Glass half full puzzle

Problem

How can I reduce a cylindrical glass that is 100% filled with water to 50% filled with water without using any kind of measurement instrument?

Solution

Tilt the glass and pour out the water, until the water just touches the bottom surface of the glass, as shown in the picture below. (Picture courtesy - http://www.quora.com/You-have-a-cylindrical-glass-100-full-of-water-How-can-you-make-it-50-half-full-without-using-a-scale-or-any-type-of-measuring-instrument)

Milkmen in India have known solution to this problem for ages. They been using cylindrical 1 Liter measuring cup to give you half liter milk. They fill up the 1 liter cup, and start pouring the milk till they are just able to see bottom of the cup emerge at one corner. That’s it, you got half cup emptied
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References

• http://www.quora.com/You-have-a-cylindrical-glass-100-full-of-water-How-can-you-make-it-50-half-full-without-using-a-scale-or-any-type-of-measuring-instrument
• http://math-fail.com/2013/02/glass-half-full-riddle.html

Prove that for every prime p, p>3. p^2-1 is always divisible by 24.

Problem:
Prove that p^2 – 1 is divisible by 24 if p is a prime number greater than 3?

Solution:
We can factorize p^2 - 1 to (p-1)*(p+1) = 2*3.4(k), k is some integer

The most elementary proof , without explicitly mentioning any number theory: out of the three consecutive numbers p–1, p, p+1, one of them must be divisible by 3; also, since the neighbors of p are consecutive even numbers, one of them must be divisible by 2 and the other by 4, so their product is divisible by 3⋅2⋅4=24 — and of course, we can throw p out since it’s prime, and those factors cannot come from it.