### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

### Have You Got It?

Can you explain the strategy for winning this game with any target?

# Product Sudoku

### Why do this problem?

This is no ordinary Sudoku because it requires mathematical knowledge in addition to logical thinking. This problem offers an engaging context in which to apply knowledge of factors, multiples and prime factor decomposition.

### Possible approach

These printable resources may be useful: Product Sudoku,
Product Sudoku a Possible 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.

Since this product Sudoku is quite challenging, you may wish to start with A First Product Sudoku and Multiples Sudoku

Display the first one without explaining anything and in silence fill in two adjacent cells.
"I'm going to fill in a few more cells but I'm not going to explain what I'm doing.
If you can work out what I'm doing and can suggest some other numbers then put up your hand.
Please don't spoil it for anyone else by giving away what's happening.
The title of the problem may give you a clue."

Add contributions from the class, in each case asking for a justification, and continue until you feel they have the idea. Provide printed copies for those who wish to complete the problem in their own time and offer them a chance to try Product Doubles Sudoku before they move onto the Product Sudoku.

Offer pairs of students one printed copy of the Product Sudoku and one copy of the blank 9 by 9 grid - the 'journey grid'.

Warn them that this is a little more challenging! On the Sudoku they are going to write the solution. On the journey grid they are going to record the order in which they fill in the cells, from 1-81.You are expecting them to convince each other of the accuracy of their suggestions before anything gets added onto their papers.

When a pair has finished, they put their journey grid on display so that everyone's journeys can be compared. As a plenary, invite comments on any similarities and differences they noticed between their own and others' solutions.

### Key questions

Some clues have lots of possibilities and some have few. Which are which?
Which are the most helpful clues to begin?

### Possible extension

Can one pair take another pair's journey grid and retrace their steps?
You may wish to try Diagonal Product Sudoku

### Possible support

Spend longer on A First Product Sudoku

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