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We received some excellent solutions to
this problem. Pupils at Ysgol Aberdyfi Gwynedd wrote to tell
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
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
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
$15$; $24$; $33$; $42$; $51$
$114$; $123$; $132$; $141$; $213$; $222$; $231$; $312$; $221$;
$1113$; $1122$; $1131$; $1212$; $1221$; $1311$; $2112$; $2121$;
$11112$; $11121$; $11211$; $12111$; $21111$
This is a very careful approach, well
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:
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
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