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## The Amazing Tao

On July 16, 1983, Terence Tao was asked the following set of seven questions (though he was not yet 8 years old).

[1] Two circles have a radii equal to 2 cm and 3 cm. The distance between their centers is 4 cm. Do they intersect?

[2] What angle does an hour hand describe in 20 minutes?

[3] A can of kerosene weighs 8 kg. Half of ther kerosene is poured out of it, after which the can weighs 4 1/2 kg. What is the weight of the empty can?

[4] What time is it now if the time which passed since noon constitutes a third of the time that remains until midnight?

[5] I walk from home to school in 30 minutes and my brother takes 40 minutes. My brother left 5 minutes before I did. In how many minutes will I overtake him?

[6] The perimeter of a right-angled triangle is 5 cm. Two of its sides are each 2 cm long. How long is the third side?

[7] A class received some regular and some special notebooks, and altogether there were 80 notebooks. A regular notebook costs 20 cents and a special one 10 cents.How many of each kind of notebook did the class receive?

At this time, Terence was being interviewed by Ken Clements as part of an evaluation of mathematically gifted children. He answered all of the questions orally...never wrote anything down...and was correct on all seven questions.

Today, at age 31, Terence Tao is a mathematics professor at UCLA, having completed his Ph.D. at age 20. Two of his specialities are prime number ideas and the compression of images. He has already won a Fields Medal and a MacArthur Fellowship.

Source: K. Chang's "Journey's to the Distant Fields of Primes," New York Times, 3/13/2007

Hint: No hints...as Terence received no hints.

Solution Commentary: Terence's responses:

[1] Yes. If they didn't intersect, the distance between their centers would be more than 5.

[2] Simple. 1/3 of 1/12 of a full circle is 1/36th of a circle. 1/36th of 360o equals 10o.

[3] You get an algebraic equation, but it's hard to work out in your head. Weight of can + weight of kero = 8. Weight of can + 1/2(weight of kero) = 4 1/2. So weight of kero = 7 kg. weight of can = 1 kg.

[4] 1 unit + 3 units = 12 hours. So 1 unit = 3 hours. So time is 3 p.m.

[5] 35 minutes. If you started at the same time as your borhter you'd arrive 10 minutes before him....Oh no. 15 minutes because then you'd both be halfway.

[6] The thrid side is 1 cm. That can't be true, by the way. Pythagoras' theorem says it has to be...SQRT(8)...It's impossible.

[7] I don't know really (laughs). r + s = 80. All you're given is the cost. It can't be done. Could be 40 regulars and 40 specials, or 50 regulars and 30 specials.