1140-1120=20
Example 3
Top Left x Bottom Right- 74 x 86= 6364
Top Right x Bottom Left- 76 x 84= 6384
6384-6364=20
General Rule For 3x2
If we look at the 3x2 square algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+12)=X2+12X
Top Right x Bottom Left- (X+2)(X+10)=X2+12X+20
Prove
X=2
Top Left x Bottom Right- (2)2+12(2)=4+24=28
Top Right x Bottom Left- (2)2+12(2)+20=4+24+20=48
48-28=20 QED
This proves the difference is always 20
4x2
I will now be looking at 4x2 rectangle and seeing if it follows a pattern.
Example 1
Top Left x Bottom Right- 17 x 30= 510
Top Right x Bottom Left- 20 x 27= 540
540-510=30
The difference is 30. I will now look at 2 more examples and if this correct for those as well.
Example 2
Top Left x Bottom Right- 31 x 44= 1364
Top Right x Bottom Left- 34 x 41= 1394
1394-1364=30
Example 3
Top Left x Bottom Right- 86 x 99= 8514
Top Right x Bottom Left- 89 x 96= 8544
8544-8514
General Rule For 4x2
If we look at the 4x2 square algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+13)=X2+13X
Top Right x Bottom Left- (X+3)(X+10)=X2+13X+30
Prove
X=2
Top Left x Bottom Right- (2)2+13(2)= 4+26=30
Top Right x Bottom Left- (2)2+13(2)+30= 4+26+30= 60
60-30= 30 QED
This proves that the difference is always 30
5x2
I will now be looking at a 3x2 rectangle and seeing if it follows a pattern
Example 1
Top Left x Bottom Right- 11 x 25= 275
Top Right x Bottom Left- 15 x 21= 315
315-275=40
The difference is 40. I will now look at 2 further examples to see if it is the same for them as well.
Example 2
Top Left x Bottom Right- 36 x 50= 1800
Top Right x Bottom Left- 40 x 46= 1840
1840-1800=40
Example 3
Top Left x Bottom Right- 54 x 68= 3672
Top Right x Bottom Left- 58 x 64= 3712
3712-3672=40
General Rule For 5x2
If we look at the 5x2 square algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+14)=X2+14X
Top Right x Bottom Left-(X+4)(X+10)=X2+14X+40
Prove
X=2
Top Left x Bottom Right- (2)2+14(2)= 4+28= 32
Top Right x Bottom Left- (2)2+14(2)+40=4+28+40=72
72-32=40 QED
This proves the difference is always 40
General Rule For MxN
An M x N rectangle can also be written like this. I have only included the four numbers that are used in the multiplication as the others are not necessary in this equation.
Top Left x Bottom Right-(X)(X+10(N-1)+(M-1)=X2+10X(N-1)+X(M-1)
Top Right x Bottom Left-(X+(M-1))(X+10(N-1))=X2+10X(N-1)+X(M-1)+10(N-1)(M-1)
T.R. X B.L. – T.L. x B.R.=X2+10X(N-1)+X(M-1)+10(N-1)(M-1)-X2+10X(N-1)+X(M-1)
=10(N-1)(M-1)
Prove
M=3 N=2
10(2-1)(3-1)=10(1)(2)=20 QED
This is correct as the difference for a 3x2 rectangle is 20 so this proves this equation can be used for any MxN square
I will now be looking at the same shapes in a different number grid. One which has row’s of 11 instead of 10, basically a 11x10 number grid.
2x2
I will be looking at a 2x2 square in this new grid and seeing if it follows any patterns.
Example 1
Top Left x Bottom Right- 27 x 39=1053
Top Right x Bottom Left- 28 x 38= 1064
1064-1053= 11
The difference is 11. I will do 2 more examples to see if this is the pattern
Example 2
Top Left x Bottom Right- 54 x 66= 3564
Top Right x Bottom Left- 55 x 65= 3575
3575-3564=11
Example 3
Top Left x Bottom Right- 79 x 91= 7189
Top Right x Bottom Left- 80 x 90= 7200
General Rule For 2x2
If we look at the 2x2 square algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+12)= X2+12X
Top Right x Bottom Left- (X+1)(X+11)=X2+12X+11
Prove
X=2
Top Left x Bottom Right- (2)2 +12(2)=4+24=28
Top Right x Bottom Left- (2)2+12(2)+11=4+24+11=39
39-28=11 QED
This proves the difference is always 11
3x3
I will now be looking at a 3x3 square in this number grid and seeing if it follows a pattern.
Example 1
Top Left x Bottom Right- 1 x 25= 25
Top Right x Bottom Left- 3 x 23= 69
69-25= 44
The difference is 44. I will do further examples to see if there is a pattern.
