Here are my results algebraically
S(S+44) = S2 + 44S
(S+40)(S+4) = S2 + 44S + 160
The difference is 160 as I predicted
Squares Difference I noticed that the underlined
2x2 (12) 10 numbers were square numbers
3x3 (22) 40 and that they were the answer
4x4 (32) 90 to the area of the previous
5x5 (42) 160 square i.e. 4x4=16 so taking the
squares side length as w I
worked out this formula
F=(w-1)2x10 (where F is the
Difference)
I can now formulate an equation to find the difference of any square in a 10x10 grid. I shall use one of my 2x2 squares to demonstrate
As you know, I already have some algebraic values for this square
W
In this square W=2
S(S+10W-10+W-1) = 1(12) =12
(S+10W-10)(S+W-1) = (11)(2) = 22
The difference is 10, which is consistent with the difference for a 2x2 square, which proves that my formula works for 2x2 squares in a 10x10 grid. I shall now have to investigate what happens with other size squares.
(3x3)
S(S+10[W-1]+[W-1]) = 55(77) = 4235
(S+10[W-1])(S+[W-1]) = (75)(57) = 4275
again, this is the correct difference
(4x4)
S(S+10[W-1]+[W-1]) = 67(100) = 6700
(S+10[W-1])(S+[W-1]) = (97)(70) = 6790
This is, once again the correct difference, if the same can be said for my next algebraic square then I have proved my formula works.
(5X5)
S(S+10[W-1]+[W-1]) = 44(88) = 3872
(S+10[W-1])(S+[W-1]) = (84)(48) = 4032
The difference is 160, which is the correct value
Now I will investigate some squares in a 9x9 grid
Again, I shall investigate 2x2 squares
Again I shall lay out my findings algebraically
S(S+10) = S2+10S
(S+9)(S+1) = S2+10s+9
The difference is nine
My investigation continues with 3x3 squares
S(S+20) = S2+20S
(S+18)(S+2) = S2+20S+36
The difference is 36
Now I am investigating 4x4 squares
S(S+30) = S2+30S
(S+27)(S+3) = S2+30S+81
the difference is 81
My second theory (see Above†) will be proved right or wrong by my investigations into 5x5 squares within a 9x9 grid.
S(S+40) = S2+S40
(S+36)(S+4) = S2+S40+144
This is the correct difference
My second theory (see Above†) was correct
I am now going to produce a formula for squares in a 9x9 grid
As before, I shall use my algebraic values to investigate 2x2, 3x3, 4x4 and 5x5 squares
W
S(S+9W-9+W-1) = 1(11) =11
(S+9W-9)(S+W-1) = (10)(2)=20
the difference is 9, which is what it should be
S(S+9[W-1]+[W-1]) = 29(49) = 1421
(S+9[W-1])(S+[W-1] = (47)(31) = 1457
the difference is 36, which is the correct difference
S(S+9[W-1]+[W-1]) = 22(52) = 1144
(S+9[W-1])(S+[W-1]) = (49)(25) = 1225
The difference is 81
S(S+9[W-1]+[W-1]) = 41(81) = 3321
(S+9[W-1])( S+[W-1]) = (77)(45) = 3465
the difference is 144
Now I am investigating 8x8 grids
Here are my investigations into 2x2 squares
This proves my first theory (see above*)
S(S+9) = S2+9S
(S+8)(S+1) = S2+9S+8
The difference is 8
I am now once again investigating 3x3 squares
S(S+18) = S2+18S
(S+16)(S+2) = S2+18S+32
I hope to be able to prove my third theory (see below‡) with my next two investigations 4x4 and 5x5 squares.
S(S+27) = S2+27S
(S+24)(S+3) = S2+27S+72
the difference is 72
Now, here are my 5x5 squares.
S(S+36) = S2+36S
(S+32)(S+4) = S2+36S+128
The difference is 128
Now I hope to be able to create a formula to find the difference for any size square in any size grid I will use C as the term for my grid size. I think that as I have already proved that these formulas work with the numbers 10, 9 and 8 then all I have to do is substitute these numbers for the letter C and if it works once for the number 10,9 or 8 it will work every time for the number 10,9 or 8.
As you can see, I have chosen to use a 2X2 square from a 10X10 grid with a starting number of fifteen and in my algebraic square, I have substituted 10 for C.
S(S+C[W-1]) = 15(26) = 390
(S+C[W-1])( S+[W-1]) = (25)(16) = 400
the difference is ten, which is correct so I have no hesitation in writing that this formula works in a 10X10 grid.
To prove this formula doesn’t only work for 2X2 squares in 10X10 grids I have chosen to test it on a 3X3 square from a 9X9 grid.
S(S+C[W-1] +[W-1]) = 13(33) = 429
(S+C[W-1])( S+[W-1]) = (31)(15) = 465
The difference is 36 this almost proves that my formula works, to prove it beyond reasonable doubt I shall conduct one more test of the formula
S(S+C[W-1]+[W-1]) = 11(38) = 418
(S+C[W-1])(S+[W-1]) = (35)(14) = 490
the difference is 72
I am now extending my investigation to include rectangles within a 10x10 grid.
First, I shall investigate rectangles with a width of two, then three, then four, and lengths ranging from three to five obviously missing out any width that is the same as the current length.
(2x3)
(2x4)
(2x5)
(3x2)
The difference for 3X2 rectangles is the same as for 2X3 rectangles I had thought they would be different, this information may be useful later on in my essay.
(3X4)
(3X5)
(4X2)
(4X3)
There seems to be another link between two sets of rectangles the 3X4 and the 4X3 sets have the same difference.
(4X5)
If I modify my formula for the difference of any size square in a 10X10 grid, I can create a formula for any size rectangle in a 10X10 grid.
I already have the terms:
- S=the starting number
- W=the width
Now I need one for the length I shall use O.
To test my formula I shall use the rectangle
For this rectangle
O=3
W=2
S=24
S(S+10[O-1]+[W-1]) = 24(45) = 1080
(S+10[O-1])(S+[W-1]) = (44)(25) = 1100
the difference is 20 which is the correct number for this rectangle
* I now have a theory ; I think that the number in the difference column in a table for a 2x2 grid is the same as the size of the grid the 2x2 square is in, I shall investigate this further after I have completed my investigation of a 9x9 grid
† I have another theory, I think that the numbers in the difference column for a square in a 9x9 grid are 90% of the difference of the corresponding size squares in a 10x10 grid just as the size of a 9x9 grid is 90% of the size of a 10x10
‡ I am extending my second theory to say that the number in the difference column of a certain size square in an 8x8 grid is 80% of the same number for the same size square in a 10x10 grid and 70% for a 7x7 grid etc.