Math

THE SHORTHAND

OF COUNTING

RECTANGLES

CABUGCABUG NATIONAL HIGH SCHOOLPresident Roxas, Capiz

S.Y. 2012-2013

Researcher:JESTONI SANTILLAN

THE SHORTHAND OF COUNTING RECTANGLES

I. INTRODUCTION

As we go on with our journey for the quest of knowledge, there are some

instances in which we can encounter problems like counting rectangles of all possible

sizes that could be fitted in the diagram. Sometimes it takes time for us to count all of

them; some overlooked; some double-counted. You cannot instantly give the precise

and accurate number of all rectangles in it.

In this investigation, the investigators want to find out if there is/are formula in

finding for the correct number of rectangles that could be placed in a diagram.

This study also aims to explore with the sequence of total number of rectangles if

there is a pattern formed from them.

Consider this one:

How many rectangles are there?

II. STATEMENTS OF THE PROBLEMS CONSIDERED

1. Does the total number of rectangles in a group of diagrams with the same

length forms a pattern? What pattern is it?

2. How could you count all the rectangles in different possible sizes that could

be fitted in a certain diagram in a shorter way?

3. Is there any formula to find for the total number of rectangles in a diagram?

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III. CONJECTURES

Conjecture #1

There is a pattern in the total number of rectangles that could be placed in the

diagram. This pattern is also known as the “Sequence of Triangular Numbers”. A

pattern of polygonal or figurate numbers in which each triangular number is obtained by

adding one more than the preceding difference. In symbols, Tn = n (n+1)/2, where

Tn= triangular number and n= number. However, for rectangular grids with length

more than one, the formula is multiplied to the triangular number of the length.

Testing/Verifying Conjecture #1

For diagrams with length equal to one

Table 1.a

Size (LxW) Total No. Of Rectangles

1x1 1

1x2 3

1x3 6

1x4 10

1x5 15

Refer to the table above. In the sequence of triangular numbers, the next number

is obtained by adding one more than the preceding difference. The first five terms in the

sequence were 1, 3, 6, 10 and 15. Therefore, the total number of rectangles in a

diagram with length equal to one follows the pattern of triangular numbers.

2

Size of The Diagram No. Of Rectangles

2x1 3

2x2 9

2x3 18

2x4 30

2x5 45

3x1 6

3x2 18

3x3 36

3x4 60

3x5 90

4x1 10

4x2 30

4x3 60

4x4 100

4x5 150

Table1

Refer from the table above. In diagrams with length 2 units, the number of

rectangles were 3, 9, 18, 30, and 45. But when each of this values were divided by

a constant value 3, the result will be the first five terms of the triangular sequence. So, it

follows the pattern of triangular numbers.

3/3=1 ; 9/3=3 ; 18/3=6 ; 30/3=10; 45/3=15

IfL = 3 units, the numbers were 6, 18, 36, 60 and 90. But if when each were

divided by a constant number 6, the results were the numbers of the first five termsin

the sequence of triangular numbers.Hence, it follows the pattern of triangular numbers.

6/6=1 ; 18/6=3 ; 36/6=6 ; 60/6=10; 90/6=15

3

If L = 4 units, the numbers were 10, 30, 60, 100 and 150. But if when each were

divided by a constant number 10, the results were the numbers of the first five termsin

the sequence of triangular numbers.Hence, it follows the pattern of triangular numbers.

10/10=1 ; 30/10=3 ; 60/10=6 ; 100/10=10; 150/10=15

Conjecture #2

There is an assigned number for every groups of rectangular grids grouped

according tothe same value of length. This number is called the

diagram’s“Triangular Number”. This number is also the triangular number of the

diagram’s length. When this constant number is used to divide the total

number of rectangles of a certain diagram, the result will be a triangular number.


Table 2

SIZE (LxW)

No. Of Rectangles(TR)

Triangular No.(DT)(assigned)

TR/DT=Tn

1x1 1 1 11x2 3 1 31x3 6 1 61x4 10 1 101x5 15 1 152x1 3 3 12x2 9 3 32x3 18 3 62x4 30 3 102x5 45 3 153x1 6 6 13x2 18 6 33x3 36 6 63x4 60 6 103x5 90 6 154x1 10 10 14x2 30 10 34x3 60 10 64x4 100 10 104x5 150 10 15

4

From the table 1.b shows how each diagram is assigned with a triangular

number. In the last column showsthe quotient of the total number of rectangles and

Their triangular number. Whereas the quotient is a triangular number.

Conjecture #3

To find for the total number of rectangles that could be possibly placed in the diagram, the formula is stated as:

TR = DT[w (w+1)/2] Where:

TR=total number of rectangles

DT=triangular number of the grid (also the triangular number of the length )

w =width of the diagram


How many rectangles are there in the following diagrams?

5

a. 1 x 2 diagram b. 2 x 2 diagram

TR= ? TR= ?

TR= DT[ w(w+1)/2] TR= DT[ w(w+1)/2]

= 1 [2(2+1)/2] = 3 [2(2+1)/2]

= 1 [2(3)/2] = 3 [2(3)/2]

= 1 [ 6/2 ] = 3 [ 6/2 ]

= 3 = 9

c.3 x 3 diagram d.. 3 x 4 diagram

TR= ? TR= ?