Example 2
Top Left x Bottom Right- 31 x 55= 1705
Top Right x Bottom Left- 33 x 53= 1749
1749-1705= 44
Example 3
Top Left x Bottom Right- 71 x 95= 6745
Top Right x Bottom Left- 73 x 93= 6789
6789-6745= 44
General Rule For 3x3
If we look at the 3x3 square algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+24)= X2+24X
Top Right x Bottom Left- (X+2)(X+22)=X2+24X+44
Prove
X=2
Top Left x Bottom Right- (2)2+24(2)= 4+48= 52
Top Right x Bottom Left- (2)2+24(2)+44= 4+48+44= 96
96-52= 44 QED
This proves the difference is always 44.
4x4
I will now look at a 4x4 square in this number grid and see if there is a pattern.
Example 1
Top Left x Bottom Right- 4 x 40= 160
Top Right x Bottom Left- 7 x 37= 259
259-160= 99
The difference is 99. I will now do 2 further examples and see if this the same for them as well.
Example 2
Top Left x Bottom Right- 34 x 70= 2380
Top Right x Bottom Left- 37 x 67= 2479
2479-2380= 99
Example 3
Top Left x Bottom Right- 74 x 110= 8140
Top Right x Bottom Left- 77 x 107= 8239
8239-8140= 99
General Rule For 4x4
If we look at the 3x3 square algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+36)=X2+36X
Top Right x Bottom Left- (X+3)(X+33)=X2+36X+99
Prove
X=2
Top Left x Bottom Right- (2)2+36(2)= 4+72= 76
Top Right x Bottom Left- (2)2+36(2)+99= 4+72+99= 175
175-76= 99 QED
This proves the difference is always 99
General Rule For NxN Square
An N x N square can also be written like this. I have only included the four numbers that are used in the multiplication as the others are not necessary in this equation.
Top Left x Bottom Right- (X)(X+11(N-1)+(N-1))=X2+11X(N-1)+X(N-1)
Top Right x Bottom Left- (X+(N-1))(X+11(N-1))=X2+11X(N-1)+X(N-1)+11(N-1)2
T.R. x B.L. – T.L. x B.R. = X2+11X(N-1)+X(N-1)+11(N-1)2-X2+11X(N-1)+X(N-1)
= 11(N-1)2
Prove
X=3
11(N-1)2=11(3-1)2=11(2)2=11(4)=44 QED
This is correct as the difference in a 3x3 square is 44.
I will now be looking at the different rectangular formations in this 11x10 number grid and seeing whether they follow any patterns.
3x2
Primarily I will be looking at a 3x2 rectangle and seeing if it follows a pattern
Example 1
Top Left x Bottom Right- 9 x 22= 198
Top Right x Bottom Left- 11 x 20= 220
220-198= 22
The difference is 22. I will now do 2 further examples to see if this is the same for them.
Example 2
Top Left x Bottom Right- 49 x 62= 3038
Top Right x Bottom Left- 51 x 60= 3060
3060-3038=22
Example 3
Top Left x Bottom Right- 89 x 102= 9078
Top Right x Bottom Left- 91 x 100= 9100
9100-9078=22
General Rule For 3x2
If we look at the 3x2 rectangle algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+13)= X2+13X
Top Right x Bottom Left- (X+2)(X+11)= X2+13X+22
Prove
Top Left x Bottom Right- (2)2+13(2)= 4+26=30
Top Right x Bottom Left- (2)2+13(2)+22= 4+26+22=52
52-30= 22 QED
This proves that the difference is always 22
4x2
I will now be looking at 4x2 rectangles and seeing if they follow a pattern.
Example 1
Top Left x Bottom Right- 30 x 44= 1320
Top Right x Bottom Left- 33 x 41= 1353
1353-1320= 33
The difference is 33. I will do 2 further examples to see if it is the same for them as well.
Example 2
Top Left x Bottom Right- 92 x 106= 9752
Top Right x Bottom Left- 95 x 103= 9785
9785-9752=33
Example 3
Top Left x Bottom Right- 58 x 72= 4176
Top Right x Bottom Left- 61 x69= 4209
4209-4176= 33
General Rule For 4x2
If we look at the 4x2 rectangle algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+14)= X2+14X
Top Right x Bottom Left- (X+3)(X+11)= X2+14X+33
Prove
Top Left x Bottom Right- (2)2+14(2)=4+28=32
Top Right x Bottom Left- (2)2+14(2)+33=4+28+33=65
65-32= 33 QED
This proves that the difference is always 33
5x2
I will now be looking at 5x2 rectangles and seeing if they follow a pattern.
Example 1
Top Left x Bottom Right- 56 x 71= 3976
Top Right x Bottom Left- 60 x 67= 4020
4560-3976=44
The difference is 44. I will now be looking at 2 further examples to see if I obtin the same result.
