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All the Digits

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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 all four agreed on:

5 6 9 4
x 3

1 7 0 8 2

Well done, particularly to Kirsty for her excellent reasoning!