It would be useful to have done some work on digital roots before
doing this activity, as this will add considerably to what the
pupils can get out of it. It is likely that the children have
already met things like the patterns that are evident in the 9
times table. When children have seen that 18, 27, 36, 45, 54
<i>etc.</i> have digits that add up to 9, then you can go further.
Adding up digits in this way is sometimes called ``digitizing''.
Any number can be digitized. So next year, 1999, would add up to 28
if the digits were summed. The 28 should then be added [2 + 8] to
give 10. The 10 should then be added as a 1 + 0 to give 1. So the
digital root of 1999 is 1. Some further examples would be:- 
<p>1564 <img src="../../mathsf/journalf/aams/images/implies.gif"
alt="--&gt;" /> 16 <img src=
"../../mathsf/journalf/aams/images/implies.gif" alt="--&gt;" />
7<br />
225864 <img src="../../mathsf/journalf/aams/images/implies.gif"
alt="--&gt;" /> 27 <img src=
"../../mathsf/journalf/aams/images/implies.gif" alt="--&gt;" />
9<br />
and so on.</p>
<p>If children are happy with this, and they usually are, then
introduce this idea, perhaps starting from what they notice about
the answers to the 9 times table, before starting this challenge,
as it adds a new dimension of things that can be studied and
enjoyed in the exploration of this set of numbers that you may
recognize as double the triangle numbers.</p>
<p>A useful thing which we have not done in number patterns before
that works well in this investigation is to look at how the numbers
that occur later on in the series can be made from the addition of
previous numbers in the set.</p>
<p><i>e.g.</i> 110 = 20 + 90, all these three numbers being found
in this number pattern from the sending of cards.</p>