Borders - find out the differences in the patterns of the colored squares.

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George Eghator 11O

BORDERS

A

Table of Results

Num.          num black                   num white                             total

1.        0        1        1

2.        1        4        5

3.        5        8        13

4.        13        12        25

5.        25        16        41

6.        41        20        61

7.        61        24        85

8.        85        28        113

9.        113        32        145

N                      Total                                       Structure

1        1        (1+1)2+1²

2        5        (2+1)2+2²

3        13        (3+1)2+3²

4        25        (4+1)2+4²

5        41        (5+1)2+5²

6        61        (6+1)2+6²

7        85        (7+1)2+7²

8        113        (8+1)2+8²

9        145        (9+1)2+9²

Order of the squares

1 + 3 + 1 = 5

1 + 3 + 5 + 3 + 1 = 13

In each of these cases, 2 have been added.

1 + 3 +5 7 + 5 + 3 + 1 = 25

As seen in the other pattern, there are additions with 2 being added on.

For the next pattern, I think that the total number of squares will be 41, using the following pattern:

1 + 3 + 5 + 7 + 9 + 7 + 5 + 3 + 1 = 41

Now I am going to test my prediction.

Number of squares = 25 + 16 = 41

Join now!

This result is made because we are just adding onto it.

Differences

 

From the quadratic sequence, we see that the main difference is 4. The first formula I will try to find is the formula for the surrounding white squares.

Formula for white squares

Pattern

'font-size:12.0pt; '>1        x 4 = 4 white squares

'font-size:12.0pt; '>2        x 4 = 8 white squares

'font-size:12.0pt; '>3        x 4 = 12 white squares

N = pattern number

D = dark squares

W = white squares

I believe that the formula is 4 x the pattern number or 4N. ...

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