1
00:00:07,989 --> 00:00:11,291
These are the first five elements
of a number sequence.
2
00:00:11,291 --> 00:00:13,031
Can you figure out what comes next?
3
00:00:13,031 --> 00:00:14,956
Pause here if you want
to figure it out for yourself.
4
00:00:14,956 --> 00:00:16,030
Answer in: 3
5
00:00:16,030 --> 00:00:16,818
Answer in: 2
6
00:00:16,818 --> 00:00:17,731
Answer in: 1
7
00:00:17,731 --> 00:00:19,358
There is a pattern here,
8
00:00:19,358 --> 00:00:22,053
but it may not be the kind
of pattern you think it is.
9
00:00:22,053 --> 00:00:26,171
Look at the sequence again
and try reading it aloud.
10
00:00:26,171 --> 00:00:29,251
Now, look at the next number
in the sequence.
11
00:00:29,251 --> 00:00:31,882
3, 1, 2, 2, 1, 1.
12
00:00:31,882 --> 00:00:37,432
Pause again if you'd like to think
about it some more.
13
00:00:37,432 --> 00:00:38,393
Answer in: 3
14
00:00:38,393 --> 00:00:39,292
Answer in: 2
15
00:00:39,292 --> 00:00:40,451
Answer in: 1
16
00:00:40,451 --> 00:00:43,882
This is what's known as
a look and say sequence.
17
00:00:43,882 --> 00:00:45,572
Unlike many number sequences,
18
00:00:45,572 --> 00:00:49,450
this relies not on some mathematical
property of the numbers themselves,
19
00:00:49,450 --> 00:00:51,471
but on their notation.
20
00:00:51,471 --> 00:00:54,312
Start with the left-most digit
of the initial number.
21
00:00:54,312 --> 00:00:58,693
Now, read out how many times
it repeats in succession
22
00:00:58,693 --> 00:01:01,603
followed by the name of the digit itself.
23
00:01:01,603 --> 00:01:06,894
Then move on to the next distinct digit
and repeat until you reach the end.
24
00:01:06,894 --> 00:01:10,103
So the number 1 is read as "one one"
25
00:01:10,103 --> 00:01:13,588
written down the same way
we write eleven.
26
00:01:13,588 --> 00:01:17,604
Of course, as part of this sequence,
it's not actually the number eleven,
27
00:01:17,604 --> 00:01:19,153
but 2 ones,
28
00:01:19,153 --> 00:01:21,804
which we then write as 2 1.
29
00:01:21,804 --> 00:01:25,414
That number is then read out
as 1 2 1 1,
30
00:01:25,414 --> 00:01:31,984
which written out we'd read as
one one, one two, two ones, and so on.
31
00:01:31,984 --> 00:01:37,765
These kinds of sequences were first
analyzed by mathematician John Conway,
32
00:01:37,765 --> 00:01:40,744
who noted they have
some interesting properties.
33
00:01:40,744 --> 00:01:46,125
For instance, starting with the number 22,
yields an infinite loop of two twos.
34
00:01:46,125 --> 00:01:48,393
But when seeded with any other number,
35
00:01:48,393 --> 00:01:51,655
the sequence grows in some
very specific ways.
36
00:01:51,655 --> 00:01:54,895
Notice that although the number
of digits keeps increasing,
37
00:01:54,895 --> 00:01:58,885
the increase doesn't seem
to be either linear or random.
38
00:01:58,885 --> 00:02:04,166
In fact, if you extend the sequence
infinitely, a pattern emerges.
39
00:02:04,166 --> 00:02:07,568
The ratio between the amount of digits
in two consecutive terms
40
00:02:07,568 --> 00:02:13,105
gradually converges to a single number
known as Conway's Constant.
41
00:02:13,105 --> 00:02:16,017
This is equal to a little over 1.3,
42
00:02:16,017 --> 00:02:19,941
meaning that the amount of digits
increases by about 30%
43
00:02:19,941 --> 00:02:22,938
with every step in the sequence.
44
00:02:22,938 --> 00:02:25,717
What about the numbers themselves?
45
00:02:25,717 --> 00:02:27,997
That gets even more interesting.
46
00:02:27,997 --> 00:02:30,296
Except for the repeating sequence of 22,
47
00:02:30,296 --> 00:02:36,106
every possible sequence eventually breaks
down into distinct strings of digits.
48
00:02:36,106 --> 00:02:38,387
No matter what order these strings
show up in,
49
00:02:38,387 --> 00:02:43,657
each appears unbroken in its entirety
every time it occurs.
50
00:02:43,657 --> 00:02:46,568
Conway identified 92 of these elements,
51
00:02:46,568 --> 00:02:50,286
all composed only of digits 1, 2, and 3,
52
00:02:50,286 --> 00:02:52,238
as well as two additional elements
53
00:02:52,238 --> 00:02:56,969
whose variations
can end with any digit of 4 or greater.
54
00:02:56,969 --> 00:02:59,447
No matter what number the sequence
is seeded with,
55
00:02:59,447 --> 00:03:02,841
eventually, it'll just consist
of these combinations,
56
00:03:02,841 --> 00:03:08,539
with digits 4 or higher only appearing
at the end of the two extra elements,
57
00:03:08,539 --> 00:03:10,969
if at all.
58
00:03:10,969 --> 00:03:12,839
Beyond being a neat puzzle,
59
00:03:12,839 --> 00:03:16,659
the look and say sequence
has some practical applications.
60
00:03:16,659 --> 00:03:18,759
For example, run-length encoding,
61
00:03:18,759 --> 00:03:23,109
a data compression that was once used for
television signals and digital graphics,
62
00:03:23,109 --> 00:03:25,647
is based on a similar concept.
63
00:03:25,647 --> 00:03:28,590
The amount of times a data value repeats
within the code
64
00:03:28,590 --> 00:03:31,592
is recorded as a data value itself.
65
00:03:31,592 --> 00:03:36,029
Sequences like this are a good example
of how numbers and other symbols
66
00:03:36,029 --> 00:03:38,700
can convey meaning on multiple levels.