Pizza Puzzle

Can your students fit the pizza in the tray?


The other day, we had pizza for dinner. Sorry, not home-made pizza, but the out-of-the-freezer-and-into-the-oven variety. The round pizza was too big to fit into our square baking dish. My wife's simple solution?

Cut it up!

This gave me the idea for this puzzle - the pizza cutting puzzle!

For reasons I'll explain later, let's imagine that instead of putting a round pizza into a square dish, we have

  • A square pizza, and
  • A round dish.
The puzzle is to find a way to cut up the pizza, and rearrange the pieces so they fit in the dish.

Whether this is a difficult puzzle or an easy one, depends on the sizes of the pizza and the dish. Imagine you have a 10 inch square pizza:

  • The distance from corner to corner is just over 14 inches. If you have a round tray, 14.2 inches across, the puzzle is too easy. There's no need to do any cutting at all!
  • On the other hand, as the tray shrinks to about 11.3 inches the puzzle quickly becomes very very hard, because there is less and less spare room on the tray!
  • Amazingly, this puzzle can still be solved even when the pizza and the tray have exactly the same area. Unfortunately, the pieces are not the kind you can make with a knife or scissors, and... wait for it... you need
    over 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 pieces!

    Not exactly a suitable puzzle for kids! (It was actually solved by a Hungarian mathematician, Miklós Laczkovich, now a professor in the UK. See this page for more details.)

I've put in some thought to figure out what size tray is best for kids. I've also prepared something you can print so you can give this puzzle to your kids to try out! I made the pizza square for a simple reason.

  • If you give this puzzle to someone to try, you'll only need one tray, but you'll need many many pizzas for them to cut up in different ways. And it's easier to cut out squares than circles - just use a guillotine, for example.
Anyway, here's my suggestions for tray sizes.
  • A square pizza, 10 inches per side, only needs to be cut into 3 pieces to fit on a tround tray 12.57 inches across. In the printable version, I give a tray 13 inches across. You can use this to introduce the puzzle to the kids in your care.
By the way, the pizza I give to print is not really 10 inches across, it's scaled down 2:1. The trays are likewise scaled. You could always try to enlarge it when you print it.
  • The next tray is only 12.5 inches across. I think this is just too small for three pieces to do. I'm not sure how many pieces will be needed, but it's probably still not many.
  • If that turns out to be very easy, try the kids on the 12 inch tray! This is likely to be more of a challenge. If you are giving this puzzle to, say, a class of kids, you could give a reward of some sort to the child who solves the puzzle with the fewest number of pieces.
  • The 11.8 inch tray will be very challenging! The pizza will fill almost 95% of the tray, and likely need to be cut into many many pieces. Reserve this puzzle for the brightest and most motivated of your kids.

Well, that's it for the pizza cutting puzzle! Hope your kids enjoy it!

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