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.