# 3 by 3 Mathdokus

## Problem

In this 3 by 3 Mathematical Sudoku, you need to use the clues available to fill the nine cells.

Clicking on the purple cog in the top right corner allows you to access twelve different puzzles at different levels of difficulty.

If you are not familiar with Mathdokus, you may like to watch this introductory video, which gives you some ideas of the strategies you may find useful.

This Mathdoku belongs to a set, which also contains 4 by 4, 5 by 5 and 6 by 6 grid sizes.

**If you are considering sending us solutions, we are particularly interested to hear about any elegant strategies you used when you were stuck, and you think are worth sharing.**

**NRICH would like to thank Tetsuya Miyamoto, a Japanese maths teacher, whose puzzles have inspired our Mathdokus.**

## Student Solutions

Thank you to everybody who sent in their solutions to this activity.

We received a couple of solutions from children in Dry Drayton, England. You can click on the pictures to see the full-size solutions.

Annabel and Alice sent in this explanation of how they solved one of the puzzles:

Michael and Harry sent in this step-by-step explanation of how they solved the same Mathdoku:

Perri-Jane at Halstead Prep School in England explained how they solved one of the Mathdokus. They use the letter x to represent an unknown number:

This is how I worked out this 3x3 mathdoku:

Firstly, I filled in the numbers that are already given to me, 1 and 2.

Since they are in the same column, I can deduce that the bottom left corner box would be 3.

Now that I know that is 3, I have to minus 1, which is 2. So, the only number left would be 1.

Next, I have to find the value of x if x-1=1. This would leave the answer of 2. That means the only number left is 3.

Now I have 2 blank spaces, one in the middle and one in the top row, middle. I can deduce that the middle one is 3 as it is the only number left for that row and 1 for the top one, following the same procedure.

Now I know that the top row is 2,1 and 3. The middle row is 1,3 and 2. The bottom row is 3,2 and 1.

Thank you all for sharing your methods with us!

We would love to hear from more of you, particularly about any strategies you used when you were stuck. Please email us if you have a solution you'd like to share with us.

## Teachers' Resources

### Why do this problem?

Mathdoku grids are a motivating context for learners to develop fluency with number bonds, and factors and multiples, as well as providing an opportunity to reason mathematically.

### Possible approach

*The 4 by 4 version of this problem featured in the NRICH Primary and Secondary webinar in October 2022.*

Display the interactivity and, without saying much else, invite learners to consider what they notice and what questions they would like to ask. Give them time to think on their own, then talk to a partner, before drawing everyone together. Facilititate a whole group discussion, using the points raised to explain how the Mathdoku grid works. It would be useful to introduce the vocabulary of 'cages' and squares. Alternatively, you might like to watch this demonstration video, which you could pause as you wish.

Ask for suggestions about where we might start. Which square might we fill in first? Emphasise that you are particularly interested in their reasoning. How do they know that the number they are offering *must* go in that square? Can they convince the rest of the class and you?

If you have not watched the introductory video, you may wish to demonstrate how to seek help from the interactivity if learners are not sure which square is possible. (Clicking on 'Show me a square I can solve' will result in a yellow box appearing around a square which is solvable. Clicking on 'Give me a hint about this square', will suggest how you might go about working out the number in that square.)

You can continue in this way with the whole group for as long as you feel is appropriate. Once everyone has got the idea, you can ask learners to complete the grid in pairs, either using the interactivity on a tablet or computer, or using a printed copy (clicking on the purple cog in the top right allows you to select a version of the grid which you can print using the browser printing option). As they work, listen out for examples of children's watertight reasoning, which could be shared with the whole class in the plenary.

You may wish to display a new grid in the plenary for the class to solve together, so they have chance to practise creating chains of reasoning using their knowledge of number and calculation.

### Key questions

What are the *possible options* for this square? How do you know?

Is there any other information in the grid that could help us narrow down the possibilities?

Can you convince me/someone else that this number *must* go in this square?

### Possible support

The interactivity has built-in hints which will help all learners access this challenge. Many children will find it useful to have paper and pencil to hand to jot down possibilities for the square they are working on (this could be a print-out of the grid, but could simply be plain paper).

### Possible extension

Once learners have tried all the grids in the interactivity (see the Settings menu), or on paper, you could offer them larger grid sizes: 4 by 4, 5 by 5 and 6 by 6. You could also challenge them to create their own Mathdoku in pairs. Their grid must have a unique solution and they can give it to another pair to solve.

Learners may also like to have a go at one of NRICH's Sudokus, which contain the numbers 1-9 in each row, column and three by three grid (currently they are paper based only). A First Product Sudoku would be a good starting point.