Times Tables and Digit Squaring

This is a great bit of fun which will focus on your knowledge of square numbers as well as on normal times tables. It is also a good exercise in looking for patterns.

The idea is to make number chains by following a simple rule and this is how this particular rule works:

Think of a number and write it down. Any number will do, but it is best to start with a two digit number as you will soon see.

I will choose the number 37.

The rule is simply to square each of the digits in the number and add them up.

The square of 3 is 9 and the square of 7 is 49 and when we add these we get 58.

Simple enough, you might think. So now we carry on the chain by doing the same to 58.

The square of 5 is 25 and the square of 8 is 64. The total of 25 and 64 is 89, so our chain now looks like this:

Occasionally the answer to the total will have just one digit, in which case just square it (so, for example the number 11 will lead to the number 2. 2 squared is 4 and 4 squared is 16 and you are back to two digits again). In some cases you may find you have three digits, in which case square all three and add these up.

Carry on with the chain until you notice some interesting things. For instance, you may notice the same number keeps popping up, so you are going round in circles. You could redraw this bit of the chain as a circle with some other numbers leading into it. I don’t want to do this here as that will be giving the game away.

It is possible to put all the numbers up to 100 (with a few larger than 100 that pop up now and then) into a large diagram, so that each number appears once and only once. The question is, ‘’Will this be one large diagram in which every number is connected to another number in the diagram, or will it have several sections that are not joined to the other sections?”

Okay, here’s one tiny clue that may help. Sooner or later you will be using the number 37 in the chain and if you square the digits and add them up, the total is 58. But, of course, 73 has the same digits as 37, so when you square them and add them up, the total will be 58 too.

So, on your diagram both 37 and 73 will lead to 58. In this way, we can gradually include all the numbers.

This will happen to lots of pairs such as 64 and 46.

You will need to work very carefully as a few simple errors can mess up the whole investigation.

And you will probably need quite a bit of paper (or you could write very small!).

If you are really nuts, you could count how many calculations you have done from the times tables facts. Have fun!