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.
https://forums.civfanatics.com/threads/about-a-test-tao-took-when-he-was-9.673888/
http://math.fau.edu/yiu/Oldwebsites/MPS2010/TerenceTao1984.pdf
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