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We received some excellent solutions to this problem. Pupils at Ysgol Aberdyfi Gwynedd wrote to tell us:

The answer we got was that there were thirty-one different numbers with digits that total the number $6$.

We started off in groups to try and find out the answer, and the answer varied between twelve and eighteen therefore we knew that there were obviously more and that we needed to be more organised.

Therefore we started with two-digit numbers, then three-digit numbers, four-digit numbers, five-digit and then six. We saw that the numbers were reverse of each other and we also saw some palindrome numbers.

The smallest number was $15$ and the largest was $111 111$.

We knew that we got them all because of the way we worked it out.

Two-digit numbers:

$15$; $24$; $33$; $42$; $51$

Three-digit numbers:

$114$; $123$; $132$; $141$; $213$; $222$; $231$; $312$; $221$; $411$

Four-digit numbers:

$1113$; $1122$; $1131$; $1212$; $1221$; $1311$; $2112$; $2121$; $2211$; $3111$

Five-digit numbers:

$11112$; $11121$; $11211$; $12111$; $21111$

Six-digit numbers:

$111111$

This is a very careful approach, well done.

Ben and Charlie from Brewood Middle School sent in exactly the same list of numbers and explained how to make sure you find all the solutions:

Work systematically.

Start with a two-digit number (that adds up to six), when you
have finished writing all the two-digit numbers go on to
three-digit, four-digit, five-digit and six-digit numbers.

For all the digits that you make, start with the smallest
number and then the second smallest and then the third smallest and
then the fourth smallest etc. This way you will not miss out any
numbers.

Swap the numbers around, for example: $11112$ then $11121$
then $11211$ then $12111$ then $21111$.

Make sure no numbers are repeated.

Well done too to Ha Young from Wesley College who also found these solutions. Children from Oakhampton Primary School decided that there are in fact thirty-two solutions because they included the single-digit number $6$ as well. Good thinking!