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# Investigating borders - Math Problem Example

## Extract of sampleInvestigating borders

1st difference
8 12 16 20 24 28 32 36 40
2nd difference
4 4 4 4 4 4 4 4

Now we will first find the formula for the white squares and after that the formula for the total squares. Subtracting from the total squares the number of white squares we will get the number of black squares.

Finding the formula for white squares
In each case we observe that if we multiply the pattern number by 4 it gives the amount of white squares.
For example, in case of pattern 1, we get 1x 4 = 4 white squares. In case of pattern 2, we get 2x 4 = 8 white squares, etc.
Using the pattern number only we can find out the total number of white squares.

Formula for white squares
We will fix the notation first, that is, we will use following symbols
N = pattern number,
D = black squares,
W = white squares, and
T = Total number of squares, throughout this discussion.

We claim that the formula for white squares is just 4N.
For example, for pattern 7, pattern number, N = 7, and the number of white squares = 4N = 47 = 28. We can verify by using the above table that this value is correct. We got
W = 4N

Finding a formula for the black squares
After analyzing the above tables, we get that the number of black squares equals the total number of squares in its previous pattern.

Pattern
Total squares
Black squares
White squares
1
5
1
4
2
13
5
8
3
25
13
12
4
41
25
16
5
61
41
20
6
85
61
24
7
113
85
28
8
145
113
32
9
181
145
36
10
221
181
40

To find a formula for the black squares, we can find a possible formula to find the total number of squares and then subtract the number of white squares.
Thus, we can write,
(Total number of squares) - (Number of white squares) = (Number of black squares)
That is, (Total number of squares) - 4N = (Number of black squares)

Finding a formula...
Now we will first find the formula for the white squares and after that the formula for the total squares. Subtracting from the total squares the number of white squares we will get the number of black squares.
Let us find the pattern in counting all the squares. In Pattern 1 we can see first two rows contain 1+3=4 squares. If we consider the last two rows then they also contain 1+3=4 squares. We are counting the three squares in the middle row two times so we have to subtract them from the total to get the total number of squares in the pattern. So we get
In Pattern 2 the first three rows contain 1+3+5=9 squares. Similarly the last three rows contain 9 squares. We are counting the five squares in the middle row two times so we have to subtract them to get the total number of squares in the pattern. So we get ...Show more

## Summary

As seen in the explanation of above tabular data, we can extend this idea further to find out the number of squares in the following patterns. To get the total number of squares in pattern 5, we add the total number of squares in the pattern 4, and new squares, following the above idea, that is, adding (11 + 9) squares, which gives the total number of squares as 61.
Author : dixie20
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