# Somebody ordered a pizza

Somebody ordered a pizza. They ordered $X$ slices with pepperoni, $Y$ slices with mushrooms, and $Z$ slices with cheese. If no two slices were identical, and no combinations of toppings were not present, how many (equal) slices was the pizza cut into?

• reasons for the downvotes? – Legotruck Nov 19 '14 at 12:22
• @Legotruck I think because the question is unclear as to exactly what counts as a topping as well as the phrase "no two slices identical" meaning within relation to each other. I didn't -1. – Raystafarian Nov 19 '14 at 12:38

7. PMC, PM, PC, MC, P, M, C. 8 if you can have one with no toppings.

Explanation:

 - 3.  There are 3 combinations where each slice has only one topping (as 3 toppings).
- 3.  There are 3 combinations where each slice has 2 toppings (more easily visualised as 3 combinations where one topping is missing)
- 1.  There is only 1 topping with all three combinations of toppings.
- 1.  There is only 1 topping with no toppings (debatable whether this is a valid "combination").
3 + 3 + 1 = 7 unique combinations of toppings excluding the empty topping
3 + 3 + 1 + 1 = 8 unique combinations of toppings including the empty topping


For each slice, each slice either has a topping or doesn't have a topping.

That's two possibilities per topping, for three toppings, or a total of 8 possibilities on a single slice.

This means that there were 8 slices on the pizza, with the following combinations of toppings:

\begin{array}{cccc} \text{Slice #} & P & M & C \\ 1 & \times & \times & \times \\ 2 & \times & \times & \circ \\ 3 & \times & \circ & \times \\ 4 & \times & \circ & \circ \\ 5 & \circ & \times & \times \\ 6 & \circ & \times & \circ \\ 7 & \circ & \circ & \times \\ 8 & \circ & \circ & \circ \\ \end{array}

So four slices have each topping.

I would say 3, but you referring to slices so every topping must be > 1. So my answer is 7:
The slices has been made in this order: A - B - A - B - C - B - C where the last C is next to the first one. So, X = 2, Y = 3 and Z = 2

• How on earth is this correct? It don't even make sense. – warspyking Nov 18 '14 at 23:13
• I think identical means we can have a pizza with a slice with a topping next to a slice with the same topping. – Emi987 Nov 18 '14 at 23:16
• @Emi Oh... The question needs rewording... – warspyking Nov 18 '14 at 23:23

Pizza's not pizza without cheese; you don't call up the place, order a pizza "with just pepperoni", and expect it to show up without cheese (or sauce). A "normal" amount of cheese is included in the cost of the pie, and you usually don't save any money ordering a cheeseless pizza. So, I will interpret Z as the number of slices that have only cheese. Given your other requirements,

Z = 1 because no two slices are alike, therefore no two slices can have only cheese.

The rest of it is simple; there is

one slice with pepperoni, one slice with mushrooms, and one slice with both pepperoni and mushrooms, for four unique equal slices of pizza.

Reminds me of school..:P

This reminds me of binary digits , combination 3 0's and 3 1's ..... So, in that case from 000 to 111 is valid for 3 binary digits . Hence (111)2 = (7)10. So 7 slices

Well, the fact none can be identical means nothing. It limits nothing but that it cannot be split into even sized portions with the same amount of toppings.

There was n slices means plural. There were 3 toppings, meaning a minimum of 3x2 slices; 6 slices

No combinations were present. Reinforcing the idea of 6.

So the minimum of slices was 6. The maximum amount of slices could depend on how precise the cutter is with sizes and if there were other types of slices with other, or no toppings (like pineapple, or hamburger meat). So anything 6+ would work.