. You can guess what the solution will look like…


Not too interesting. So we need to add another rule. Perhaps it should be that we need to use at least four shapes… or perhaps that the largest shape should be smaller than area 10.

I like the first idea more. Rule: You must use at least four shapes.


So find a set of four or more shapes that are not the same so that the percentages are equal. I found a solution after twenty minutes of searching. The next page will give my solution, but you should find your own.


I used two shapes the same, but they are oriented differently so you might give me a thumbs up for good effort, but nevertheless tell me that my solution is wrong 😉

Challenge #2: Look for a set of five shapes so that the percentage covers are all different… and the highest percentage is as small as possible.

Dot to Dot Median Path Puzzles

(MathPickle, 2012)

This puzzle is excellent to work at with a WHOLE class. It is less good for students to work at in small groups, because errors propagate… One error will wreck the solution.

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)