Imp2 Pow 17: Cutting The Pie

Problem Statement

Given a pie, you are to make cuts to produce the max amount of slices. Figure out how many slices 4, 5, and 10 cuts would produce. Make an In-Out table, diagrams, and anything that would help make a formula to find out the max amount of slices with x cuts. The slices may be different sizes. You are looking for the maximum amount of slices produced by x cuts, not the minimum.

Process

When I first looked at this problem I noticed the pattern in the increasing of maximum number of pieces based on a 1-unit increase of the number of cuts. I substituted the number of cuts with X, and the maximum amount of pieces with Y. I noticed that placement of the cuts could change the number of pieces created. I drew circles and made as many intersections as possible with only four lines.

My solution was found by continuing the In-Out table based on the pattern that I found in the Y variable. "Y" was increasing by one more than the preceding addition to the "y" variable as "x" increased by one. What I mean is that one more value was added to the previous addend to come up with the next value. This wouldn't have worked since the independent variable is the number of cuts, not the largest amount of pieces.

After awhile I noticed how the range between the two variables increased as the number of cuts increased. I came to the realization that the number of cuts had to be squared, and then added to another value, and then divided. Dividing the value just after squaring it wouldn't work, since not all "y" variables correspond with the factors and/or multiples of the "x" variable. My first step was to square the number, which in my example I will use 7 cuts. Seven squared is 49. Then I added 7 to the original number, which came out to be56. The only step left to do was to divide. I divided 56 by 2 giving me 28. Then I added one to have 29. Adding would make sense because dividing had now set up a starting point to perform any operation to the "x" variable. I used the formula as: [(xÐ'І + x)/2] + 1. I used this equation to solve the problem of ten cuts. Substituting ten in place of "x" in the formula shows the following:

[(10Ð'І + 10)/2] + 1=

(110/2) + 1=

55 +1=56.

The largest possible number of pieces from ten cuts is fifty-six.

Solution

x

(number