Can You Get More Of The Pizza?

 

Is it possible to share a pizza so one person gets more of the pizza?  Let's see how it all works out.  We know that most pizza's are cut with an even number of pieces so if two people share a pizza, they will both end up with an equal number of slices. If the slices are of equal size, no problem.  If as in reality, the slices are not equal, then the person will go after the largest piece available to end up with the most.

Now to look at this from a more mathematical point of view since mathematics can explain the world.  When taking slices from a pizza, you either take the next one over - a shift - or you skip the empty spot - a jump.

If the person's strategy includes a number of jumps, then it is referred to as a j-jump strategy.  In addition, the pizza can be represented by a circular sequence with each piece being given a certain weight or the weight of P.  Now a player can have gain g if the strategy they use results in a certain number of pieces whose sum of weights is equal to g.

If a person does not have a jump strategy, there is only a 1/3 possibility of making it but if they have a one jump strategy, it jumps to a 7/16 possibility. If the person has a two jump strategy, it moves up to a 4/9 possibility.  Each jump increases the chances of coming out ahead. This is all based on an even number of pieces.

If there is an odd number of pieces, then it is a characteristic cycle instead of a circular cycle.  The characteristic cycle has an arc defined as a certain number of consecutive pieces with a specific length and weight. Furthermore, the arc of length defined as (n + 1)/2 is called a half circle. In the zero jump strategy, it has a potential of 1/3 as the lower bound.  This means it covers three half circles since the potential of v is the minimum number of weights of half circles covering v.  The upper bound is 100100100.

For a one jump strategy, the results depend on how and when the jump is made and could be a 1 if n is one,  4/9 if n is an odd number beginning at 15 or 1/2 for everything else,  a nice piecewise function.   It is best to use a one jump strategy when n is even or 9, 11, 13 or a two jump strategy for the odd numbers beginning at 15. 

The best two numbers of pieces the pizza should be cut into is 15 or 21 so one person gets more pieces than the other. I admit, it's been a while since I've had to read anything written in formal mathematical language so it took me a bit.  As a teacher, I don't spend much time reading things like this because I'm just trying to fill in missing foundational knowledge.  Let me know what you think, I'd love to hear.  Have a great day.