# Mathdoku

The idea of this task is to complete the four by four grid so that the numbers 1, 2, 3 and 4 appear only once in each row and only once in each column.

You will see that some squares, or groups of squares, are outlined with a thick black line. These groups of squares are called 'cages'.

In the corner of one of the squares in a cage, you will see a small target number and usually a mathematical operation too. For example, if a cage of three squares has '8x' in the corner of one of the squares, it means that the three numbers in that cage multiply together to make 8.

If the cage is made up of only one square, which has a number in its corner but not an operation, that number is telling you exactly what goes in that square.

Have a go at the grid in the interactivity below. Clicking on a square allows you to type in a number, or choose a number from the options given. You can change your mind just by typing/choosing another number instead.

**How would you convince someone else that the number you have put in a square must be correct?**

Click on the Settings menu (the purple cog in the top right-hand corner) to select another grid to try. You can increase the level of difficulty in the Settings menu too.

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.

You may also like to take a look at the mathematical Sudokus feature, which includes also 3 by 3, 5 by 5 and 6 by 6 Mathdokus.

Are there any 'cages' which are single cells? The labels for these squares simply tell you the number inside the square, so that can be a good starting point.

Is there a cage containing three squares in a line? Can you work out the three numbers, even if you don't know their order? How does that help you work out the value of the remaining square in that row or column?

If you have cages of two squares, you may not be able to know for certain what the value of a particular square is, but can you tell what the *other two* squares in the row or column must be? You will need to use information from other cages to be sure of the values.

If you are stuck, click on the button 'Show me a square I can solve'. A question mark will then appear in a square which is solvable.

If you need to, you can then click on 'Give me a hint about this square', which will suggest how you might go about working out the number in that square.

Well done to everybody who solved these puzzles. We had lots of correct solutions sent in, but we've only included the solutions where children explained their reasoning for how to solve the puzzles.

Dhruv from St Anne's Primary School in the UK sent in a picture of his solution to the first puzzle. You can click on the picture below to enlarge it:

Thank you for sending in this solution, Dhruv - it's interesting to see the order in which you filled in the cages!

Viva from Halstead Preparatory School in the UK sent in her solution to the first puzzle, and she explained how she solved this:

- Row 4 column 4 must be 4
- Row 1 column 4 must be 2
- Because row 4 columns 1 and 2 must be 1 or 2, column 3 must be 3
- To get a total of 7 in row 2 column 3, then row 2 column 4 must be 3 because there is already a 4 in column 4
- Therefore row 3 column 4 must be 1
- Row 3 column 3 must be a 2 (2 x 3 x 1 = 6)
- Row 1 column 3 is 1 and therefore row 1 column 2 = 3
- Row 2 column 2 must be 1 because it can’t be 4 but must be either (4 divided by 1 = 4)
- Row 4 column 2 must be 2
- Row 4 column 1 must be a 1 to get 1 x 2 = 2
- The rest of the numbers in column 1 are obvious to get the sequence for each row

Well done for writing out your reasoning step by step, Viva!

Harriet from Halstead Preparatory School gave some general advice for how to solve the puzzles:

I put the numbers in that I knew only had one square, then I worked out which numbers were possible for the other squares using the sums which were shown. When I had 3 numbers in a row, I knew which the 4th number was in any row or column.

Thank you for sharing your thoughts with us, Harriet - it's difficult to give general advice as each puzzle is so different!

Ci Hui Minh Ngoc from Kong Hwa School in Singapore sent in solutions to all of the puzzles as a PDF. Ci Hui Minh Ngoc's full solution shows how you can use the factors of a number to work out all the possibilities when the cage has a multiplication operation in the corner. Well done for solving all of these puzzles!

### Why do this problem?

Mathdoku grids are a motivating context for learners to develop fluency with number bonds, factors and multiples, as well as providing an opportunity for learners to reason mathematically. Having experience of creating chains of reasoning provides an excellent basis on which to create proofs.

### Possible approach

*This problem featured in an NRICH Primary webinar in January 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.

Ask for suggestions about where we might start. Which square might we fill in first? (There are two cages which are each single cells - top right and bottom right - so they could be completed straightaway in either order.) Invite learners to suggest which square we might fill in now. At this point, 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?

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 question mark appearing in 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 (this sheet contains three different grids, corresponding to the three grids which are rated as 'difficulty level 1' in the interactivity). 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). This feature also contains 3 by 3 Mathdokus, which some learners might find useful to try before the 4 by 4 grids.

### Possible extension

Once learners have tried all the grids in the interactivity (see the Settings menu), or on paper (Mathdoku2.pdf and Mathdoku3.pdf), you could 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.

Some of the Mathdokus in this feature offer further challenge due to their increased grid size.

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.