Proof:
- (37 x 55) – (35 x 57) = 40
- (75 x 93) – (73 x 95) = 40
Equation:
- This algebraic equation also proves that were ever this particular cross is situated on the grid, that the out come of the top right number multiplied by the bottom left number, subtract the top left number multiplied by the bottom right number is always 40.
Finding numbers:
(a) 35 37 (a+2) (b-2) 35 37 (b)
(a+11) 46 (b+9) 46
(a+20) 55 57 (a+22) (b+18) 55 57 (b+20)
(c-22) 35 37 (c-20) (d-20) 35 37 (d-18)
(c-11) 46 (d-9) 46
(c-2) 55 57 (c) (d) 55 57 (d+2)
(x-11) 35 37 (x-9)
(x) 46
(x+9) 55 57 (x+11)
Finding numbers in equations:
= (a+2)(a+20) – (a)(a+22)
= (a +20a+2a+40) – (a +22a)
= (a +22a+40) – (a +22a)
= 40
= (b)(b+18) – (b-2)(b+20)
= (b +18b) – (b +20b-2b-40)
= (b +18b) – (b +18b-40)
= -40
= (c-20)(c-2) – (c-22)(c)
= (c -2c-20c+40) – (c -22c)
= (c -22c+40) – (c -22)
= 40
= (d-18)(d) – (d-20)(d+2)
= (d -18d) – (d +2d-20d-40)
= (d -18d) – (d -18d-40)
= -40
= (x-9)(x+9) – (x-11)(x+11)
= (x +9x-9x-81) – (x +11x-11x-121)
= (x +x-81) – (x +x-121)
= 40
- This shows implies that if numbers b or d are known that the equation is changed, so the answer to (b x d) – (a x c) = 40, positive or negative pending on the number known.
- Also, this cross can link back to the original:
Rule:
-
(Top left + bottom left)(top right + bottom right) – (top left + top right)(bottom left +bottom right) = 99
4
- This rule states that if the top left number add the bottom left number multiplied by the top right number add the bottom right number, subtract the top left plus the top right multiplied by the bottom left plus the bottom right number. With number, it’s divided by 4 to give 99.
Proof:
-
(35 + 55)(37 + 57) – (35 + 37)(55 + 57) = 99
4
-
(73 + 93)(75 + 95) – (73 +75)(93 + 95) = 99
4
Equation:
-
(a + d)(b + c) – (a + b)(d + c) = 99
4
- This algebraic equation also proves that were ever this particular cross is situated on the grid, that the out come of the top left number add the bottom left number multiplied by the top right number add the bottom right number, subtract the top left plus the top right multiplied by the bottom left plus the bottom right number, then that total number divided by four always gives 99
Symmetrical cross
26 54 a1
36 64 a2
44 45 46 47 48 72 73 74 75 76 d1 d2 x b2 b1
56 84 c2
66 94 c1
Rule:
-
(1st left number x end right number) – (top number – bottom number) = 396
-
This rule states if the 1st left number multiplied by the end right number, subtract the top number multiplied by bottom number the answer is always 396.
Proof:
- (44 x 48) – (36 x 66) = 396
- (72 x 76) – (54 – 94) = 396
Equation:
- (d1 x b1) – (a1 x c1) = 396
-
This algebraic equation shows were ever this particular cross is situated on the grid that if the 1st left number multiplied by the end right number, subtracted by the top number multiplied by the bottom number always gives 396.
Finding numbers:
26 (a) 26 (b-22)
36 36
(a+18) 44 45 46 47 48 (a+22) (b-4) 44 45 46 47 48 (b)
(a+20) 56 (b-2) 56
66 (a+40) 66 (b+18)
26 (c-40) 26 (d-18)
36 36
(c-22) 44 45 46 47 48 (c-18) (d) 44 45 46 47 48
(c-20) 56 (d+2) 56
66 (c) 66 (d+22)
26 (x-20)
36
(x-2) 44 45 46 47 48
(x) 56
66 (x +20)
Finding numbers in equations:
= (a+18)(a+22) – (a)(a+40)
= (a +22a+18a+396) – (a +40a)
= (a +40a+396) – (a +40)
= 396
= (b)(b-4) – (b-22)(b+18)
= (b -4b) – (b +18b-22b-396)
= (b -4b) – (b -4b-396)
= -396
= (c-22)(c-18) – (c)(c-40)
= (c -18c-22c+396) – (c -40c)
= (c -40c+396) – (c -40c)
= 396
= (d)(d+4) – (d-18)(d+22)
= (d +4d) – (d +22d-18d-396)
= (d +4d) – (d +4d-396)
= -396
= (x-2)(x+2) – (x+20)(x-20)
= (x +2x-2x-4) – (x -20x+20x-400)
= (x +x-4) – (x +x-400)
= 396
- This also implies that the sum (d1 x b1) – (a1 x c1) = 396 can come out positive or negative, pending on the number used to solve the equation.
- This equation also relates back to the first equation of (d x b) – (a x c) = 99 as 369 is a multiple of 99 as it equals 99 multiplied by 4 equals 396. This also shows that the cross is proportional, as the length of the cross has increased by 2, 2 multiplied by 2 equals 4.
Proof
-
(44 x 48) – (36 x 66) = 99
4
-
(72 x 76) – (54 – 94) = 99
4
Equation
-
(d1 x b1) – (a1 x c1) = 99
4
- This supports the new rule, which links the crosses together to shows the relationship of numbers in the cross equalling 99.
