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# Diagonal Differences

Extracts from this document...

Introduction

Diagonal

Differences

## Diagonal Differences

Examples

I will try a 2X2, 3X3, 4X4 and 5X5 square in a 5X5 grid

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

2X2 square = 2X6=12,       1X7=7            12-7=5

3X3 square = 3X11=33,    1X13=13         33-13=20

4X4 square = 4X16=64,    1X19=19         64-19=45

5X5 square = 5X21=105,  1X25=25         105-25=80

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

2X2 square = 12X16=192,     11X17= 187    192-187=5

2X2 square = 4X8=32,            3X9=27          32-27=5

Middle

Also the number of rows in the 4X4 square are timed by 9 And in a 5X5 square the rows are timed by 16.

I then noticed that if you take the number of rows in the square and take away one and square the answer you can times this by the number of rows in the grid you have the diagonal difference. This equation is written

(rows in area used - one)squared X rows in grid = diagonal difference

or

(x-1)2

Conclusion

>

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

Yellow =a
Green =b
Blue    =c
Purple =d

D= diagonal differences

In “a” D is (2X6)-(1X7)=4
In “a+b” D is (3X6)-(1X8)=12

I noticed that D of “a+b” was 3 times “a”.

In “c” D is (13X21)-(11X23)=20

In “c+d” D is (15X21)-(11X25)=315-275=40

I noticed that D of “c+d” was 2 times “c”.

I will now try this on some different squares.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Yellow=a

Green=b

Blue=c

Purple=d

14X

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