### 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?

### Bracelets

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

### Sweets in a Box

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

# All the Digits

##### Age 7 to 11Challenge Level

Zachary, Stephen and Natasha who are all pupils at Trinity Middle School in Newport, Isle of Wight sent us correct answers to this problem.Stephen said:

I decided that if you don't use a 9 last in the 4 numbers at the top it would work out.

Yes - we agree Stephen, it is the third digit in the 4-figure number which is the sum of consecutive numbers. Well spotted!

Natasha looked up the clues. She wrote the numbers on pieces of card and then jiggled them about.

Phoebe from Watton Junior School said:

I wrote down 1-9 so I could cross off the ones I had used.

I knew that I couldn't use 7, 8, 9 as my consecutive numbers as they are too big. I also couldn't use 1, 2, 3 because 3 had already been used. So it had to be 4, 5, 6.

I knew I couldn't put the 5 as the last digit on the top row because 5 x 3 = 15 which puts another 5 in the total which I can't do.

So then I shuffled the rest of the numbers round until it worked. I then check it all with the instructions and it all was right!

Well done, Phoebe. Thank you for sharing your reasoning.

Kirsty from St Aldhelms School in Poole agreed that it must be the third digit of the 4-digit number which is the sum of consecutive numbers. She explained very clearly how she arrived at her answer:

The only three consecutive numbers that can go in the 4-figure number are 4, 5 and 6. 7, 8 and 9 are too big. The sum of any two of these is greater than 9. For example:
7 + 8 = 15
8 + 9 =17
9 + 7 = 16
0, 1 and 2 cannot go on the first line because:
0 x 3 = 0 (same number twice)
1 x 3 = 3 (same number twice)
Therefore the third number must be 9 (5 + 4) beause 6 + 5 and 6 + 4 are both too big.
The fourth number in the 4-figure number cannot be 5 as 5 x 3 = 15 (repeat digit 5).
The fourth number also cannot be 6 as then we would get 8 twice, so it must be 4.
So, the last two digits must be 5 then 6 so they're not in order.

This is the answer everyone agreed on:

 5 6 9 4 x 3 1 7 0 8 2

Well done, to you all!