NUMBER OF PIECES

Kanishk Malhotra

003566-035 (May 2012)

In physics and mathematics, the ‘DIMENSION’ of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it (for example, to locate a point on the surface of a sphere you need both its latitude and its longitude). The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces. *taken from www.Wikipedia.com

AIM: - TO INVESTIGATE THE MAXIMUM NUMBER OF PEICES OBTAINED WHEN AN ‘N’ DIMENSIONAL OBJECT IS CUT.

1st -DIMENSION

The first dimension is best and the easiest to start with. Lets us take an example of a line segment which is a finite one dimensional object.

Now if there is 1 cut on the line segment, then

The number of pieces obtained are 2

If there are two cuts on a line segment, then

2

The number of pieces obtained are 3

If there are three cuts on the line segment, then

The number of pieces obtained are 4

If there are four cuts on the line segment, then

Then the number of pieces obtained are 5

If there are five cuts in the line segment

The number of pieces obtained are 6.

This information is represented in a tabular form below: -

3

Number of cuts

1

2

3

4

5

Pieces

2

3

4

5

6

Thus while observing the pattern it has been seen that for the number of cuts ‘N’ the pieces obtained has been ‘N+1’

Hence the conjecture for the number of pieces obtained for one dimensional object is ‘N+1’ which means that the number of pieces for a particular number of cuts will be equal to the number of cuts + 1. This conjecture according to the question has to be defined as (S).

2nd - DIMENSION

Now let us observe a finite two dimensional figure which is a circle.

Now if there is 1 cut in the circle, then

4

The number of pieces obtained are 2

If there are two cuts in the circle, then

The number of pieces obtained are 4

If there are 3 cuts in the circle, then

Then the number of pieces obtained are 7

If there are 4 cuts in the circle, then

5

The number of pieces obtained are 11

If there are 5 cuts in the circle, then

The number of pieces obtained are 16

This information has been represented in a tabular form below: -

6

Number of

Pieces for

Pieces for 2-D (R)

cuts

1-D (S)

1

2

+

2

2

3

+

4

3

4

+

7

4

5

+

11

5

6

16

By analysing the above table, the pattern can be found which is recursive so the Recursive formula for 2-D figures for ‘n’ number of cuts is Rn= S(n-1) + R(n-1) We form the conjecture for the above by a particular method which is – Consider the general sequence with terms U1, U2, U3, U4 and U5, and then the difference array is as follows:

Sequence

U1

U2

1st difference

U3

1U1

2nd difference

U4

1 U2

U5

1U3

2U1

2U2

1U4

2U3

We can now form a conjecture that an expression for Un in terms of U1 is given byUn= U1 + (n- 1 )

1U1 +

2U1+......+

rU1+.....+

n-1U1

Applying this for the sequence 2, 4, 7, 11, 16

Sequence

1st difference

2nd difference

2

4

7

2

3

1

11

16

4

1

5

1

7

U1=2,

1U1= 2,

2U1= 1

Un= 2+ 2(n-1) + 1

= 2+ 2(n-1) +

=

=

Hence this is the conjecture for the sequence. Thus the number of pieces made by the cuts in a second dimensional object (which is a circle) can be found out by the conjecture

. This conjecture according to the question has

to be defined as (R). Thus writing it as Rn=

.

The above conjecture can hence be proved by the principle of Mathematical Induction.

P(n): the number of pieces made by n cuts can be...