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The Sausage Conjecture

What is ther strangest math idea you know? I vote for the Banach-Tarski Paradox. But, Author Ian Stewart casts his vote for the Sausage Conjecture. Any idea what it is?

Suppose you need to wrap up a gift involving multiple congruent circles. How should you arrange them so that you minimize the amount of string needed to tie them together?

This is the case of n=2, where the conjecture refers to method for wrapping n-dimensional spheres using (n-1)-dimensional wrapping paper, minimizing voulme. Here, a circle is a 2-dimensional sphere, the string is 1-dimensional wrapping paper, and the "volume" is area interior to the wrapped string.

Now, the odd thing. If you have one to six circles, it is best to line them in a row and wrap the string around them (i.e. a sausage). But, when you have seven circles, it is best arrange them in the shape of a hexagon.

When n=3, the sausage shape should be used for up to 56 spheres. But, for 57 spheres, arrange them tightly packed as in a bag of potatoes.

And for n=4? An answer is not known, except that the sausage "break" occurs somewhere between 50,000 and 100,000.

And for dimensions greater than n=4? Formulated by Laszlo Tóth in 1975, the conjecture is that the sausage shape will always be the optimal arrangement for "spheres," regardless of the number to be wrapped. That is, the cases n=2, 3, 4 supposedly differ from the higher dimensions.

And perhaps the oddest thing? Mathematicians have proven the conjecture of always being a sausage shape is true for n greater than n=41. Thus, only the cases of dimensions n=4 through n=41 remain. Now, I find this odd!

Source: Adapted from Ian Stewart's "It's a Wrap," New Scientist, Dec. 2008