Upon elaboration of the number grid on Page 2 (Fig. 1) we can call the number of each block going along the side of the grid (the coefficient of w in the formulae) “y” and call the number of each block going along the bottom (the number after the addition sign in the formulae) “x”.
x
y
Fig. 1
When we incorporate this into the formulae that solve the numbers in each block on a number grid we can say that the formula for any block on any number grid is:
wy-w+x
When the keystone of a step stair is the block in the bottom left hand corner of any size step stair. (For example, the keystone of the 3-step stair on Fig. 1 on page 1 is 25) the formula for the keystone is:
wy-w+x
and the formula for the block just right of the keystone is:
(wy-w+x)+0w+1
and therefore the formula for the block just above the keystone is:
(wy-w+x)+w+0
From this we can say that the formula for a block in a step stair is the same as the formula for the keystone with an integer and a multiple of w added to it. The integer added is the number of places the block is to the right of the keystone and the multiple of w added is the number of places the block is higher than the keystone (for example, if the block in question is two places higher and two places to the right of the keystone the formula would be: (wy-w+x)+2w+2).
Thus, we can calculate the stair total of a step stair given the coordinates of the keystone (see page 5).
Fig. 1
The formula for a 3-step stair is:
(wy-w+x) + [(wy-w+x)+1] + [(wy-w+x)+2] + [(wy-w+x)+w] + [(wy-w+x)+w+1] + [(wy-w+x)+2w]
This can be simplified to:
6wy-2w+6x+4
From these formulae, when x is substituted with 5 and y is substituted with 3 (5 and 3 being the coordinates of the keystone of the step stair positioned lower on the grid above), an answer of 194 is calculated.
25+26+27+35+36+45 = 194
Also from these formulae, when x is substituted with 7 and y is substituted with 7 (7 and 7 being the coordinates of the keystone of the step stair positioned higher on the grid above), an answer of 446 is calculated.
67+68+69+77+78+87 = 446
This confirms the formula 6wy-2w+6x+4 can be used to calculate the stair total of a 3-step stair on a 10 by 10 grid.
Fig. 1
From the formula (6wy-2w+6x+4), when x is substituted with 2 and y is substituted with 2 (2 and 2 being the coordinates of the keystone of the step stair positioned lower on the grid above), an answer of 86 is calculated.
9+10+11+16+17+23 = 86
From the formula (6wy-2w+6x+4), when x is substituted with 4 and y is substituted with 6 (5 and 6 being the coordinates of the keystone of the step stair positioned higher on the grid above), an answer of 266 is calculated.
39+40+41+46+47+53 = 266
This confirms that the formula 6wy-2w+6x+4 can be used to calculate the stair total of a 3-step stair on a number grid of any size.
Fig. 1
The formula for the 4-step stair is:
(wy-w+x) + [(wy-w+x)+1] + [(wy-w+x)+2] + [(wy-w+x)+3] + [(wy-w+x)+w] + [(wy-w+x)+w+1] + [(wy-w+x)+w+2] + [(wy-w+x)+2w] + [(wy-w+x)+2w+1] +
[(wy-w+x)+3w]
This can be simplified to:
10wy-0w+10x+10
From these formulae, when x is substituted with 2 and y is substituted with 2 (2 and 2 being the coordinates of the keystone of the step stair positioned lower on the grid above), an answer of 230 is calculated.
12+13+14+15+22+23+24+32+33+42 = 230
Also from these formulae, when x is substituted with 4 and y is substituted with 6 (4 and 6 being the coordinates of the keystone of the step stair positioned higher on the grid above), an answer of 650 is calculated
54+55+56+57+64+65+66+74+75+84 = 650
This confirms the formula 10wy-0w+10x+10 can be used to calculate the stair total of a 3-step stair on a 10 by 10 grid.
Fig. 1
From the formula (10wy-0w+10x+10), when x is substituted with 5 and y is substituted with 1 (5 and 1 being the coordinates of the keystone of the step stair positioned lower on the grid above), an answer of 140 is calculated.
5+6+7+8+13+14+15+21+22+29 = 140
From the formula (10wy-0w+10x+10), when x is substituted with 2 and y is substituted with 6 (2 and 6 being the coordinates of the keystone of the step stair positioned higher on the grid above), an answer of 510 is calculated.
42+43+44+45+50+51+52+58+59+66 = 510
This confirms that the formula 10wy-0w+10x+10 can be used to calculate the stair total of a 4-step stair on a number grid of any size.
Fig. 1
The formula for the 5-step stair is:
(wy-w+x) + [(wy-w+x)+1] + [(wy-w+x)+2] + [(wy-w+x)+3] + [(wy-w+x)+4] +
[(wy-w+x)+w] + [(wy-w+x)+w+1] + [(wy-w+x)+w+2] + [(wy-w+x)+w+3] +
[(wy-w+x)+2w] + [(wy-w+x)+2w+1] + [(wy-w+x)+2w+2] + [(wy-w+x)+3w] +
[(wy-w+x)+3w+1] + [(wy-w+x)+4w]
This can be simplified to:
15wy+5w+15x+20
From these formulae, when x is substituted with 3 and y is substituted with 4 (3 and 4 being the coordinates of the keystone of the step stair on the grid above), an answer of 585 is calculated.
27+28+29+30+31+35+36+37+38+43+44+45+51+52+59 = 585
This confirms the formula 15wy+5w+15x+20 can be used to calculate the stair total of a 5-step stair on a number grid of any size.
Fig. 1
The formula for the 2-step stair is:
(wy-w+x) + [(wy-w+x)+1] + [(wy-w+x)+w]
This can be simplified to:
3wy-2w+3x+1
From these formulae, when x is substituted with 2 and y is substituted with 3 (2 and 3 being the coordinates of the keystone of the step stair positioned lower on the grid above), an answer of 63 is calculated.
18+19+26 = 63
This confirms the formula 3wy-2w+3x+1 can be used to calculate the stair total of a 2-step stair on a number grid of any size.