How To Find The Next Number In The Sequence?

I recently encountered this puzzle type and I really struggled to find the next number in the sequence. So, I thought I would write about it.

Interestingly, this sequence belongs to a family of sequences. The beauty about this family is that it presents very interesting and practical properties in theoretical as well as applied mathematics.

If this sort of thing is right up your alley, read along. But I suggest that you give this sequence a shot on your own first. It can be challenging, but at the same time, rewarding as well.

Beyond this point, I will be explicitly discussing the solution to this sequence. So, consider this a spoiler alert if you wish to solve it on your own. Without any further ado, let us begin.

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Misguided Adventures

My initial approaches to solving this puzzle involved a lot of mathematical manipulation. Multiplication; addition; exponentiation; you name it. I tried it all with no dice.

I even tried to approach the sequence with a cryptographic drift, and converted the sequence numbers to binary form by replacing 2s with 0s. Again, no luck.

One approach that showed a little promise was digit-position-manipulation. But I couldn’t extend the logic beyond one pair of numbers in the sequence; failure yet again.

In the end, the answer turned out to be so remarkably simple, yet so remarkably genius!

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How to Find the Next Number in the Sequence?

The trick to finding the next number in the sequence is to realise that the sequence is not mathematical. Instead, it follows a simple pattern that just involves counting and reading the numbers aloud.

I will reveal the answer straight away. See if you can pattern match and figure out what is going on here.

If you are having difficulties recognising the pattern, worry not. I will explain what is going on here.

Let us start with the first number in the sequence: 3. What do you see? Well, a three. In other words, “one three”. Write this out in decimal numbers, and you have: 13, which makes up the second number in the sequence.

Now, apply this logic recursively once again to 13. What do you see? “One one, one three.” So, the next number in the sequence is: 1113.

Applying the logic recursively yet again, we arrive at next number: “Three ones, one three” (3113). This leads us directly to our solution with the next number: “One three, two ones, one three” — 132113.

There we go. A simple sequence, right? But hold on. This sequence has more to it that that meets the eye. Next, let us see what it has up its sleeve.

The Look and Say Sequence — Revealed

This family of sequences was originally popularized by John Conway under the name “Look-and-say sequence”, after one of his students challenged him at a party.

While analysing the sequences, Conway noticed some interesting properties. For instance, each new number in the sequence has roughly 30% more digits than the previous number.

All sequences, but one, grow indefinitely. As the number of digits increases, the ratio of the number of digits in the nth generation number to that of the (n − 1)th generation number converges on one number.

Since Conway proved this property, this number came to be known as Conway’s constant (λ = 1.303577269034…).

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Final Thoughts

A cryptographer named Robert Morris also helped popularise this sequence family via his contribution to a puzzle in Clifford Stoll’s puzzle book titled The Cuckoo’s Egg.

Based on this, when the sequence begins with a seed (first number) of 1, it is known as the Morris number sequence.

While we are on the topic of the seed, let me ask you an interesting question. What do you think happens when we start the sequence with a seed of 22? Let us find out:

22, 22, 22, 22, 22, …

This is the lone degenerate series that I alluded to earlier; the only series that does not grow indefinitely.

Even considering all these interesting properties, you would think that this sequence family would not be practically relevant. But this is not the case.

For instance, the properties of this sequence family are used in the field of signal processing to extract data from repeating signal data.

I might cover some of the other properties of this sequence family in a future essay. But for now, what other interesting property did you note from this sequence that I did not cover already?

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Further reading that might interest you: 

• Tricky Logic Puzzle (VII) – How To Really Solve It?
• The Strong Law Of Small Numbers
• How To Really Solve This Fun Geometry Puzzle? (II)

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