Sunday, 29 September 2013

Three Onions Holding Up A Melon

Who Ordered That?

A favourite Twitter follow is @maanow, with their daily MAA Minute Math problems. These are usually simple little scenarios that take, as implied by the name, a minute or so to solve. But one of the best so far, I don't mind you knowing, took me a little longer than that. It involved three touching 1" radius spheres supporting another sphere of radius 2", and asked: what's the height of this arrangement?
What surprises me about the solution to this one is the appearance of the square root of 69 in the answer. Doesn't that strike you as more than a little bit literally odd?

The solution is arrived at by first noticing that a line connecting the centre of the large sphere to the centre of one of the smaller, touching ones, has length 3 (all dimensions may as well be in inches; it's immaterial to the result). Now, this line is the hypotenuse of a right triangle, whose third vertex is the midpoint of an equilateral joining the centres of the three smaller spheres. This second triangle has side length 2, so the distance from its midpoint to its centre is 2/√3. Squaring both this length, and that of the earlier hypotenuse, en route to the ultimate solution, we discover that curious √69 emerging from the fact that 3³ - 2² = 23 = 69/3.

That's all, just wanted to draw your attention to this little oddity today.

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