# The Pizza Cutting Challenge

You and two friends decided to order a large circular pizza to split between each other. Unfortunately at the moment you didn't have enough to pay your entire third of the bill. Your friends decide that they will cover the remainder of your bill but for a cost! Instead of just giving you less pizza, they will give you a challenge to earn as much as you can. This is the challenge:

The pizza will be set up on a large cutting board.

You will be given 1 pizza cutter and are allowed to make 3 cuts in total.

Each cut must go from one end of the board to the other. (i.e no 'half cuts' or stopping in the middle of the pizza)

You are allowed to set it up how you please before and after each cut.

When you are finished cutting the pizza, you're friends will pick all their slices first, and they will take the higher number if distribution isn't 'fair'

(i.e if there are 4 slices, [Friend 1] will take 2, [Friend 2] will take 1 and then [You] will take 1)

(5 slices: [Friend 1] takes 2 then [Friend 2] takes 2 then You take 1)

**You must not cut it into the standard 6th's as done by normal pizza places (60° from each cut)

(Sorry for the last rule but your hungry friends have became cranky)

Your friends are really hungry and will always try to take as much pizza as possible.

What is the maximum amount of pizza you can get if you follow these rules?

Bonus:

If you are still restricted to 3 cuts, but you are allowed to distribute the pizza however you please (as long as your friends always have equal or more than you) how can you cut and distribute the pizza such that you all get even portions (The max as possible for you)?

• In case any of you were wondering, in the 4 slices example, [Friend 1] will split his extra slice with [Friend 2]. ;) – Mark N May 12 '15 at 20:35
• Do you have to make 3 cuts, or can you make 2 or 1? Also, lets say you cut it into 9 pieces, will your friends each take 4 pieces and leave you with one?? – MisterEman22 May 12 '15 at 20:37
• @MisterEman22 You have to make 3 cuts. With 9 pieces you would each get 3. (But you get the 3 left behind) – Mark N May 12 '15 at 20:38

I'm probably missing something but couldn't you just

Cut the pizza into 6ths so each of you gets 2 pieces? (Diameter + 60° off through center + 60° off through center)

Edit If you can't do the center 60° cut, just use parallel cuts to cut into 3rd, then a perpendicular cut to get 6ths again. The parallel cuts have an arc of 2.605 rads, or about .267 radius from the center.

Thinking
So, I thought about how to handle this issue if you have N friends:
1 friend: 6ths still works
3 friends: Diameter, then 2 parallel cuts to give areas of 1/4
4 friends: Uneven X with a bar across the bottom. I'd guess the bottom bar is 1/2 from center, X crosses 1/8 from center (center is between X and bar)
5 friends: 6ths still works
6 friends: Cut out a triangle from the middle, total of 7 pieces, max.

• ....I realized after making the puzzle that it was much easier than intended :( ...BUT if i disallow this set up then it is much harder (don't know if it's too late for that) – Mark N May 12 '15 at 20:41
• Sorry, but I'm going to change the question a little bit to remove this (obvious) answer...I wanted to see some creativity. (Accept my +1 as my gift) – Mark N May 12 '15 at 20:49
• @MarkN Shouldn't you give him the answer regardless and then create another post, make it cutting challenge 2 or something. Seems unfair. – Daedric May 12 '15 at 22:55

first, cut the pizza in half
The second and third cut are on the friends' throat

tada! The pizza is all mine now!

• Remind me to never go out for pizza with you. – Aggie Kidd May 13 '15 at 3:57
• @AggieKidd LOL no worries we will order enough pizza to shares :D – Alex May 13 '15 at 14:02

Scrape all of the toppings into a bowl. Cut pizza in half, let them take the 2 pieces. Cook noodles, dump topping in noodles, enjoy.

More onto the puzzle itself:

Best answer is still to find a way to cut the pizza in 6 slices as identical as possible. Instead of trying to cut it traditionally try stacking the pizza after each cut. This means that cut #2 will make 2 identical 'sixths', its the hardest cut. Cut # 3 will slice in half the 2 thirds remaining making (hopefully) 4 identical 'sixths'

Or a bit of a mix of both, cut it in half (blue), stack the two halves, then cut along the yellow line (doing 2 cuts at once), and green line. The larger pieces are the ones with mostly only crust, so your friends end up with 2 pieces of crust, and you end up with the goodies :)

• Now that's a clever use for the toppings! – Nefer007 Jul 23 '15 at 13:17
• Best answer - you can easily maximize your toppings in exchange for your friends maximizing surface area. By calories consumed (roughly approximates satiety) you can actually come out ahead. – Iiridayn Oct 23 '18 at 22:30

Am I missing something here? Assuming you are a perfect cutter and can eyeball with precision, it's two parallel cuts and a perpendicular cut:

(drawing is not perfect)

• The only thing you will be missing is the good part of the pizza, when you get all that crust :p – Mark N May 13 '15 at 12:49

Given that the friends will always take the biggest pieces and more pieces, if the number of pieces is not divisible by three, the absolute most you can get is one third (1/3) of the pizza.

Following the rules this is achieved as follows:

1. Cut the pizza in half with one horizontal cut
2. Rotate the lower half 180 degrees.
3. Put the lower half on top of the upper half, so that they completely overlap
4. Make a cut from the top of the pizza, exactly 60 degrees along the (now half-circle double-layered) pizza, down through the center point of the pizzas horisontal line.
5. Repeat 4, but with 120 degrees.
6. Profit

Voila, 6 equally sized slices.

%%%    %%%
%%%              %%%

%%%       1/3             %%%
_.-*''       A,C   ~ 1/3 of pizza
%%%             _.-'        %%%
(two cuts)  ============|_      1/3
%%%              `-._       %%%
|_
%%%        1/3       |_    %%%
|____    B  ~ 1/3 of pizza
%%%                  %%%

%%%     %%%

Pretend the picture is to proportion. Thus, you approximate $A$ to be $\frac13$ of the pizza, $B$ to be the same (except going upwards), and $C$ to overlap $A$ (technically having 3 cuts, though only $2$ legit cuts). Thus, three slices of equal proportion are left.

Alternatively, you could split the pizza into six equal slices.

# Bonus

Still workin' on it

# Edits

## 1

The first scenario would work with your "restriction."

## 2 ~ Extra solution

This cuts the pieces into (roughly) equal pieces (cutting along the blue lines). Hopefully, your friends will trust you in that they are equal, if you do a bad cutting job:

• If the pieces are equal, then it doesn't actually matter whether or not your friends trust you that they are equal - if they believe the pieces are unequal, they'll take the ones they think are bigger, but since you know they're equal, you're ok with that. – Dan Henderson Oct 19 '18 at 16:15

Cut one line horizontally through the center. Then make two more cuts vertically, such that all three pieces above (or below) the original cut have equal area.

Update: Per the comment: the vertical cuts should go through points about 15 degrees from the center. This would be a smidge over 1/4 radius. Thus, the central pieces would be about 1/2 radius wide, and the others would be 3/4 radius wide.

• I would recommend a more 'precise' answer rather than use of 'such that' if possible :) – Mark N May 12 '15 at 20:59