Sharing food with a friend (math riddle)

Zippy the Wonderslug: You and your friend decided to order an extra large (round) pizza online.

When it was delivered though, you sadly find that it was carelessly cut 8 times.

Not even one of the cuts went through the exact center of the pizza.

Thus creating 16 extremely wild and lopsided sizes for each slice.

The pizza looks really good to the both of you, but because all of the slices are completely different in size, you wonder how to share this equally with your friend.

What is the easiest method to ensure that you both receive a final total of equal portions to this pizza?

wildrose62: let one take one slice and the other take the one opposite it - whoever got the bigger slice of that turn gets the smaller of the next pair of opposite pieces.... repeat.

Ho Lee Sheet: Easiest method is to cut the pizza all over again into equal portions. ?

Confusedsouls: you cut it in half a 9th time, this time the line will go through the center
and each of you will eat a half, no matter how many slices or shape they will contain

Zippy the Wonderslug: @Confusedsouls

You find that there are no knives at home to cut it for the 9th time.

The pizza is now getting very cold while you and your friend attempt to problem solve this.

@Wildrose62

Interesting theory and you're quite close to the simple solution.

Confusedsouls: the 1st person will eat the biggest and smallest piece
the 2nd person will now eat the biggest and smallest piece of the remaining pizza
and so on and so forth.

like if the 16 pieces were named from A to P, and A>B>C.....>P
then the 1st person will eat A and P
and 2nd will eat B and O
then 1st person will again eat C and N ....etc until the pizza is over

Ho Lee Sheet: How about this, Lets say there are 16 pieces in total, out of which we have 8 big pieces and 8 small pieces, each person takes a turn and gets the big piece first.. 8 turns and 8 big pieces are eaten by both, then they take turns to finish off the 8 smaller ones. By this way both get 8 pieces each and both are mentally satisfied that it was a fair trade policy as they got 4 big pieces and 4 small pieces each

Wonderbunny: Weigh the pieces. But what if some of the pieces are nearly all crust and some are nice juicy pieces with topping on them?

Wonderbunny: Got it! Though I had to resort to drawing it on a bit of paper to see it. Each person takes it in turns to take a random piece and the piece immediately opposite it. These 2 pieces add up to exactly an eighth. So each person will end up with 4 eighths or one half of the pizza.

Zippy the Wonderslug: The only trouble I can find with these ideas (and they are quite brilliant!) is knowing fully what each pizza slice size is when comparing one to another.

A visual comparison might not always be correct since there is no center cut in the pizza.

To be absolutely certain your method(s) are indeed 100% accurate, you would have to calculate the surface area (the radius and degree) of each slice so that no mistakes were to be made.

That would take extra time while the pizza continues to get cold.

Zippy the Wonderslug: To keep things simple as possible, here's an example of how the line cook cut the pizza.

Picture

By the way, he should be fired for such hateful nonsense.

Zippy the Wonderslug: @Wonderbunny

Oh. You're correct with your second response.

Scientific answer:
Let p be an interior point of the disk, and let n be a multiple of 4 and greater than or equal to 8. Form n sectors of the disk with equal angles by choosing an arbitrary line through p, rotating the line n/2 − 1 times by an angle of 2π/n radians, and slicing the disk on each of the resulting n/2 lines.

Easy to follow answer:
Number the sectors consecutively in a clockwise or anti-clockwise fashion.
The sum of the areas of the odd-numbered sectors equals the sum of the areas of the even-numbered sectors.

So either have you or your friend take the first slice.

Then alternate between you two either clockwise or counterclockwise on every slice thereafter.

Taking two slices at a time, one and then the other from the opposite side also works.

Good job Wonderbunny.

Wonderbunny: So it only works with equal angles. You never said anything about equal angles.

Your answer only describes how the pizza is sliced. It doesn't prove that the areas are equal, but I'll take your word for it.

Also my answer is incorrect because if the first person chooses, say, 2 opposite odd numbered slices, the second person might still choose odd numbered slices as there are still 6 left.