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Math Chat: Roads, Tennis, and the Sun

Old challenge

What is the shortest road network connecting the four corners of a square? What about the six vertices of a regular hexagon?

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The "double Y" of Figure 1 with length 3 + 1 or about 2.73 (on a unit square) beats the popular answer of the X of length 22 or about 2.83, as well as the U of length 3.

It is a general principle that at junctions in shortest networks the roads always meet in threes at angles of exactly 120 degrees. Perhaps our interstate highways should meet in threes to save concrete, and decrease congestion as a side benefit!

Incidentally, soap bubbles also meet in threes at 120-degree angles for the similar reason of minimizing surface energy.

Ron Douglass, Darrell Krulce, John Kruschke, W. T. Pu, and Erik Randolph all realized that the shortest road network connecting the six vertices of the regular hexagon of Figure 2 is just five of the sides, which can get you from any vertex to any other, although by a very circuitous route from the upper left to the upper right. The alternative pictured branching network has length 6.

This example of "Graham's problem" is not easy to prove and stumped mathematicians for years, until J. H. Rubinstein and D. A. Thomas published a long, complicated proof in 1992. If you can find a simple proof that the five sides are best, you win instant mathematical fame.

Tennis paradox

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Joe Shipman claims that for two evenly matched tennis players, the one behind, "Tortoise," may sometimes have a better probability of winning the match than the one ahead, "Rabbit."

He's right. Suppose each has very high probability of winning the point when serving, so that usually the server wins all points and the match ends in a tie. Now if Rabbit is serving and somehow Tortoise wins one of the first three volleys, it turns out that Tortoise now has a better chance of winning this game against Rabbit's serve than Rabbit has of winning the next game when Tortoise serves.

So even though Rabbit is ahead, right now Tortoise has a larger probability of winning the match.

Infinitely many primes

Euclid (300 BC) proved that there are infinitely many prime numbers. Recently two mathematicians, John Friedlander of the University of Toronto and Henryk Iwaniec of Rutgers University, proved that there are infinitely many primes of the form a2 + b4, such as 52 + 24 = 25 + 16 = 41.

New challenge (Harlan Durand)

Why does the earliest sunset come around Dec. 10 and the latest sunrise come around Dec. 31, compared to shortest day around Dec. 21? And how can I remember (from year to year) which comes first?

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