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Today Puzzle #593

Puzzle No. 593鈥� Monday 21 October

I inflate one of two identical spotted balloons until all its spots are an inch across, and the other until all its spots are a foot across. Now I join the necks of the balloons, letting air flow from one into the other, pinching them off again when the spots on the larger balloon have decreased to 10 inches across. How big now are the spots on the other balloon?

Today’s #PuzzleForToday has been set by School of Mathematics and Statistics at the University of Sheffield

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The spots are 9 inches across. Volume scales as the cube of linear dimensions. So if at the beginning we call the volume of the smaller balloon one unit, then the volume of the larger balloon is 12x12x12=1728 units, since the spots are 12 times as wide. That is a total volume of 1729 units. After the exchange of air, the larger balloon has 10x10x10=1000 units volume, leaving 729 for the (still just) smaller balloon. Since 729=9x9x9, this implies that the spots on the smaller balloon are now 9 inches across (perhaps surprisingly large).

Writing in 1940, Cambridge professor G.H. Hardy recounted of the Indian mathematical prodigy Ramanujan:

''I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. ''No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." (1729=12x12x12+1x1x1=10x10x10+9x9x9.)

The Unitarian chapel in Fulwood, Sheffield, has the date 1729 above the door. One of our colleagues likes to attend for that reason.

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