We think that you’ll find a pattern in the numbers below.

8 x 8 + 13 =

88 x 8 + 13 =

888 x 8 + 13 =

8888 x 8 + 13 =

88888 x 8 + 13 =

Does the pattern extend indefinitely?

In this problem students are challenged to find a number pattern. They may be able to generalise the pattern more readily if they first work without a calculator and then confirm using calculator.

Whilst much of mathematics is about finding patterns, the greater challenge, as in this problem, is to see how long they continue.

### The Problem

We think that you’ll find a pattern in the numbers below.

8 x 8 + 13 =

88 x 8 + 13 =

888 x 8 + 13 =

8888 x 8 + 13 =

88888 x 8 + 13 =

Does the pattern extend indefinitely?

#### Teaching sequence

- Pose the students the problem and have them complete a couple of calculations.
- Have them continue the calculations in their groups as they look for a pattern.
- Help groups as required though this is a pattern that most groups should be able to see.
- Pose the Extension problem to those who are ready.
- Have a few groups report on what they have done. Make sure that everyone understands the pattern. Take care over the explanation.
- Give all students time to write up their solution. This should consolidate their understanding of what is happening in the pattern.

#### Extension to the problem

Have a look for the pattern here.

1 x 1 =

11 x 11 =

111 x 111 =

1111 x 1111 =

11111 x 11111 =

Does the pattern extend indefinitely?

### Solution

The answers are 77, 717, 7117, 71117 and 711117.

So the conjecture (guess) is that 8…8 x 8 + 13 = 71…17. But we need to be a little more accurate than that. How many 8s will give us how many 1s? Well one 8 gives us no 1s, two 8s gives us one 1 and so on. So it looks as though if we have six 8s we’ll have five 1s, if we have ten 8s we’ll have nine 1s and f we have n 8s we’ll have n - 1 1s. In other words we’ll get one less 1 than we have 8s. But how can we show this?

Let’s forget about the 13 for a minute. Then 8 x 8 = 64, 88 x 8 = 704, 888 x 8 = 7104, 8888 x 8 = 71104 and so on. Each time we add another 8 we add 64 to the first digit of the previous answer. The result is to change 71…104 with say m – 2 1s (coming from m 8s) into 71…104 with m – 1 1s. So one new 1 is added at each step. Now adding 13 at each stage we change the 04 into 17, to give another 1. So if m 8s give m – 2 1s in the 71…104 they give m – 1 1s in the 71…17. This is what we had guessed.

#### Solution to the Extension:

Here the answers are 1, 121, 12321, 1234321 and 123454321. This pattern continues up to 12345678987654321 when carry-overs start to occur and mess up the pattern.