0:00:07.989,0:00:11.291
These are the first five elements[br]of a number sequence.
0:00:11.291,0:00:13.031
Can you figure out what comes next?
0:00:13.031,0:00:14.956
Pause here if you want [br]to figure it out for yourself.
0:00:14.956,0:00:16.030
Answer in: 3
0:00:16.030,0:00:16.818
Answer in: 2
0:00:16.818,0:00:17.731
Answer in: 1
0:00:17.731,0:00:19.358
There is a pattern here,
0:00:19.358,0:00:22.053
but it may not be the kind[br]of pattern you think it is.
0:00:22.053,0:00:26.171
Look at the sequence again[br]and try reading it aloud.
0:00:26.171,0:00:29.251
Now, look at the next number[br]in the sequence.
0:00:29.251,0:00:31.882
3, 1, 2, 2, 1, 1.
0:00:31.882,0:00:37.432
Pause again if you'd like to think[br]about it some more.
0:00:37.432,0:00:38.393
Answer in: 3
0:00:38.393,0:00:39.292
Answer in: 2
0:00:39.292,0:00:40.451
Answer in: 1
0:00:40.451,0:00:43.882
This is what's known as[br]a look and say sequence.
0:00:43.882,0:00:45.572
Unlike many number sequences,
0:00:45.572,0:00:49.450
this relies not on some mathematical[br]property of the numbers themselves,
0:00:49.450,0:00:51.471
but on their notation.
0:00:51.471,0:00:54.312
Start with the left-most digit[br]of the initial number.
0:00:54.312,0:00:58.693
Now, read out how many times [br]it repeats in succession
0:00:58.693,0:01:01.603
followed by the name of the digit itself.
0:01:01.603,0:01:06.894
Then move on to the next distinct digit[br]and repeat until you reach the end.
0:01:06.894,0:01:10.103
So the number 1 is read as "one one"
0:01:10.103,0:01:13.588
written down the same way [br]we write eleven.
0:01:13.588,0:01:17.604
Of course, as part of this sequence,[br]it's not actually the number eleven,
0:01:17.604,0:01:19.153
but 2 ones,
0:01:19.153,0:01:21.804
which we then write as 2 1.
0:01:21.804,0:01:25.414
That number is then read out[br]as 1 2 1 1,
0:01:25.414,0:01:31.984
which written out we'd read as[br]one one, one two, two ones, and so on.
0:01:31.984,0:01:37.765
These kinds of sequences were first[br]analyzed by mathematician John Conway,
0:01:37.765,0:01:40.744
who noted they have [br]some interesting properties.
0:01:40.744,0:01:46.125
For instance, starting with the number 22,[br]yields an infinite loop of two twos.
0:01:46.125,0:01:48.393
But when seeded with any other number,
0:01:48.393,0:01:51.655
the sequence grows in some[br]very specific ways.
0:01:51.655,0:01:54.895
Notice that although the number[br]of digits keeps increasing,
0:01:54.895,0:01:58.885
the increase doesn't seem[br]to be either linear or random.
0:01:58.885,0:02:04.166
In fact, if you extend the sequence[br]infinitely, a pattern emerges.
0:02:04.166,0:02:07.568
The ratio between the amount of digits[br]in two consecutive terms
0:02:07.568,0:02:13.105
gradually converges to a single number[br]known as Conway's Constant.
0:02:13.105,0:02:16.017
This is equal to a little over 1.3,
0:02:16.017,0:02:19.941
meaning that the amount of digits[br]increases by about 30%
0:02:19.941,0:02:22.938
with every step in the sequence.
0:02:22.938,0:02:25.717
What about the numbers themselves?
0:02:25.717,0:02:27.997
That gets even more interesting.
0:02:27.997,0:02:30.296
Except for the repeating sequence of 22,
0:02:30.296,0:02:36.106
every possible sequence eventually breaks[br]down into distinct strings of digits.
0:02:36.106,0:02:38.387
No matter what order these strings[br]show up in,
0:02:38.387,0:02:43.657
each appears unbroken in its entirety[br]every time it occurs.
0:02:43.657,0:02:46.568
Conway identified 92 of these elements,
0:02:46.568,0:02:50.286
all composed only of digits 1, 2, and 3,
0:02:50.286,0:02:52.238
as well as two additional elements
0:02:52.238,0:02:56.969
whose variations[br]can end with any digit of 4 or greater.
0:02:56.969,0:02:59.447
No matter what number the sequence[br]is seeded with,
0:02:59.447,0:03:02.841
eventually, it'll just consist[br]of these combinations,
0:03:02.841,0:03:08.539
with digits 4 or higher only appearing[br]at the end of the two extra elements,
0:03:08.539,0:03:10.969
if at all.
0:03:10.969,0:03:12.839
Beyond being a neat puzzle,
0:03:12.839,0:03:16.659
the look and say sequence[br]has some practical applications.
0:03:16.659,0:03:18.759
For example, run-length encoding,
0:03:18.759,0:03:23.109
a data compression that was once used for[br]television signals and digital graphics,
0:03:23.109,0:03:25.647
is based on a similar concept.
0:03:25.647,0:03:28.590
The amount of times a data value repeats[br]within the code
0:03:28.590,0:03:31.592
is recorded as a data value itself.
0:03:31.592,0:03:36.029
Sequences like this are a good example[br]of how numbers and other symbols
0:03:36.029,0:03:38.700
can convey meaning on multiple levels.