Can you find all the sets of five positive whole numbers that have

a mean of 4

a median of 3

and a mode of 3?

One solution is 3, 3, 3 ,5 and 6 and there are several other solutions.

There are 11 solutions with a single mode and 3 solutions with two modes.

If the range is 10, what solutions can you find?

How many sets of 5 positive integers can you find with mean 8, median 7 and mode 7? There are more than 50.


  1. The teacher thinks of any 5 positive numbers and tells the class their mean, median and mode.
  2. The class are given 2 minutes to find sets of 5 numbers with the given mean, median & mode.
  3. After 2 minutes players score a point for each solution that they have found.

See  Notes for Teachers – M, M and M

This activity is adapted from the NRICH task M, M and M with permission of the University of Cambridge. All rights reserved.

7 Responses to M, M and M

  1. Buyi Taho says:

    I gave it to Grade 10 learners. Most of the learners easily identified that the sum five numbers must be 20 for them to get the mean of 4.

    One learner identified that if the mode is 3, it means the numbers are repeated. They found 6 solutions including one with a range of 10.

  2. toni says:

    Good work.

    There are a few more solutions. Can your class find them? Can they find a method of listing the solutions so they can check from their list that they have found them all?

  3. Chris Clarke says:

    Were learners able to find one solution quickly at least? Perhaps they can all try to find one each, then work in pairs or groups to see how many different ones they can find in a group. Finally you could get the whole class working together to list all of the solutions. I think you are correct though if they do not have a good understanding of what the mean, median and mode are then they will surely struggle!

  4. Elizabeth Turok says:

    We need to use different skills as teachers when we are asking learners to try problem solving.
    We need to have a strategy in our head of how to guide them in the right direction.
    Starting with a simple example so that they understand what they are trying to do is a good introduction.
    Encouraging learners to work on the problem in small groups is effective with regular sharing of progress with the whole class.
    Its a learning experience for the teachers as well as the learners.With practise it gets easier.

  5. Nyameka Boya says:

    I gave my grade 8 learners this problem and they tackled it with ease.They enjoyed it.

  6. toni says:

    Did they find other solutions? Once the learners realise that there are many solutions it is not at all difficult to find the solution with the range of 10.

  7. toni says:

    Grade 11 should be able to find all the solutions.

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