- If the cross had a length of three numbers, the rule would still work.
Proof
- (43 x 49) – (16 x 76) = 891
9
- (61 x 67) – (44 x 94) = 891
9
Equation
-
(b1 x d1) – (a1 x c1) = 99
(length )
-
This equation shows that if the first left number is multiplied by the end right number, subtract the top number multiplied by the bottom number. With that number divide by the length of the arm of the cross from x, squared i.e. 3 = 9. So making the equation (b1 x d1) – (a1 x c1) = 99 (length )
- This proves that this equation is proportional to the first equation.
Y
- K gives the number that divides into another number to give 99.
Asymmetrical cross
36 64 a
44 45 46 47 48 72 73 74 75 76 d2 d1 x b2 b1
56 84 c
Rule:
-
(1st left number x end right number) – (top number – bottom number) = 96
-
This rule states that were ever this particular cross is situated on the cross that if the 1st left number multiplied by the end right number, subtract the top number multiplied by the bottom number always equals 96.
Proof
- (44 x 48) – (36 x 56) = 96
- (72 x 76) – (84 x 64) = 96
Equation
-
This algebraic equation shows were ever this particular cross is situated on the grid that if the 1st left number multiplied by the end right number, subtracted by the top number multiplied by the bottom number always gives 96.
Finding numbers:
36 (a) 36 (a-12)
(a+8) 44 45 46 47 48 (a+12) (b-4) 44 45 46 47 48 (b)
(a+10) 56 (a+20) (b-2) 56 (b+8)
36 (c-20) 36 (d-8)
(c-12) 44 45 46 47 48 (c-18) (d) 44 45 46 47 48 (d+2)
(c-10) 56 (c) (d+2) 56 (d+12)
36 (x-10)
(x-2) 44 45 46 47 48 (x+2)
(x) 56 (x+10)
Finding numbers in equations:
= (a+8)(a+12) – (a)(a+10)
= (a +12a+8a+16) – (a +10a)
= (a +10a+96) – (a +10)
= 96
= (b)(b-4) – (b-12)(b+8)
= (b -4b) – (b +8b-12b-96)
= (b -4b) – (b -4b-96)
= -96
= (c-12)(c-8) – (c)(c-20)
= (c -8c-12c+96) – (c -20c)
= (c -20c+96) – (c -20c)
= 96
= (d)(d+2) – (d-8)(d+12)
= (d +4d) – (d +12d-8d-96)
= (d +4d) – (d +4d-96)
= -96
= (x-2)(x+2) – (x+10)(x-10)
= (x +2x-2x-4) – (x -10x+10x-100)
= (x +x-4) – (x +x-100)
= 96
- This also implies that the sum (d1 x b1) – (a x c) = 96 can come out positive or negative, pending on the number used to solve the equation.
General equation
Rule
-
(Left number + right number) = middle number
2
- This rule states that were ever this particular cross is situated on the cross that if the left number plus the right number, divided by 2 always equals the middle number.
Proof
99
99
Equation
2
-
This algebraic equation shows were ever this particular cross is situated on the grid that if the left number plus the right number, divided by 2 always gives the middle number. This can also be done on any type of cross, and with also using (a + c) = x to also find the middle number.
2
- This also proves that: (a + c) – (d + b) = 0 as they both give the middle number squared.
- Example: (44 + 48) – (36 + 56) = 0 , (72 +76) – (64 + 84) = 0
Proof of 99
- (a + 9)(a + 11) – (a)(a+20) = 99
- (a + 19)(a + 21) – (a)(a + 40)
(a +21a +19a +399) – (a + 40a)
a +40a + 399 – a - 40a = 399
- (a + 29)(a + 31) – (a)(a + 60)
(a + 31a + 29a + 599) – (a +60a)
a + 60a +899 – a - 60a = 899
- (a + 39)(a + 41) – (a)(a + 80)
(a +41a +39a +1599) – (a + 80a)
a +80a + 1599 – a - 80a = 1599
1st = 300 500 700
2nd = 200 200
Relationship for a general formula
100 n
s = 99 399 899 1599
100n =100 400 900 1600
-100n = -1 -1 -1 -1
Formula = 100n -1
Changing the grid
- When changing the grid, the only thing it will change is the numbers in the cross. This will make no difference to the equations as even if the numbers are negative the outcome is positive. Although, if the grid went up in 0.5, it would make a difference.
- Example: (11.5 x 12.5) – (2 x 22) = 99.75
Cross with no same length
- I found that if the original cross was developed to have no two sides the same that there wouldn’t be a constant number able to be found, and that it wouldn’t relate at all back to the original cross.
Conclusion
- From this coursework, I found that (Left x right number) – (top x bottom number) = 99, with changes, can apply to any and every cross, made on the 10 x 10 grid of squares. I have also concluded and proved my hypnosis’s, and found a conclusion to all. I have also extended my work to find solutions to finding the number 99 in different ways, this has also proved my first equation, and supports and proves the equation. I also found that if the cross had two or more sides the same, that it was possible to relate back to the original cross or equation, pending on whether the cross developed length ways or width ways.
Evaluation
- If I did this coursework again I wouldn’t change the way I set out my work, or disclose any of the work I have done already. I would do more research into grids, to see if this at all effected the cross, and find different rules that apply to all crosses through out. I would also do the cross more extended i.e. increasing in one length side and one width side, to see if the equation could apply to more, and that if the original cross could relate to any cross made on the 10 x 10 square grid.
