Picasso’s cuboid puzzle uses volume in an unequal battle between a Greedy and Generous Cuboid.

20 can be written as the sum of the volumes of two cuboids… a Greedy one and a Generous one…

All three dimensions of the Greedy Cuboid {3,3,2} are at least as big as the three dimensions of the Generous Cuboid {2,1,1}. This must always be true.

How small can you make the Greedy Cuboid? 18 is too big. 

This is best. We’ve managed to limit the Greedy cuboid to 3x2x2 = 12.

What about for 25? Split 25 between the Greedy and Generous Cuboid. Remember that all the dimensions of the Greedy Cuboid must be greater or equal to all the dimensions of the Generous Cuboid…

 Sadly – we cannot help much. The Greedy Cuboid gets nearly everything ;-(

What about for 26?

 26 we can help a lot! The Greedy Cuboid takes only 18 – leaving 8 for the Generous Cuboid.

What about 27?

Oooh – that is ugly. Greedy Cuboid got nearly everything.

What about 30-40?

Picasso’s Cuboids (MathPickle, 2016)

This puzzle sequence gives practice in volume calculations. Here is a classroom presentation.

After looking at the presentation above, you may choose instead to use an area variant of the puzzle. In this case, for an integer n, you would be searching for four variables

a ≧ b ≧ c ≧ d

such that

a*b + c*d = n

and c*d is as large as possible.

You may also wish to experiment with a hybrid of volume and area… a ≧ b ≧ c ≧ d ≧ e  such that a*b*c + d*e = n  or a*b + c*d*e = n

Please be the first to email me about how this puzzle worked in your classroom: gord at mathpickle dot com

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)