Example 2
Top Left x Bottom Right- 12 x 27= 324
Top Right x Bottom Left- 16 x 23= 368
368-324= 44
Example 3
Top Left x Bottom Right- 78 x 93= 7254
Top Right x Bottom Left- 82 x 89= 7298
7298-7254=44
General Rule For 5x2
If we look at a 4x2 rectangle algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+15)=X2+15X
Top Right x Bottom Left- (X+4)(X+11)=X2+15X+44
Prove
X=2
Top Left x Bottom Right- (2)2+15(2)= 4+30= 34
Top Right x Bottom Left- (2)2+15(2)+44=4+30+44=78
78-34=44 QED
This proves the difference is always 44.
General Rule For MxN
An M x N rectangle can also be written like this. I have only included the four numbers that are used in the multiplication as the others are not necessary in this equation.
Top Left x Bottom Right-(X)(X+11(N-1)+(M-1))=X2+11X(N-1)+X(M-1)
Top Right x Bottom Left-(X+(M-1))(X+11(N-1))=X2+11X(N-1)+X(M-1)+11(N-1)(M-1)
T.R. x B.L.-T.L. x B.R.= X2+11X(N-1)+X(M-1)+11(N-1)(M-1)- X2+11X(N-1)+X(M-1)
=11(N-1)(M-1)
Prove
M=3 N=2
11(N-1)(M-1)=11(2-1)(3-1)=11(1)(2)=22 QED
*the difference for a 3x2 rectangle is 22
I will now be looking at the same shapes in a different number grid. One which instead of increasing in 1 per term it increases by 5. It is still a 10x10 grid though
2x2
I will now be looking at a 2x2 square in this new grid.
Example 1
Top Left x Bottom Right- 50 x 105= 5250
Top Right x Bottom Left- 55 x 100= 5500
5500-5250=250
The difference is 250. I will now look at 2 further examples to see if this is the same for them as well.
Example 2
Top Left x Bottom Right- 240 x 295= 70800
Top Right x Bottom Left- 245 x 290= 71050
71050-70800= 250
Example 3
Top Left x Bottom Right- 110 x 165= 18150
Top Right x Bottom Left- 115 x 160= 18400
18400-18150=250
General Rule For 2x2
If we look at a 2x2 square algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+55)= X2+55X
Top Right x Bottom Left- (X+5)(X+50)= X2+55X+250
Prove
X=2
Top Left x Bottom Right- (2)2+55(2)=4+110=114
Top Right x Bottom Left- (2)2+55(2)+250= 4+110+250= 364
364-114= 250 QED
This proves the difference is always 250
3x3
I will now be looking at a 3x3 square in this grid
Example 1
Top Left x Bottom Right- 255 x 365= 93075
Top Right x Bottom Left- 265 x 355= 94075
94075-93075= 1000
The difference is 1000. I will now do 2 further examples to see if this is a pattern.
Example 2
Top Left x Bottom Right- 15 x 125= 1875
Top Right x Bottom Left- 25 x 115= 2875
2875-1875=1000
Example 3
Top Left x Bottom Right- 185 x 295= 54575
Top Right x Bottom Left- 195 x 285= 55575
55575-54575= 1000
General Rule For 3x3
If we look at a 3x3 square algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+110)= X2+110X
Top Right x Bottom Left- (X+10)(X+100)= X2+110X+1000
Prove
X=2
Top Left x Bottom Right- (2)2+110(2)=4+220=224
Top Right x Bottom Left- (2)2+110(2)+1000=4+220+1000=1224
1224-224= 1000 QED
4x4
I will now be looking at a 4x4 square.
Example 1
Top Left x Bottom Right- 320 x 485= 155200
Top Right x Bottom Left- 335 x 470= 157450
157450-155200= 2250
The difference is 2250. I will do 2 further examples to see if this is constant.
Example 2
Top Left x Bottom Right- 200 x 365= 73000
Top Right x Bottom Left- 215 x 350= 75250
75250-73000= 2250
Example 3
Top Left x Bottom Right- 65 x 230= 14950
Top Right x Bottom Left- 80 x 215= 17200
17200-14950= 2250
General Rule For 4x4
If we look at a 4x4 square algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+165)=X2+165X
Top Right x Bottom Left- (X+15)(X+150)= X2+165X+2250
Prove
X=2
Top Left x Bottom Right- (2)2+165(2)=4+330=334
Top Right x Bottom Left- (2)2+165(2)+2250=4+330+2250=2584
2584-334=2250 QED
This proves that the difference is always 2250
General Rule For NxN
An N x N square can also be written like this. I have only included the four numbers that are used in the multiplication as the others are not necessary in this equation.