TR= DT[w(w+1)/2] TR= DT[ w(w+1)/2]

= 6[3(3+1)/2] = 6 [4(4+1)/2]

= 6 [3(4)/2] = 6 [4(5)/2]

= 6 [ 12/2 ] = 6 [ 20/2 ]

= 36 = 60

e.4 x 4diagram f. 4 x 5 diagram

TR= ? TR= ?

TR=DT[w(w+1)/2] TR=DT[w(w+1)/2]

= 10 [4(4+1)/2] = 15 [4(4+1)/2]

= 10 [4(5)/2] = 15 [4(5)/2]

= 10 [ 20/2 ] = 15 [ 20/2 ]

= 100 = 150

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Table 3. Total Number of Rectangles in an mxn diagram.

SIZE(LXW)

1x1

1x2

1x3

1X4

1x5

2x1

2x2

2x3

2x4

2x5

3x1

3X2

3x3

3x4

3x5

4x1

4x2

4x3

4x4

4x5

Total

1x1 1 1

1x2 2 1 3

1x3 3 2 1 6

1x4 4 3 2 1 10

1x5 5 4 3 2 1 15

2x1 2 1 3

2x2 4 2 2 1 9

2x3 6 4 2 3 2 1 18

2x4 8 6 4 2 4 3 2 1 30

2x5 10 8 6 4 2 5 4 3 2 1 45

3x1 3 2 1 6

3x2 6 3 4 2 2 1 18

3x3 9 6 3 6 4 2 3 2 1 36

3x4 12 9 6 3 8 6 4 2 4 3 2 1 60

3x5 15 12 9 6 3 10 8 6 4 2 5 4 3 2 1 90

4x1 4 3 2 1 10

4x2 8 4 6 3 4 2 2 1 30

4x3 12 8 4 9 6 3 6 4 2 3 2 1 60

4x4 16 12 8 4 12 9 6 3 8 6 4 2 4 3 2 1 100

4x5 20 16 12 8 4 15 12 9 6 3 10 8 6 4 2 5 4 3 2 1 150

This table shows the list of Different size diagrams and how many of the smaller

diagrams could be fitted in it.

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IV. JUSTIFICATION

Conjecture #1

Sizes Total no. of Rectangles

1x6 21

1x7 28

1x8 36

1x10 45

1x11 55

1x12 66

1x13 78

1x14 91

1x15 105

The numbers 21, 28, 36, 45 55, 66,78, 91 and 105 were the numbers of the total

number of rectangles. And if every term is to be obtained you just need to add 1 more

than the preceding difference. Thus, this sequence must follow the pattern of triangular

numbers and conjecture# 1 is correct.

Consider this sequence: 21, 28, n.

Find for n or the next term.

To find for n or the next term, you must first get the preceding difference of the

first and second term.28-21= 7 , hence 7 is the difference. Now to get the next

term, add one to the preceding difference then add the sum to the preceding

term.

7+1=8. Then, 28+8= 36. n=36 is the next triangular number.

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Conjecture #2

Table 4

Size (LxW) No. Of Rectangles(TR)

Triangular No.(DT)(assigned)

TR/DT=Tn

5x6 315 15 21

5x7 420 15 28

5x8 540 15 36

5x9 675 15 45

5x10 825 15 55

10x6 1155 55 21

10x7 1540 55 28

10x8 1980 55 36

10x9 2475 55 45

10x10 3025 55 55

15x6 2520 120 21

15x7 3360 120 28

15x8 4320 120 36

15x9 5400 120 45

15x10 6600 120 55

20x6 4410 210 21

20x7 5880 210 28

20x8 7560 210 36

20x9 9450 210 45

20x10 11550 210 55

21, 28, 36, 45 and 55 in the last column are Triangular numbers. Therefore,

conjecture # 2 is justified.

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Conjecture #3

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V.SUMMARY OF CONJECTURES

Conjecture #1

There is a pattern in the total number of rectangles that could be placed in the

diagram. This pattern is also known as the “Sequence of Triangular Numbers”. A

pattern of polygonal or figurate numbers in which each triangular number is obtained by

adding one more than the preceding difference. In symbols, Tn = n (n+1)/2, where

Tn= triangular number and n= number. However, for rectangular grids with length

more than one, the formula is multiplied to the triangular number of the length.

Conjecture #2

There is an assigned number for every groups of rectangular grids grouped

according tothe same value of length. This number is called the

diagram’s“Triangular Number”. This number is also the triangular number of the

diagram’s length. When this constant number is used to divide the total

number of rectangles of a certain diagram, the result will be a triangular number.

Conjecture #3

To find for the total number of rectangles that could be possibly placed in the diagram, the formula is stated as:

TR = DT[w (w+1)/2] TR=total number of rectangles

DT=triangular number of the grid (also the triangular number of the length )

w =width of the diagram

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VI. RECOMMENDATION

1. Investigate further with figures like L-shape figure that could be fitted in an

mxn rectangular diagram. Find out if there is also formula for it.

VII. BIBLIOGRAPHY

Dilao, Soledad Jose, Bernabe, Julieta. Geometry, textbook for Third year,

SD Publications, Inc. 2009.

Rod Pierce DipCE BEng, Triangular Numbers, 2013.

Math

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