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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Product Sudoku

##### By Henry Kwok

### Rules of Product Sudoku

### A Short Demonstration

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### Adding All Nine

### DOTS Division

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Age 11 to 16

Challenge Level

* Product Sudoku printable worksheet
Product Sudoku printable worksheet for recording your journey
Product Sudoku printable worksheet to see a possible journey*

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

1. Each column, row and $3 \times 3$ subgrid must have the numbers 1 to 9.

2. No column, row or subgrid can have two cells with the same number.

The puzzle can be solved with the help of the numbers in the top parts of certain cells. These numbers are the products of the digits in all the cells horizontally and vertically adjacent to the cell.

The cell in the top left corner of this Sudoku contains the number 20. 20 is the product of the digits in the two adjacent cells, which therefore must contain the digits 4 and 5. The 5 cannot go in the cell below the top left hand corner because 5 is not a factor of 96 (the product shown in the third cell down on the left hand side of the puzzle). Therefore 5 must be entered into the cell to
the right of the cell containing 20 and 4 in the cell below.

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!

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.