Observations:
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The left box is p, the position of the T-shape.
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The three boxes going across are all consecutive (they are one after another), which means if the first box was p, the next box is p + 1
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Which also means that the box after that is p + 2
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The top box subtracts g (the grid size) because it is decreasing. But, subtracting the grid size means that the box is right above. For example, in this example:
p – g = 31 – 9
= 22
22 is directly above 31 because each level increases or decreases by the grid size number. So, since the number I want it two spaces over, I add 2. The equation for that box is p – g + 2
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The bottom box adds g (the grid size) because it is increasing on the grid. But, since every level increased or decreased means you have to add or subtract the grid size number, adding g (which is 9 in this example) would give me the box directly underneath p. In this example:
p + g = 31 + 9
= 40
40 is directly underneath 31 because each level increases or decreases by the grid size number. So, since the box I want is 2 spaces over, I add 2. The equation for this box is p + g + 2
Working out the Formula:
Using the equations I have come up with in my observations, I can add then together to come up with a final formula for a 90º rotation about the p to find out the T-total of the new T-shape if we only know the position of the T-shape.
T = p (p + 1) + (p + 2) + (p + g + 2) + (p – g + 2)
T = 5p + 7
*The (+2) and the (-2) cancel each other out
Testing the Formula:
5 x 5 grid:
Using the formula:
T = 5p + 7
T = 5(12) + 7
T = 60 + 7
T = 67
T = 9 + 12 + 13 + 14 + 19
T = 67
6 x 6 grid:
Using the Formula:
T = 5p + 7
T = 5(22) + 7
T = 110 + 7
T = 117
T = 22 + 23 + 14 + 18 + 30
T = 117
7 x 7 grid:
Using the Formula:
T = 5p + 7
T = 5(24) + 7
T = 120 + 7
T = 127
T = 24 + 25 + 26 + 19 + 33
T = 127
180º Rotation
Observations:
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The top box p, is the position of the T-shape
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The box right underneath is one level down, which means the grid size number (g) must be added from p (because this grid increases as it goes down). Therefore, the equation for this box is p + g
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The box directly underneath that is two levels down from p, which means you multiply the grid size (g) by 2 (because you moved 2 levels), and then add it to p. The equation for this box is p + 2g
- Each time you go up a level the number decreases by the gird size, so therefore, each time you go down a level, the number increases by the grid size
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The three bottom boxes are also two levels down from p, which means you multiply the grid size number (g) by 2, and then add it to p (because it has decreased levels on the grid). Since these three numbers are consecutive numbers, the box on the left would be one less than the middle (p + 2g – 1), and the box on the right would be one more than the middle box ( p + 2g +1)
Working out the Formula:
Adding up all these small equations, I can come up with a final formula for a 180º rotation about p, to find T if we only know p.
T = p (p + g) + (p +2g) + (p + 2g – 1 ) + (p + 2g + 1)
T = 5p + 7g
*the (+1) and the (-1) cancel each other out
Testing the Formula:
5 x 5 grid:
Using the formula:
T = 5p + 7g
T = 5(12) + 7(5)
T = 60 + 35
T = 95
T = 12 + 17 + 21 + 22 + 23
T = 95
6 x 6 grid:
Using the Formula:
T = 5p + 7g
T = 5(22) + 7(6)
T = 110 + 42
T = 152
T = 22 + 28 + 33 + 34 + 35
T = 152
7 x 7 grid:
Using the Formula:
T = 5p + 7g
T = 5(24) + 7(7)
T = 120 + 49
T = 169
T = 24 + 31 + 37 + 38 + 39
T = 169
270º Rotation
Observations:
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p is the position of the T-shape
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The three numbers in the middle row are consecutive. So, this means that one box is p – 1 (it is one less than p because it is 1 box over) and the next one over would be p – 2 (it is 2 less than p because it is 2 boxes over)
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The top box is one level up from p, which means it decreases. Subtract the grid number (g) from p because it has increased one level, which means it has decreased (because the grid increases as it goes down). This equation is p – g – 2
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The bottom box is one level down from p, which means that it increases. Add the grid number to p, which gives you p + g – 2
Working out the Formula:
I can use these small equations and add them together, to find the final formula for a 270º Rotation about p, to find T if we only know p.
T = p + (p – 1 ) + (p – 2) + (p – g – 2) + (p + g – 2)
T = 5p – 7
*the (+g) and (-g) cancel each other out
Testing the Formula:
5 x 5 grid:
Using the formula:
T = 5p – 7
T = 5(14) – 7
T = 70 – 7
T = 63
T = 7 + 12 + 17 + 13 + 14
T = 53
6 x 6 grid:
Using the Formula:
T = 5p – 7
T = 5(22) – 7
T = 110 – 7
T = 103
T = 14 + 20 + 26 + 21 + 22
T = 103
7 x 7 grid:
Using the Formula:
T = 5p – 7
T = 5(24) – 7
T = 120 – 7
T = 113
T = 15 + 22 + 29 + 23 + 24
T = 113
Formulas for rotation about p:
