# Opposite Corners

Opposite corners

On a 10*10 square grid, choose any 2*2 square, multiply the corners in that grid and then find the difference between the two corners investigate.

12  13           12       13                    16  17             16        17

22  23        * 23    * 22                    26  27          * 27     * 26

276     284                                         332      342

284-276=10                                     342-332=10

Difference=10

The two answers are the same. I think it would be the same for any 2*2 square. To prove this I will use algebra to show that in any 2*2 square the difference will 10.

z- number in the top left corner

z     z+1                                                                              z(z+11)=z²+11z

z+10z+11                (z+1)(z+10)=z²+11z+10      (z²+11z+10)-(z²+11z)=10

Difference = 10

This proves that with any 2*2 square the corners multiplied then subtracted always = 10

To further my investigations I am now going to use a 3*3 square and do the same as I did with the 2*2 square.

3     4      5               3          5               1      2      3              3          1

13   14    15        * 25     * 23               11    12    13       * 21     * 23

23   24    25           75      115               21    22    23          63        23

115-75=40                    Difference = 40                          63-23=40

Opposite corners

These answers are the same; just as the answer for the 2*2 squares are the same. I think that any 3*3 square would have a difference of 40. To prove this I will use algebra.

z     z+1   z+2                                                                  z(z+22)=z²+22zz

z+10 z+11z+12                                                      (z+2)(z=20)=z²+22z+40

z+20 z+21z+22                                                    (z²+22z+40)-(z²+22z)=40

This proves that with any 3*3 square the corners multiplied the subtracted always = 40

Now I am going to further my investigations again. I am now going to use a 4*4 square and do the same, as I did with the 2*2 and 3*3 square.

1      2      3     4             1           4           7      8     9     10          7         10

11    12    13   14       * 34      * 31          17    18   19    20     * 40      * 37

21    22    23   24          34       124          27    28   29    30      280       370

31    32    33   34       124-34=90            37    38    39   40     370-280=90

Difference = 90

The answer for the 4*4 squares are the same, just as the 2*2 and 3*3 squares had the same difference. I think that any 4*4 square would have a difference of 90. To prove this I will use algebra.

z-number in the top left corner

z     z+1  z+2   z+3                                                           z(z+33)=z²+33z

z+10z+11 z+12z+13                                       ...

#### Here's what a teacher thought of this essay

This is an excellent pieces of mathematical investigation. It is well structured moving from the concrete to the algebraic easily. The are a few small mathematical errors which limit the piece to four stars. There are specific strengths and improvements suggested throughout.