Top Left x Bottom Right-(X)(X+(50N-50)+(5N-5))=X2+(50NX-50X)+(5NX-5X)
Top Right x Bottom Left-(X+(5N-5))(X+(50N-50))=X2+(50NX-50X)+(5NX-5X)+(50N-50)(5N-5)
T.R.xB.L.–T.L.xB.R.=X2+(50NX-50X)+(5NX-5X)+ (50N-50)(5N-5)- X2+(50NX-50X)+(5NX-5X)
= (50N-50)(5N-5)=250N2-250N-250N+250=250N2-500N+250
Prove
N=2
250(2)2-500(2)+250=1000-1000+250=250 QED
* the difference for a 2x2 square is 250
I will now be looking at the different rectangular formations in this particular grid and seeing whether they follow any patterns.
3x2
I will now be looking at a 3x2 rectangle in this particular grid.
Example 1
Top Left x Bottom Right- 400 x 460= 184000
Top Right x Bottom Left- 410 x 450= 184500
184500-184000=500
The difference is 500. I will do 2 further examples to see if they are the same as well.
Example 2
Top Left x Bottom Right- 35 x 95= 3325
Top Right x Bottom Left- 45 x 85= 3825
3825-3325=500
Example 3
Top Left x Bottom Right- 325 x 385= 125125
Top Right x Bottom Left- 335 x 375= 125625
125625-125125=500
General Rule For 3x2
If we look at a 3x2 rectangle algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+60)=X2+60X
Top Right x Bottom Left- (X+10)(X+50)= X2+60X+500
Prove
X=2
Top Left x Bottom Right- (2)2+60(2)= 4+120=124
Top Right x Bottom Left- (2)2+60(2)+500=4+120+500=624
624-124=500 QED
This proves the difference is always 500
4x2
I will now be looking at a 4x2 rectangle in this particular grid.
Example 1
Top Left x Bottom Right- 175 x 240= 42000
Top Right x Bottom Left- 190 x 225= 42750
42750-42000= 750
The difference is 750. I will do 2 more examples to see if this is the same for them.
Example 2
Top Left x Bottom Right- 430 x 495- 212850
Top Right x Bottom Left- 445 x 480= 213600
213600-212850= 750
Example 3
Top Left x Bottom Right- 255 x 320= 81600
Top Right x Bottom Left- 270 x 305= 82350
82350-81600=750
General Rule For 4x2
If we look at a 4x2 rectangle algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+65)=X2+65X
Top Right x Bottom Left- (X+15)(X+50)= X2+65X+750
Prove
X=2
Top Left x Bottom Right- (2)2+65(2)=4+130=134
Top Right x Bottom Left- (2)2+65(2)+750=4+130+750= 884
884-134=750 QED
This proves the difference is always 750
5x2
I will now be looking at a 5x2 rectangle in this particular grid.
Example 1
Top Left x Bottom Right- 50 x 120= 6000
Top Right x Bottom Left- 70 x 100= 7000
7000-6000=1000
The difference is 1000. I will do 2 further examples to see if this is the same for those as well.
Example 2
Top Left x Bottom Right- 25 x 95= 2375
Top Right x Bottom Left- 45 x 75= 3375
3375-2375=1000
Example 3
If we look at a 4x2 rectangle algebraically it would look like this:-
Top Left x Bottom Right- 405 x 475= 192375
Top Right x Bottom Left- 425 x 455= 193375
193375-192375=1000
General Rule For 5x2
If we look at a 4x2 rectangle algebraically it would look like this:-
Top Left x Bottom Right- (X)(X+70)=X2+70X
Top Right x Bottom Left- (X+20)(X+50)= X2+70X+1000
Prove
X=2
Top Left x Bottom Right- (2)2+70(2)=4+140=144
Top Right x Bottom Left- (2)2+70(2)+1000=4+140+1000=1144
1144-144=1000 QED
This proves the difference is always 1000
General Rule For MxN
An M x N rectangle can also be written like this. I have only included the four numbers that are used in the multiplication as the others are not necessary in this equation.
Top Left x Bottom Right- (X)(X+50N-50+5M-5)=X2+50NX-50X+5MX-5X
Top Right x Bottom Left- (X+5M-5)(X+50N-50)= X2+50NX-5X+5MX-5X+(50N-50)(5M-5)
T.R.xB.L.-T.L.xB.R.=X2+50NX-5X+5MX-5X+(50N-50)(5M-5)- X2+50NX-50X+5MX-5X
= (50N-50)(5M-5)= 250MN-250N-250M+250
Prove
M=3 X=2
250MN-250N-250M+250=250(3)(2)-250(2)-250(3)+250=1500-500-750+250=500
*the difference for 3x2 is 500
Conclusion
In this piece I have proved that in any number grid, as long as the number sequences follow a regular pattern, there is a NxN and MxN pattern which will work and have separate rules and results for separate grids.
