About a test Tao took when he was 9

Discussion in 'Off-Topic' started by Kyriakos, Oct 27, 2021.

  1. Kyriakos

    Kyriakos Creator

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    Terence Tao is a famous mathematician, now 46 years old (going by wiki).
    When he was 9, he was presented with a test, and while he answered most of the questions correctly, there were some he didn't get right. Among the later was the following:

    If you write all integers, from 1 to 99,999, how many times will "1" appear?

    Tao made a calculation, essentially trying to tell for each order of magnitude distinctly.The approach was not optimal though, and the question is of the kind that you'd answer far more easily if you had already come across such a thing. There are a few approaches and usually the method used is formulaic (it's also hinted at in the video) : you notice that from 0 to 99999 there are 100000 numbers, also notice (if you ever took probability theory, or even more suitably power sets, this would be your first thought) that any digit (0-9) has to appear an equal number of times as any other digit in this list of numbers, and then all you have to do is calculate how many total digits there'd be in those 100.000 numbers. The total digits are 500000 (because you write each of the numbers as a five digit one; eg 1= 00001), so "1" appears there 1/10 of the time: 50.000 times in total.

    I thought of a different approach, which is a bit more in the style of Tao at the time, but without his attempt to count 1s from all five digits in a towering progression instead of finding a pattern (that led him to come up with a wrong answer). More importantly, this approach doesn't require you to know of the tidbit from powersets about equal distribution (ok, you could think of it on your own, but it's still cheating if you knew :) ).

    For each of the 5 digits, you have 10.000 of 1s, eg from 10.000 to 19.999 you already have 1 in 10.000 numbers by only counting the 1 in the highest order digit, and since each appearance of 1 from the immediately lower digit is 10 times less frequent (has to run through 0-9 entirely to get consequent hits) but also in 10 times more numbers overall (since unlike with the top order 1, the second order 1 appears also before 10000 and after 19999), the step for the second digit also numbers 10K in total. The same follows for the third, fourth, and fifth and final digit, so all five steps number each 10.000 appearances of 1, for a total of 50000.

    You can read a few of the questions (including the one mentioned here) in the video:
    (starts at around 4.25 for the 99999 one)

    There's also a pdf of the complete test (as noted in the Lazy thread, I am too lazy to look it up, but you can :) )
     
    Last edited: Oct 27, 2021
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  2. Senethro

    Senethro Overlord

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    This sort of thing highlights to me that even though I use and work with numbers more than the average human, I'm still just using them as a model for things in the real world and have little intuitive sense for their strangenesses.. Whereas, this guy lives there.
     
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  3. Kyriakos

    Kyriakos Creator

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    He certainly had some incentive (not support; obviously he was a genius) from his parents due to their professions also being in science. That can be really the difference sometimes - if the child is already gifted :)

    A nice quote by Tao is that (I paraphrase, but not much) every true statement in math is a tautology, but you can't always see that because the types of different math objects are so many.
     
  4. Ferocitus

    Ferocitus Deity

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    Quite a remarkable family!
    Terry's brother Trevor is very much "on the spectrum" and was studying for his PhD when I was at Adelaide.
    He used to sign himself as Happy Starman in all of emails back then. :)

    Terry cites Basil Rennie as one of his influences. Basil was a very odd man. After moving to Queensland
    he decided to invent a pair of shorts that would lie flat on an ironing board to make ironing them easier.
    Unfortunately, the shorts had a thick gusset that made it look like a shorts/diaper combo.
    He lectured wearing just those shorts, sans shirt and shoes, for many years.

    Here's a beautiful photo of Terry with the remarkable Paul Erdos that was taken in our Maths tea-room when
    I was there.
    Paul_Erdos_with_Terence_Tao.jpg
     
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  5. amadeus

    amadeus Living in my own private I-dunno

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    Easy, once. Gimme my scholarships now!
     
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  6. Kyriakos

    Kyriakos Creator

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    Sure, it's this way: (points to the alley) :D
     
  7. Hrothbern

    Hrothbern Deity

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    If the numbers you use are RL numbers, then they are quite differently from for example the collection of numbers Tao got.
    In RL numbers, in the natural collections of numbers, the probability that the leading digit is a 1 is approx 30% and in Tao's set approx 11%.
    (Benford's Law)
    IIRC there was a financial fraud detected that way, because the made up numbers were random and not in compliance with Benford's Law.
     
    Last edited: Oct 29, 2021
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