Eights

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

1. Show the students the problem and let them do a couple of calculations.
2. Send them off into their groups to continue the calculations and look for a pattern.
3. Help the groups that need it though don’t be too generous with your help. This is a pattern that most groups should be able to see.
4. Groups that finish early with a good explanation should go on to the Extension.
5. Get a few groups to report on what they have done. Make sure that everyone understands the pattern. Take care over the explanation.
6. Give all students time to write up their answers. This should help them to understand what is going on.

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.