### 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?

# Missing Multipliers

##### Stage: 3 Challenge Level:

Ewan from Wilson's School sent us this solution explaining how he worked out the headings after  revealing eight of the cells.

Bradley from Bream Bay College shared his strategy for working out the headings after revealing seven of the cells:

I revealed the first 4 horizontal answers and the first 4 vertical answers. That enabled me to work out the headings of the first column and row. Once I had worked out those headings I could work out the rest of the grid.

Here is an example:

 X 2 3 8 10 4 8 12 32 40 6 12 9 18 8 16

Mollie, Jasmine, Zander, Thomas, Nicholas and Geor, from St Peters CEVC Primary School in Easton sent us this solution:

We found out that if you start with 4 diagonal numbers from top left to bottom right and two other numbers, you can always solve the problem, unless one or more of the numbers is a square number- (except 16 & 36) - in which case you could do it in less.

Square numbers are very useful because both the multiplying numbers are the same. We have always solved the problem in 6 or less.

Michael from Wilson's School also had a strategy for working out the headings after revealing just 6 cells:

The way I do it is to reveal the diagonal from the top-left to the bottom-right, then the second up on the left and the top-right.

Here is an example of this:

 X ? ? ? ? ? 24 22 ? 14 ? 36 18 ? 132

This method provides 2 columns and 2 rows with 2 numbers.

You then look for a common factor in one of the rows or columns with two numbers:
e.g. 22 and 132 = 11.
From this, you could work out that the heading for the bottom row is 132/11 = 12, and that the heading for the top row is 2.

This then makes it very easy to find the other '?'s
e.g. 24/2 = 12, therefore the heading of the first column is 12.

When all of this example has been worked out, it looks like this:

 X 12 2 6 11 2 24 22 7 14 3 36 18 12 132

Alexander, also from Wilson's School, sent us this solution that showed that a similar strategy could be used when the six exposed cells are in a different position.

We also received good solutions from Annie, Sean, Jake G. and Julie T., all from East Vincent Elementary School in the United States, and Shaun and Jack from Wilson's School.

Editor's comment: it is possible to reveal six cells which do not include cells on the leading diagonal (from top left to bottom right) and still work out the headings. Some of you may want to think about which combinations of six cells you would want to reveal.