# A pizza dilemma

You are a waiter at a restaurant. The restaurant is known for its signature dish: the Donut Pizza. The Donut Pizza is a 5-inch square pizza with a 1-inch square hole in the middle. After several customer requests, your boss has decided to create the Large Donut Pizza: a 7-inch square pizza with a 1-inch square hole in the middle. He has put it on the menu and informed you. You were supposed to inform the chefs, but you forgot to. Someone placed an order for a Large Donut Pizza. When you asked the chefs for a Large Donut Pizza to give to the customer, they gave you a regular one, and you then realize that you forgot to inform the chefs of this new addition. You quickly ask for another Donut Pizza, having decided to cut up one or both Donut Pizzas and rearrange the pieces into a Large Donut Pizza. You plan to cut up the pizzas in the following arrangement:

However, right before you start cutting, your boss, thinking that you are just slicing an already-made Large Donut Pizza, reminds you that a big pizza has to have big slices: all slices must be at least eight square inches. How should you cut up the pizzas and save your job?

Pieces can be rotated but not reflected.

• Any considerations about wastage, number of cuts or how complex a cut is (how many straight-line sub-cuts it requires)?
– smci
Mar 4 at 22:27
• @smci There can't be any wastage, but there are no restrictions on the other stuff. Mar 5 at 0:11
• I know that, I'm saying some answers interpreted "slices must be at least 8 sq.in." to mean "larger slice sizes considered better". There are multiple solutions, you didn't give any guidance on which criteria (if any) are considered optimal. (Intuitively, I'd go for minimum number and complexity of cuts).
– smci
Mar 5 at 0:51
• @EngineerToast: each slice must be >= 8 sq.in., and a pizza is 24 squares, so each pizza can only legally be cut into 2 or 3 slices.
– smci
Mar 5 at 22:53
• @smci Got it now, thanks. There isn't a target on number of slices but it's bounded by the size of each. Total pieces will be 4, 5, or 6 (two smaller pizzas each cut into 2 or 3 pieces). You're allowed to leave one pizza intact - meaning it counts as only 1 piece - but you can't do that while also keeping the pieces of the other pizza ≥8 in². Mar 6 at 13:01

I found a solution where every slice is strictly larger than 8 square inches:

Now, each slice is exactly 12 square inches:

I found a solution, and after comparing it to @spherical-wug-in-a-vacuum's solution, I found a solution which uses fewer, larger slices, by combining pairs of slices from @spherical's solution:

• Solution 1 even works without any rotation. Mar 4 at 9:53
• Solution 1 gives perfect utilization and converts 2 DPs to 1 LDP. Also only requires two straight cuts (and due to the symmetry, we could perform 2x one long cut across both DPs). Other solutions will require wastage, asymmetric cuts and/or multiple cuts.
– smci
Mar 4 at 22:23
• @smci: Asymmetrical, sloped, curved, or even fractal cuts can’t change the total area available! Short of going full Banach-Tarski and using non-measurable sets, no way of dissecting two area-24 shapes and rearranging into an area-48 shape can have any wastage. Mar 5 at 10:12
• @smci It IS a hard limit, but I was not able to convey that well with the backstory. Mar 5 at 16:22
• @smci This is Google Sheets, I don't know of a way to use it to draw sloped lines. Mar 6 at 1:19

This is a solution that I have found: