Unfair Thrones

(MathPickle, 2013)

This was MathPickle’s first puzzle to get into the New York Times. It is perfect to give students motivation to subtract fractions and turn the result into a percentage.

Start the class by naming an empress and getting her to introduce her bickering children. It’s all fun until the children start squabbling over the kingdom.

The chance that civil war ensues equals the most comfortable minus the least comfortable throne.

Download a printable version here.

Drop a stone into water. It makes a sound, “glop” for a big stone, “splitch” for a small stone. Can you predict the pitch of the sound from the size of the stone? The usual teaching of mathematical theories is like a pyramid. Young people tend to become passive (if passionate) admirers of a structure built by old people, and problems they are taught to solve make them walk straight up to the peak. But what if we want to explore a natural mountain range, whose peaks are invisible among clouds, whose trails among trees are unknown? The problem of the sound of a stone falling into water is natural, so natural that every child knows the phenomenon and can wonder about it. The mathematics involved is extremely hard, so hard that it is not taught at any mathematics department in the world.

Tadashi Tokeida

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