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!