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There is much here for the amateur to investigate. Are the conjectured fences truly the shortest possible, or is there a way to shorten them further? Can anything be proved about the conjectured solutions? What about other shapes – arbitrary polygons (convex or not), ellipses, semicircles . . And what about the same problem in three dimensions: the opaque cube and sphere? Now the aim is to minimize the total area of the fence. PURSUING POLYGONAL PRIVACY | 25 FEEDBACK Martin Gardner raised the problems of the opaque cube and sphere in 1990, and Kenneth A.
For instance, when x = 5 the gaps are 1, 2 and each occurs once. After that, the sole jumping champion is 2 until we reach x = 101, when 2 and 4 are tied for the honour. After that, the jumping champion is either 2, 4, or both until x = 179, when 2, 4, and 6 are involved in a three-way tie. At that point the challenge from 4 and 6 dies away, and 2 reigns supreme until x = 379, where it is tied with 6. From x = 389 the jumping champion is mostly 6, occasionally tied with 2 and/or 4, but in the range x = 491 to 541 the jumping champion reverts to 4.
Could, with minimal effort, give birth to anything as bafﬂing as the prime numbers 2, 3, 5, 7, 11, . .? The pattern of natural numbers is simple and obvious: whichever one you’ve got, it’s easy to work out the next one. You can’t say that for the primes, yet it is a simple step from natural numbers to primes: just take those that have no proper divisors. We know a lot about the primes, including some powerful approximate formulas that provide good estimates even when exact answers aren’t forthcoming.