Can you find the next number in this sequence? - Alex Gendler

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1, 11, 21, 1211, 111221. These are the first five elements of a number sequence. Can you figure out what comes next? Alex Gendler reveals the answer and explains how beyond just being a neat puzzle, this type of sequence has practical applications as well.

Lesson by Alex Gendler, animation by Artrake Studio.

These are the first five elements
of a number sequence.

Can you figure out what comes next?

Pause here if you want
to figure it out for yourself.

Answer in: 3

Answer in: 2

Answer in: 1

There is a pattern here,

but it may not be the kind
of pattern you think it is.

Look at the sequence again
and try reading it aloud.

Now, look at the next number
in the sequence.

3, 1, 2, 2, 1, 1.

Pause again if you'd like to think
about it some more.

Answer in: 3

Answer in: 2

Answer in: 1

This is what's known as
a look and say sequence.

Unlike many number sequences,

this relies not on some mathematical
property of the numbers themselves,

but on their notation.

Start with the left-most digit
of the initial number.

Now, read out how many times
it repeats in succession

followed by the name of the digit itself.

Then move on to the next distinct digit
and repeat until you reach the end.

So the number 1 is read as "one one"

written down the same way
we write eleven.

Of course, as part of this sequence,
it's not actually the number eleven,

but 2 ones,

which we then write as 2 1.

That number is then read out
as 1 2 1 1,

which written out we'd read as
one one, one two, two ones, and so on.

These kinds of sequences were first
analyzed by mathematician John Conway,

who noted they have
some interesting properties.

For instance, starting with the number 22,
yields an infinite loop of two twos.

But when seeded with any other number,

the sequence grows in some
very specific ways.

Notice that although the number
of digits keeps increasing,

the increase doesn't seem
to be either linear or random.

In fact, if you extend the sequence
infinitely, a pattern emerges.

The ratio between the amount of digits
in two consecutive terms

gradually converges to a single number
known as Conway's Constant.

This is equal to a little over 1.3,

meaning that the amount of digits
increases by about 30%

with every step in the sequence.

What about the numbers themselves?

That gets even more interesting.

Except for the repeating sequence of 22,

every possible sequence eventually breaks
down into distinct strings of digits.

No matter what order these strings
show up in,

each appears unbroken in its entirety
every time it occurs.

Conway identified 92 of these elements,

all composed only of digits 1, 2, and 3,

as well as two additional elements

whose variations
can end with any digit of 4 or greater.

No matter what number the sequence
is seeded with,

eventually, it'll just consist
of these combinations,

with digits 4 or higher only appearing
at the end of the two extra elements,

if at all.

Beyond being a neat puzzle,

the look and say sequence
has some practical applications.

For example, run-length encoding,

a data compression that was once used for
television signals and digital graphics,

is based on a similar concept.

The amount of times a data value repeats
within the code

is recorded as a data value itself.

Sequences like this are a good example
of how numbers and other symbols

can convey meaning on multiple levels.