The big square is squeeze-packed according to a strict set of rules called an algorithm. Let’s call this algorithm Square Sardine Packing. Can you figure out how it works in this big 9×9 square?

Here is Square Sardine Packing on a 12×12 square.

Try to let your students figure out how Square Sardine Packing works by themselves. It is not easy! Don’t feel badly if they can’t do it. Given these two examples I don’t think I would figure it. On the next slide there will be a hint.

Here is the Square Sardine Packing for a 14×14 square.

Hint: The first small square added was the 1×1 square in the top left. The second square added was the 2×2 square next to it. The next square added was the 3×3 square in the top row. That’s true for all the Square Sardine Packings that you’ve seen. A coincidence?

Try one last time – you might need to look back at the previous two slides.

The 14×14 square starts being built with these smaller squares… We stop at 6×6 just because we can’t add any larger square. How are the small squares added? Each one is added by placing it as high as possible. In the case of a tie place it as left as possible.

Now we do it all again. Starting at 1×1. Trying to place it as high as possible; and in the case of a tie placing it as left as possible. Then a 2×2. Then a 3×3.

This second pass is possible up to 4×4 -larger squares cannot be squeezed in.

Are you ready for the third pass. Again we will start at 1×1…

A third pass is again starts with 1×1 and gets up to 4×4 before being unable to squeeze in anything else.

A fourth pass starts with 1×1 and gets up to 3×3. No larger squares can be squeezed in.

After many passes it is completed:

12%    1×1 squares

14%     2×2 squares

18%     3×3 squares

25%     4×4 squares

13%     5×5 squares

18%     6×6 squares

Choose any square to pack as a class. Before you begin packing a big square – Get your students to guess which of the small square sizes will cover the most area. I like to put the guesses into percentages. In this eight by eight square the 3×3 squares win.

1×1  covers  14/64 = 22%

2×2  covers  16/64 = 25%

3×3  covers  18/64 = 28%

4×4  covers  16/64 = 25%

The 9×9 solution. The 5×5 square covers a whopping 25/81 = 31% of the big square. Is that a record?

The 10×10 solution. The 3×3 squares win here with 27% covered.

Enjoy packing other squares!

Square Sardine Packing

(MathPickle, 2017)

Students try to deduce the rules governing the packing of big squares by little squares.

Math class is where students should first be exposed to the Scientific Method as they try to uncover the truth of some “Mathematical Mini Universe” like this. As complex as the square packing might look at the start it is a lot simpler and less messy than the real world! The scientific Method should be taught by getting kids mucky in the real world in tandem with the nice crisp Mathematical Mini Universes.

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)