Cuboid computation
You are given a ruler which can be extended indefinitely, and an unlimited supply of 1-inch sticks of negligible diameter. They may be glued together to form a cuboid.
If you construct a cuboid, you can measure the length from one vertex to another, and in doing so "compute" the square root of a certain number by measuring a particular diagonal.
What is the natural density (or asymptotic density) of positive integers whose square roots cannot be "computed" in this way?
The answer is 0.166666.
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1 solution
The trick is to notice the question is really asking what natural numbers can't be writen as the sum of three squares.
Great, luckily for me Legendre worked this one out already. See Legendre's three-square theorem .
It states a number can't be writen as the sum of 3 sqaures iff it's of the form 4 a ( 8 b + 7 ) .
We can see from this that the density should be 8 1 n = 0 ∑ ∞ 4 n 1 = 6 1