Fermat's Last Pizza

Close Art, at the Sandwich bar he asked the server to "make me one with everything"  (even worse than deep pan crisp and even)

Remarkably on-topic!

One common misunderstanding about pizzas concerns how toppings are categorised. A ham & pineapple pizza, for instance, might contain 1% ham to 99% pineapple, or vice versa, or 50% of each (food standards regulations not withstanding). In order to numerate pizza toppings more precisely, I suggest we should digitize the proportions of the topping elements. Let's say we take 1/64 oz. as the basic unit, and stipulate that the maximum weight of topping is 1 ounce. In this way, a pizza with 1/64 oz. of ham and 2/64 oz. of pineapple is clearly not the same as one with 2/64 oz. of ham and 1/64 oz. of pineapple.

# of possible board permutations 10 to the 121st 10121 (Cannot get it to the top)

The proof of the pizza is in the eating:

In the case of a chessboard, each of the 64 squares can contain ONLY one of 13 items: K, Q, B, N, R, P, k, q, b, n, r, p or nothing.

Total number of chess board positions: 13 to the power 64. (Some of these will be reflections, some will be unreachable and some will be illegal).

Inscribe an 8X8 square (not unlike a chessboard) on top of a pizza.

Each of the 64 squares can contain one of at least 14 ingredients: e.g. cheese, ham, pineapple, chicken, beef, sausage, pepperoni, mushroom, tuna, salmon, broccoli, olive, capers and onion.

Total number of pizza toppings: 14 (at least) to the power 64. (Some of these will be reflections. some will be uneatable and a quite few will be ill-advised - though only illegal in certain jurisdictions).

QED

You can put chess pieces on top of a pizza, but you can't use pizza toppings for chess variations.

The different coloured olives lend themselves for use a pawns. Since you'll end up eating your opponent's pieces (as you capture them), make sure you're playing with green.

huh?

I like green olives even less than I like black olives (if that's possible).

While I see what you are trying to do here, your analysis is incorrect. Namely, while one COULD section a pizza into 64 sections and then count the pizza as different if any single section differs in quantity or type of toppings, this is simply not how the pizza community at large quantifies its pies. Generally, the pizza only has two quadrants that are used to determine its makeup.

What about the set of all subsets of all pizza combinations?

This has been discussed oh so man times...there is more chess positions!

There are only so many ways to place chess pieaces on a board. However, there are infinite ways to make a simple cheese, pepperoni, mushroom pizze. 99.982% pepperoni and 0.018% mushrooms and so on.

I suspect you're behind the times. Here's a typical online pizza ordering menu ...

Oh no, there aren't! As the earlier proof demonstrates.

I'm not denying the vast variety of pizza toppings, only that their placement is not determined by a 64 part grid, but generally by putting them on the whole pie or on only 1/2 of the pie.

Even allowing for the constraints that DW_Batty introduces, the number of pizza toppings still comfortably exceeds the number of chess positions.

Assume each ingredient is sprayed over the entire pizza top surface. Assume the ingredients are applied in a fixed sequence. For illustration purposes, assume 6 different topping ingredients. How many pizza toppings using 6 ingredients?

= sum of pizzas with 0 toppings + pizzas with 1 topping + pizzas with 2 toppings + pizzas with 3 toppings + pizzas with 4 toppings + pizzas with 5 toppings + pizzas with 6 toppings

The number of combinations  nCr = n! / (r!(n-r)!)

The number of pizza toppings with 6 ingredients is therefore the sum of

6C0 = 6! / 0!(6!) = 1

6C1 = 6! / 1!(5!) = 6

6C2 = 6! / 2!(4!) = 15

6C3 = 6! / 3!(3!) = 20

6C4 = 6! / 4!(2!) = 15

6C5 = 6! / 5!(1!) = 6

6C6 = 6! / 6!(0!) = 1

= 64

= 2 exp 6

The number of chess positions is a fixed, finite number: 13 raised to power 64 (i.e. 13 exp 64). This is approximately 10 raised to power 71 (i.e. 10 exp 71)  (http://www.calculatorfreeonline.com/calculators/exponent.php)

The question then becomes:

if n (i.e. the number of ingredients) can grow arbitrarily large: is there a value of n for which the SUM(1..n) nCr exceeds 10 exp 71 ?

i.e. 2 exp n > 10 exp 71

Woohoo!  I got the joke AND the Daily Puzzle!

And you didn't claim you were the FIRST to do so.

It turns out that using 235 ingredients for pizza toppings, yields about the same number of pizza toppings as chess positions. Reach 240 or more ingredients, and the chess board starts to look distinctly cheesy and uninteresting.

Quad Erat Pepperoni

13⁶⁴ (my calculation of the number of board positions) is around 10⁷¹  - not 10¹²¹

Whichever of these values is used, it is still trivial to find a value n (the number of ingredients), such that 2ⁿ  > 10¹²¹