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!

This is problem solving where our students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP3 Work together!

This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

MP6 Be precise!

This is where our 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!

One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist.


Please use MathPickle in your classrooms. If you have improvements to make, please contact us. We'll give you credit 😉

Gordon Hamilton

(MMath, PhD)


Lora Saarnio