# How many pieces? In this study, the maximum number of parts obtained by n cuts of a four dimensional object will be analyzed by looking at the patterns of the maximum number of segments made by n cuts on a one dimensional line, the maximum number of regio

IB Mathematics HL Internal Assessment

How many pieces?

Candidate Number:

Emily Wang

Mr. Dramble

30 January 2012

Much of mathematics involves finding rules by looking at patterns. In this study, the maximum number of parts obtained by n cuts of a four dimensional object will be analyzed by looking at the patterns of the maximum number of segments made by n cuts on a one dimensional line, the maximum number of regions made by n cuts on a two dimensional circle, and the maximum number of parts made by n cuts on a three dimensional cuboid. The sketches, cuts and results, conjectures and proofs will be shown, and the conjecture of the formula for the maximum number of cuts for the four dimensional objects will be made according to the previous studies.

• One Dimensional Object

For a line segment, to obtain the rule which relates the maximum number of segments (S) obtained from n cuts, we could look at examples and draw our results on a table.

n = 1, S = 2

n = 2, S = 3

n = 3, S = 4

n = 4, S = 5

n = 5, S = 6

It can be seen that this shows an arithmetic sequence, and by using the arithmetic sequence formula, an = a1 + (n – 1)d, (an = S) it can be seen that the common difference, d, is 1 since the values of S increase by 1 as n increases by 1. If the numbers for the variables are plugged into the equation, an = 2 + (n – 1)1, the equation for an would be an = n + 1, so the rule for S to obtain the maximum amount of cuts per segment would be S = n + 1. This result also shows a linear regression line, so the equation to find the maximum amount of cuts for a one dimensional object would use a linear regression.

• Two Dimensional Circle

n = 1, R = 2                                                                n = 2, R = 4

n = 3, R = 7                                                                                        n = 4, R = 11

n = 5, R = 16

To find a recursive rule to the maximum amount of regions, we have to view the R cases.

R1 = 2

R2 = 4 → R1 + 2

R3 = 7 → R2 + 3

R4 = 11 → R3 + 4

R5 = 16 → R4 + 5

So the recursive formula for generating the maximum amount of regions is Rn=Rn-1+ n.

Relationship between maximum number of regions (R) and number of chords (n) using technology:

1.        2.

3.

Conjecture for maximum number of regions (R) and number of chords (n): ...