In this mini-competition your students try to find a set of non-negative integers for which the mean, median and mode are integral. There are six separate challenges… one for each permutation. The one here is for median < mode < mean. Does it work?


The median and mode are the same. This set fails.

There are many sets which will succeed. How do we determine which is best?  The best is the one whose largest element is as small as possible. If there is still a tie, I’ll let your students figure out the way to break the ties.


Does this work?


No – the median is not an integer.


Does this work?


No – the mode is not integral – it is a set of integers.


Does this work?


Yes – this works.

The largest integer in the set is 22. Is that good?

No – it is not good. The set worked, but don’t forget that this is a mini-competition. Your objective is to try to get that largest element as small as possible. That 22 is too high. If you can find a set with a lower high value, it will beat the set we’ve just discovered.
Here is an attempts to find a solution to median < mean < mode. Does it work?
Yes – this works although the 21 is too high.
Now it is the time for your students to compete with one another and collaboratively against me, and other classes. Do not show the next slides at the start of the period. You may choose never to reveal them. They are my best results after an hour of searching.
These are my best results after an hour of searching. It is likely that some improvements are still possible…

Mean, Median and Mode mini-challenge (MathPickle, 2015)

This puzzle gets students involved in calculating mean, median and mode. As well as teaching these statistical terms, it is also appropriate for an algebra challenge.

Compared to other MathPickle challenges this one has a relatively high entry level. Top students will get into it, but some average students will not know where to begin.  Instead of starting with the challenge above; try experimenting with just two of mean, median and mode.  For example, just use Median < Mode.  The set with the smallest largest number might be {0,1,3,3}, but you might want to ask the class to improve upon something that looks like this: {0,0,1,2,3,4,5,6,7,7,7}. An ugly looking solution like this can be obtained by asking students to contribute elements to the set or say “stop.” Most sets acquired in this way will be horribly poor and that’s where you want to begin.

If you want a super hard challenge, investigate the same challenge with mean, median, mode and standard deviation.

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)