Murderous Maths
Chopping Pizzas

This video was filmed at BBC Scotland in front of a live audience, and it was also streamed to schools across the UK.

See more highlights here: Kjartan's Appearances

See the full 40 minute presentation here: BBC Scotland Authors Live

If you cut it with THREE straight lines you get a maximum of SEVEN bits. If you cut it with FOUR straight lines you get a maximum of ELEVEN bits. If you cut it with FIVE straight lines you get a maximum of SIXTEEN bits.
(And if you're lucky you might get a star!)

The formulas for chopping pizzas, crescents and 3-d objects are all covered in a lot more detail in The Perfect Sausage and other Fundamental Formulas
There are two things to try here:

1/ If Kjartan didn't explain, can you spot the pattern between the number of cuts and the maximum number of bits you get? Clue: think of triangle numbers!

2/ Can you draw a circle and then using a very sharp pencil, draw lines and see if you can always get the maximum number of bits? With EIGHT lines it's theoretically possible to get 37 bits!

Dividing up circles (or pizzas) leads on to a lot more interesting things to think about. See below for the formulas involved and the more complicated CHEESE chopping details!




CHEESE CHOPPING


This lump of cheese has been cut three times to produce eight pieces.
When chopping across the top of a pizza it behaves like a 2 dimensional object, but if you have a lump of cheese, the sums are very different. This is because when you cut the cheese a third time, you can cut it horizontally to get 8 pieces rather than 7.

The formula for the number of bits you get when cutting pizza is the triangle number formula +1. The formula for cutting cheese is more complicated, but they are linked in a surprising way!

PIZZA CUTTING FORMULA:

The maximum number of bits of pizza you can make is always one more than the triangle number. So if the number of cuts= c, then the maximum number of pieces of pizza is
c(c+1) + 1
   2

CHEESE CUTTING FORMULA:

The maximum number of bits of cheese you can make with "c" cuts
c3 + 5c + 1
   6

Comparision of
cuts and pieces:

Number of cuts:

Pizza pieces *:
Difference:

Cheese pieces:
Difference *:

 
 

    0       1       2       3       4       5       6       7       8

    1       2       4       7     11      16     22      29     37
        1       2       3       4       5       6       7       8     

    1       2       4       8     15      26     42      64     93
        1       2       4       7      11     16      22      29     

* The differences in the number of pieces of cheese are the same sequence of numbers as the pieces of pizza! (Thanks to John Bibby who first made us aware of this.)

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