As a “words person”, you may like the fact that I am asking you to look at each line and say it out loud. No arithmetic is required! Figure 1 below shows the first six rows of a number sequence. Can you guess the next row of numbers?
How the look-and-say sequence works
I have maths populariser, Adam Spencer, to thank for drawing my attention to this puzzle. These rows of numbers make up what is known as the look-and-say sequence. This is how it works:
- The first row is a given. We simply decided to start at 1.
- For the second row, look at the first row and say what you see out loud: One 1. Write: 1 1.
- For the third row: Look at the second row. What do you see? Two 1s. Ignore the plural and write 2 1.
- For the fourth row: Look at the third row and say what you see out loud: One 2, one 1. Hence, we write: 1 2 1 1
- For the fifth row, we read the contents of the fourth: One 1, one 2, two 1s:
1 1 1 2 2 1 - The sixth row is a read out of the fifth: Three 1s, two 2s, one 1: 3 1 2 2 1 1
Notice that, when a digit is repeated, e.g. 1 1, we read this repetition as two ones and write 2 1.
Now that you know how the sequence works, have another guess for the next row and then read on to verify your answer.
Solution
The seventh row comes from reading out out of the sixth row:
- One 3, one 1, two 2s, two 1s: 1 3 1 1 2 2 2 1
Let’s start somewhere else
So far, we have built our rows of the look-and-say sequence starting at 1. The one number that occupies the first row is called the seed. Let’s see what happens if we started with 2 as the seed number.
I will get you started and ask you for the fifth row.
- Row 1: We simply write down the seed number: 2
- Row 2: Read row 1: One 2. Write: 1 2
- Row 3: Read row 2: One 1, one 2. Write: 1 1 1 2
- Row 4: Read row 3: Three 1s, one 2. Write: 3 1 1 2
The curious case of the number 22
What happens if we use 22, twenty-two, as our seed? Notice the sequence:
- Row 1: 22
- Row 2: Read two 2s. Write: 22.
- Row 3: Read two 2s. Write: 22.
- etc.
Further reading
This video shows mathematician John Conway telling the story of the look-and-see sequence. Another sequence you might find interesting is the Fibonacci sequence. It is a famous sequence of numbers that occurs in nature and that has spawned a genre of poetry. Check out this post about it.
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