June 10, 2014

Horse 1690 - Pizza Maths

Yesterday was the Queen's Birthday Weekend Holiday, which is amusing as her birthday is really on the 21st April. If for no other reason than to keep a public holiday, hurrah for the Queen's Birthday Weekend!
I had a lovely sleep in; in a nice warm bed and as if to cap off a lovely morning, Mrs Rollo made buttermilk pancakes.
This is where the story gets weird...

I cut across a pancake with my knife and noticed that it cut into 2 pieces (okay, that's blatantly obvious). I then made a second cut across the pancake and that second cut, made 4 pieces. I was then thinking about making a third cut when I was told that it was going to get cold.
I'm sure that I must at times drive Mrs Rollo bonkers at times with my incessant wondering; so I kept this one quiet.

What is the maximum number of bits that you can cut pancakes, pizza, lahoh, chapati... into, with every new cut?
I don't really care about the size of the bits.

On the train this morning, I sat doodling; thinking of Blind Freddy working away for Hypothetical Pizza Inc. and I bet he'd know how many bits he could get for every cut of his blade. Being the kind of numbers person, I am, I thought I'd take note of the number of bits I could get for every new cut.

For 0 cuts, you get 1 bit
For 1 cut, you get 2 bits.
For 2 cuts, you get 4 bits.
For 3 cuts, you can get 7 bits.
For 4 cuts, you can get 11 bits.

If we let c be the number of cuts and b be the number of bits that you get, then this should resolve nicely into a simple equation. I also noticed that for every new cut, the number of new bits increased by 1 every time; this means that whatever equation arises, it must be of degree 2.

The general formula for any equation of degree 2 is ax²+bx+c = y
In this case x is the number of cuts and y is the number of bits.
Since the number of bits that I have with 0 cuts is 1, then it's a safe bet to assume that the final constant is 1.

This is where brute force came into play; since I couldn't remember how to actually find a specific equation from the data it generates but I did remember the basic quadratic formula; so I punched in a stack of numbers and fiddled with things until I magically found my answer.

The number of bits that you get from a given number of cuts is defined by the equation:
(c²+c)/2 + 1 = b
I figure that for 10 cuts, you should get 56 bits; for 13 cuts, you should get 92 bits etc. I have no idea what each of those very small bits would look like but to be honest, I don't care all that much. The best number of cuts to make in a pancake, pizza, lahoh, chapati etc... is none.
I can eat a whole pizza.

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