Investigating Stair Shapes
Introduction
This piece of coursework is going to investigate and look into the
relationship between stair numbers, and the stair total. It is
basically all about drawing patterns and conclusions from the
relationships, and constructing formulas based on them. Initially, I
will begin by looking into 3 stair shapes, and how the stair numbers
relate to the stair total in a grid of that size. Then I will broaden
the coursework by looking into different sized stair shapes, and
investigating their relationships. Near to the end of this
coursework, I will conclude my results by finding formulas for the
individual sized stair shapes, and then I will find the general
formula which works for any sized stair shape on any sized grid.
Part 1 - 3 stair shapes
If a stair shape is moved once to the right, its stair total is
increased by 6. Similarly, it is 6 smaller when moved once to the
left.
Stair number : 1
1+2+3+11+12+21 = 50
Stair number: 2
2+3+4+12+13+22 = 56
When a stair shape is moved once down, its stair total is decreased by
60. Similarly, it increases by 60 if moved up once.
Stair number : 1
Stair total: 1+2+3+11+12+21 = 50
Stair number : 11
Stair Total: 11+12+13+21+22+31 = 110
These rules are easily explainable as we can see that on the grid the
number increases from left to right starting in the left hand-side
corner in rows of ten.
So it is obvious that when each of the numbers is moved down or up a
row of ten then the stair number will lose a total value of 60, 10 for
each of the numbers of the shape.
The same is true for sideways movement, except that they will lose a
total value of 6, 1 for each of the numbers of the shape.