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These are the first five elements of a number sequence.
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Can you figure out what comes next?
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Pause here if you want to figure it out for yourself.
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Answer in: 3
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Answer in: 2
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Answer in: 1
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There is a pattern here,
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but it may not be the kind of pattern you think it is.
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Look at the sequence again and try reading it aloud.
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Now, look at the next number in the sequence.
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3, 1, 2, 2, 1, 1.
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Pause again if you'd like to think about it some more.
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Answer in: 3
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Answer in: 2
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Answer in: 1
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This is what's known as a look and say sequence.
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Unlike many number sequences,
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this relies not on some mathematical property of the numbers themselves,
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but on their notation.
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Start with the left-most digit of the initial number.
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Now, read out how many times it repeats in succession
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followed by the name of the digit itself.
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Then move on to the next distinct digit and repeat until you reach the end.
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So the number 1 is read as "one one"
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written down the same way we write eleven.
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Of course, as part of this sequence, it's not actually the number eleven,
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but 2 ones,
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which we then write as 2 1.
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That number is then read out as 1 2 1 1,
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which written out we'd read as one one, one two, two ones, and so on.
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These kinds of sequences were first analyzed by mathematician John Conway,
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who noted they have some interesting properties.
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For instance, starting with the number 22, yields an infinite loop of two twos.
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But when seeded with any other number,
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the sequence grows in some very specific ways.
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Notice that although the number of digits keeps increasing,
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the increase doesn't seem to be either linear or random.
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In fact, if you extend the sequence infinitely, a pattern emerges.
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The ratio between the amount of digits in two consecutive terms
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gradually converges to a single number known as Conway's Constant.
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This is equal to a little over 1.3,
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meaning that the amount of digits increases by about 30%
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with every step in the sequence.
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What about the numbers themselves?
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That gets even more interesting.
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Except for the repeating sequence of 22,
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every possible sequence eventually breaks down into distinct strings of digits.
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No matter what order these strings show up in,
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each appears unbroken in its entirety every time it occurs.
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Conway identified 92 of these elements,
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all composed only of digits 1, 2, and 3,
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as well as two additional elements
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whose variations can end with any digit of 4 or greater.
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No matter what number the sequence is seeded with,
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eventually, it'll just consist of these combinations,
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with digits 4 or higher only appearing at the end of the two extra elements,
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if at all.
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Beyond being a neat puzzle,
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the look and say sequence has some practical applications.
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For example, run-length encoding,
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a data compression that was once used for television signals and digital graphics,
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is based on a similar concept.
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The amount of times a data value repeats within the code
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is recorded as a data value itself.
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Sequences like this are a good example of how numbers and other symbols
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can convey meaning on multiple levels.
