4 by 4 Mathdokus
1 out of 3
Can you use the clues to complete these 4 by 4 Mathematical Sudokus?
Problem
In this 4 by 4 Mathematical Sudoku, you need to use the clues available to fill the sixteen 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 3 by 3, 5 by 5 and 6 by 6 grid sizes.
If you would prefer to work away from the computer, you can print out these grids: Mathdoku grids of difficulty 1, Mathdoku grids of difficulty 2, Mathdoku grids of difficulty 3.
NRICH would like to thank Tetsuya Miyamoto, a Japanese maths teacher, whose puzzles have inspired our Mathdokus.
Student Solutions
Lots of children shared their tips for completing these Mathdokus - thank you all for sending in your ideas!
Tamsyn from Frederick Irwin Anglican School in Australia sent us this explanation:
One very helpful strategy for me is to use the the numbers in the top left corner. This helps me know which numbers are possible in each square so that I don't have to choose from as many options and the answer becomes clearer to me.
e.g. If multiplied together two numbers that have to equal 4, I know that it has to be a 4 and a 1. This is because I can't do 2x2 or something like that. I will also often find that the one has to go there because all the other squares that are in a row don't have a one which means that number needs to be a 4. This tells me that the other number is a one.
Good ideas, Tamsyn - often you can use the fact that you need the numbers 1, 2, 3 and 4 in every row and column to fill in some missing boxes. I wonder why it isn't possible to have 2 and 2 in a cage with 4x in the corner?
Uday from Pate's Grammar School in England sent in a solution explaining their first couple of steps. Have a look at Uday's PDF solution and see if you can work out what to do next after the third picture, using the general method that Uday describes.
Nicolas from St Charles Catholic Primary School in Ryde, Australia sent in an explanation of how to solve the first 4 by 4 Mathdoku. Take a look at Nicolas' full PDF solution and see if you can work out how to finish the Mathdoku by starting from the last picture on the first page and using the 'domino effect' that Nicolas talks about. Does this strategy work for all of the 4 by 4 Mathdokus? Have you spotted any Mathdokus where more strategies are needed to fill in all of the boxes without making any guesses?
We also received a step-by-step explanation for the first 4 by 4 Mathdoku from the children at the Maths' Big Questions club at Lady Margaret School in the UK. Have a look at the Maths' Big Questions club's full PDF solution. Can you see how they knew that the numbers they chose were definitely correct at every step?
We would love to hear from more of you! We are particularly interested to hear about any elegant strategies you used when you were stuck, and you think are worth sharing. Email us if there's a solution you'd like to share with us.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
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
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. Facilitate 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: 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. A First Product Sudoku would be a good starting point, followed by Multiples Sudoku, Product Sudoku and LCM Sudoku.