# Multiples Sudoku

## Problem

*Multiples Sudoku printable worksheet*

By Henry Kwok

#### Rules of Multiples Sudoku

Like a conventional Sudoku, this Multiples Sudoku has two basic rules:

- Each column, row, and 3 x 3 subgrid must have the numbers 1 to 9.
- No column, row, or subgrid can have two cells with the same number.

The puzzle can be solved with the help of the numbers which are placed on the border lines between selected pairs of neighbouring cells.

These numbers are the product of the two digits in the cells to the left and right of the clue.

For example, where there is a 12 on the line between two neighbouring cells, the cells must contain 2 and 6, or 6 and 2, or 3 and 4, or 4 and 3.*You may be interested in the other problems in our Playful Puzzling Feature.*

## Getting Started

In the third row, 30 and 18 appear on adjacent border lines. Which set of three numbers could be placed in the first three cells of that row?

In the fifth row, 8 and 18 appear on adjacent border lines. Which set of three numbers could be placed in the last three cells of that row?

Alison and Charlie kept a record of the order in which they filled the Sudoku.

They filled the cell marked 1 first, then the cell marked 2, then the cell marked 3...

You might like to retrace their route to fill in the cells in the same order, though this is just one possible route through the problem.

*You can print their journey here.*

## Student Solutions

Thank you to everyone who sent in a solution. There were too many good solutions to mention everybody here, but this is a selection of what we received.

Most people said they began by using the clues that are the products of the numbers on their left and right. Spandan from MGIS Ahmedabad in India, George and James from Walton High School in the UK and Hannah and Amy from Priestlands School in the UK described listing factor pairs to help them. George and James said:

To start with, we listed all of the factors [of] the numbers in between two cells, provided they were both between 1 and 9 inclusive. We started with numbers with only one pair of factors, like 20, 21 and 54. This was made easier when we realised that some numbers that had two different factor possibilities could only have one solution because of all of the numbers around them that we had already filled in; some wouldn't work with the rest of the puzzle's rules. The cell with 8 in the middle could be 2 and 4 or 1 and 8. However, 2 was already in the same line.

Clara from Bangkok Patana School in Thailand, Alistair from King's School Grantham and Danielle from Greenfield Community College used common factors, and Clara also used some other properties of the numbers. Clara said:

I started off looking for prime and square numbers. 2 is a prime number so then one of the squares next to it had to be 2 and the other one 1. Then moving on to square numbers there was 9 and 4. You can not use the square root of these numbers because then the same number will be repeated twice which is one of the most important rules of sudoko. Therefore you must go with 1 and 9 (for an example) instead of 3 and 3.

If I had two answers that were next to each other such as the 9 and 18, I would see what common factors they had. In this case it was 9 and I knew that the two boxes next to 9 had to be either 1 or 9 as I said before. This then meant that I could put 9 between the 2 boxes and 1 and 2 on either side.

Daniella also used this method for some of the other numbers:

Then I did the 30 and 18 in the top left box. 5 6 3 (since the common multiples of 18 and 30 that are less than 10 are 3 and 6, and 30 as 3$\times$10 is not allowed).

Clara then said:

After that, I looked for numbers with few factors such as 21. 21's factors are 1,3,7,21. I could not use 1 and 21 for this so I had to go for 3 and 7. I could also do this for many other larger numbers because many of them have only one set of factors where both numbers are under 10. After having found quite a few numbers I was able to solve the puzzle just as I would solve a normal sudoko.

This is Hannah and Amy's completed sudoku:

## Teachers' Resources

### Why do this problem?

This Sudoku offers an engaging context which requires students to think logically and apply their knowledge of factors and multiples.

### Possible approach

*These printable resource may be useful: Multiples Sudoku*

*Multiples Sudoku Journey*

If your students do not know the rules of Sudoku then set aside a little time for them to become familiar with the 'standard' Sudoku.

Work together with the class filling in a few cells to make sure everyone understands the rules - the 3rd, 5th and 7th rows offer opportunities for filling cells easily at the early stages.

Then hand out the sheets and invite students to work in pairs, emphasising that they must convince each other that their suggestions are correct, before anything gets added onto their papers.

### Key questions

Some clues have lots of possibilities and some have few. Which are which?

Which are the most helpful clues to begin?

### Possible support

Provide students with this possible journey through Multiples Sudoku and suggest they try to retrace the route.

For other Factors and Multiples problems that might help to prepare your students for this task, see Missing Multipliers, Dozens, the Factors and Multiples Game, and the Factors, Multiples and Primes Short Problems collection.

### Possible extension

For more challenges on Factors and Multiples, see Gabriel's Problem or the Factors, Multiples and Primes Short Problems collection.