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• 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.

• 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.

• 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.

• 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.

These are the first five elements of a number sequence.

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# 【TED-Ed】你可以在這個序列中找到下一個數字嗎？ - 亞歷克斯·甘德勒 (Can you find the next number in this sequence? - Alex Gendler) ()

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X69599596 に公開 2017 年 07 月 29 日