I think it's Spanish?

Probability Level 5

A pizza in the shape of a regular $n$ -gon is sliced in an unusual way:

Each cut is a straight line segment
No cuts intersect inside the pizza
Cuts are only made from one vertex to a non-adjacent vertex
Only $n - 3$ cuts are made

How many ways can a 16-gon pizza be sliced?

Details and Assumptions

The pizza has non-uniform topping placement; rotated or flipped solutions are counted separately.

For example, a pentagon pizza can be sliced 5 ways, as depicted in the image.

The pieces don't necessarily need to be of equal area.

This is not an original problem.

The answer is 2674440.

2 solutions

Pi Han Goh
Mar 10, 2015

This is just Catalan number . Because we want to cut a non zero area of pizza, the number of ways is simply $C_{16-2} = C_{14} = \frac {1}{14+1} \cdot { 2 \cdot 14 \choose 14 } = \boxed{2674440}$

Yes, this is right. Now it's bonus question time: in the problem, I stated this is not an original problem. Do you know which famous mathematician worked on this problem? (although they probably didn't have pizza on mind)

Caleb Townsend - 6 years, 3 months ago

You could have used the word 'triangulations' for this problem.

Nihar Mahajan - 6 years, 3 months ago

Frank Aiello
Aug 29, 2017

Number of ways a regular n-gon can be divided into n - 2 triangles if different orientations are counted separately is the Catalan number $C_{n-2}$ , where the Catalan numbers $C_n$ are defined as:

$C_n$ = $\frac{1}{n+1}$ ${{2n}\choose{n}}$ = $\frac{(2n)!}{n!(n+1)!}$

Thus the number of ways to divide a regular 16-gon into n - 2 triangles if different orientations are counted separately is:

$C_{16 - 2}$ = $C_{14}$ = $\frac{1}{14+1}$ ${{2(14)}\choose{14}}$ = $\frac{(2(14))!}{14!(14+1)!}$ = 2674440

Bonus: the mathematician who worked on this problem was Euler.

I think it's Spanish?

The answer is 2674440.

2 solutions

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