This thread goes over a possible Python implementation for the look-and-say sequence.

  • 1
  • 11
  • 21
  • 1211
  • 111221
  • 312211
  • ???

I challenged you with this sequence earlier.

Now I'll explain the answer.

I will also show you a nice Python ๐Ÿ implementation for this sequence.

Here we go ๐Ÿš€

https://twitter.com/mathsppblog/status/1503823606899519489

First, how does the sequence work?

This sequence is called the look-and-say sequence.

Why?

Because the next term is created by looking at the previous one and saying what you are seeing.

We start with 1. What do you see?

You see โ€œone 1โ€ โ†’ 11.

Now you have 11. What do you see?

You see โ€œtwo 1sโ€ โ†’ 21.

Now you have 21. What do you see?

You see โ€œone 2 and one 1โ€ โ†’ 1211.

Keep doing this until you reach 312211. What do you see?

You see โ€œone 3, one 1, two 2s, and two 1sโ€ โ†’ 13112221.

Thus, the next term in the sequence would be 13112221.

Trivia fact: all terms in this sequence have even length!

(Well, except the starting 1 ๐Ÿคช)

Another cute fact:

As the sequence grows, the lengths of the terms tends to increase by ~30% each time.

What does this mean?

Say you walk along the sequence and find a term with length 1000.

Then, the next term will have length ~1300!

Isn't it insane that this is a sequence described by words AND maths can prove this fact?

I wrote about it in a TIL article of mine ๐Ÿ‘‡

https://mathspp.com/blog/til/look-and-say-sequence-growth

But hey, let's jump straight into the implementation.

The key here is understanding that we want to look at groups of consecutive equal digits, right?

That's how we go from 111221 to 312211 to 13112221.

111 | 22 | 1 31 22 11 โ†“ 3 | 1 | 22 | 11 13 11 22 21

How can we do this in Python?

This grouping functionality is perfect for one tool from the itertools module:

groupby

groupby returns consecutive keys and groups from an iterable.

The keys are the unique elements and the groups are the runs of unique elements.

Here are some examples to show how groupby works.

  • Notice how the keys are the unique consecutive elements.

  • If we don't convert groups to lists first, we can't get the length of the group.

So, do you see where this is going? ๐Ÿ‘‡

>>> from itertools import groupby

# The keys are the unique elements:
>>> [k for k, _ in groupby("AAABAADDDCC")]
['A', 'B', 'A', 'D', 'C']

# The groups are iterables with the consecutive elements:
>>> [list(g) for _, g in groupby("AAABAADDDCC")]
[['A', 'A', 'A'], ['B'], ['A', 'A'], ['D', 'D', 'D'], ['C', 'C']] 

# We can compute the length of a group:
>>> [len(list(g)) for _, g in groupby("AAABAADDDCC")]
[3, 1, 2, 3, 2]

# We can pair keys and length of groups to count elements:
>>> [(len(list(g)), k) for k, g in groupby("AAABAADDDCC")]
[(3, 'A'), (1, 'B'), (2, 'A'), (3, 'D'), (2, 'C')]

By using a similar structure as the last line of code, we can get pretty far with our look-and-say implementation ๐Ÿ‘‡

>>> def look_and_say(digits):
...     return [(len(list(g)), k) for k, g in groupby(digits)]
...
>>> look_and_say([1, 1, 1, 2, 2, 1])
[(3, 1), (2, 2), (1, 1)]

However, the result isn't a flat list of digits... It's a list of tuples.

How can we flatten this?

itertools to the rescue again!

One of the best ways to flatten a list of lists is with itertools.chain.from_iterable:

>>> from itertools import chain

>>> list(chain.from_iterable(
...     [(3, 'A'), (1, 'B'), (2, 'A'), (3, 'D'), (2, 'C')]
... ))
[3, 'A', 1, 'B', 2, 'A', 3, 'D', 2, 'C']

So, by putting together chain and groupby, we have ourselves an implementation!

Bonus Q: what if you wanted your function to accept an integer and return an integer?

E.g., look_and_say(111221) โ†’ 312211.

Can you modify the function below to handle that case?

>>> from itertools import chain, groupby

>>> def look_and_say(digits):
...     return list(chain.from_iterable( 
...         (len(list(g)), k) for k, g in groupby(digits)
...     ))
...
>>> look_and_say([1, 1, 1, 2, 2, 1])
[3, 1, 2, 2, 1, 1]

I hope you learnt something from this thread!

Follow me @mathsppblog for more educational threads like this ๐Ÿ˜„

Also, FYI, you can read this thread โ€“ and all my other threads โ€“ on my blog:

https://mathspp.com/blog/twitter-threads

TL;DR:

  • the look-and-say sequence works by โ€œreading out loudโ€ the digits;
  • itertools has a tool groupby that groups consecutive elements in an iterable; and
  • itertools has a tool chain that you can use to flatten a list of lists.
from itertools import chain, groupby

def look_and_say(digits):
    return list(chain.from_iterable( 
        (len(list(g)), k) for k, g in groupby(digits)
    ))

If you want to read more about the look-and-say sequence, you can go through this article.

I hope you learned something new! If you did, consider following the footsteps of the readers who bought me a slice of pizza ๐Ÿ•. Your small contribution helps me produce this content for free and without spamming you with annoying ads.

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