Eccentric Pizza Slices

Suppose six friends visit a pizzeria and have enough cash for just one big pizza. They need to divide it fairly into six equal pieces. That is simple: cut the pizza in the usual way into six equal sectors.

But suppose there is meat in the centre of the pizza and some of the friends are vegetarians. How can we cut the pizza into slices of identical shape and size, some of them not including the central region?

A pizza with various toppings. Image: Pizza Masetti Craiova, Romania (Flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons.

A pizza with various toppings. Image: Pizza Masetti Craiova, Romania (Flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)%5D, via Wikimedia Commons.

Have a think about this before reading on. There is more than one solution.

More formally, here is the mathematical problem to be solved:

Divide a disc into a finite number of congruent pieces in such a way that not all of them touch the centre.

Clearly, since some of the six pieces must contain arcs of the outer edge, all pieces must have similar arcs on their boundaries. Drawing arcs of the same radius as the pizza, with centres on the edge, we get the pattern on the left below. Each of the six pieces is a curvy triangle, reminiscent of a yacht’s spinnaker. All pieces are congruent, that is, identical in shape and size.

But this does not solve the problem, since all six slices touch the centre. But now we bisect each slice as shown on the right. Then we get twelve pieces, all identical in shape and size, six of which exclude the central region. Now the three vegetarians can each choose two outer pieces, leaving the inner ones for the carnivores.

Problem: In addition to the solution shown above, there is an infinite family of other solutions. Can you find them? [Hint: Start from the pattern in the left panel above and sub-divide each piece].

The solution we have shown is the basis for the logo of the annual Mathematics Advanced Study Semesters (MASS) Program at Penn State University, which takes place every Fall.

See also Slice of Pizza with no Crust at Math Stack Exchange.

[Posted from Ragusa, Sicily following exhaustive testing and validation]

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