### 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?

### Adding All Nine

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!

### Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

# Product Sudoku

### The Basic Rules of "Product Sudoku"

Like the conventional Sudoku, this Sudoku variant consists of a grid of nine rows and nine columns subdivided into nine $3 \times 3$ subgrids. Like the Sudoku Classic, it has two basic rules:
1. Each column, each row, and each box ($3 \times 3$ subgrid) must have the numbers 1 to 9.
2. No column, row or box can have two squares with the same number.

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

### A Short Demonstration

The square in the top left corner of this Sudoku contains the number 20. 20 is the product of the digits in the two adjacent squares, 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.

A word document containing the problem can be found here, with an explanation for classroom use.
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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