Marked on the grid is a stair shape; this is called a 3-Step Stair.The total of the numbers inside the stair shape is:

Authors Avatar

NUMBER STAIRS

This is a 10 by 10 Number Grid:

Fig. 1

Marked on the grid is a stair shape; this is called a 3-Step Stair.

The total of the numbers inside the stair shape is:

25 + 26 + 27 + 35 + 36 + 45 = 194

The Stair Total for this 3-step stair is 194.

Part 1

For other 3-step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid.

Part 2

Investigate further the relationship between the stair totals and other step stairs on other number grids.

w = width

Fig. 1

On every size of number grid the formula in each block above will be the right formula for the corresponding number on any grid. However, on larger or smaller number grids some blocks may not exist or more blocks may exist. The formulae in the furthest right hand side column are always in the furthest right hand side column on every size of grid as these define what the number is at the end of each row. However, the height and width of the number grid may be more than, or less than, five. Additional blocks on a larger grid will follow the same pattern as that which can be seen above and if on a smaller number grid there will just be less blocks (for example, on a 3 by 3 number grid the block in the top right hand corner will be 3w and on a 8 by 8 number grid the block in the top right hand corner will be 8w).

Join now!

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 ...

This is a preview of the whole essay