Squares right = 0
Squares up = 0
Abbreviations:
Total = n
Number of squares = s
Squares right = r
Squares up = u
For this position, in 3 step stairs, the formula would be:
n = 50 + ( ( s x r ) + ( 10s x u ) )
By using the same method as before, I can work out the formulas for the other positions of 3 step stairs.
For this position, in 3 step stairs, the formula would be:
n = 90 + ( ( s x r ) + ( 10s x u ) )
For this position, in 3 step stairs, the formula would be:
n = 94 + ( ( s x r ) + ( 10s x u ) )
For this position, in 3 step stairs, the formula would be:
n = 54 + ( ( s x r ) + ( 10s x u ) )
4 Step Stairs
This is a 4 step stair on a 10 x 10 number grid:
All 4 step stairs consist of 10 squares and its total is found by adding all the numbers inside the squares.
Total = (25+26+27+28+35+36+37+45+46+55)
Total = 360
This 3 step stair is in a certain position on the grid. This position is called P1 (see below)
This position is called P1
This position is called P2
This position is called P3
This position is called P4
Formulas
There are 4 different formulas to find the total of the 4 step stairs, each one depending on the position of the stairs.
Number added = (1+2+3+411+12+13+21+22+31)
Number added = 120
Number of squares = 10
Squares right = 0
Squares up = 0
For this position, in 4 step stairs, the formula would be:
n = 120 + ( ( s x r ) + ( 10s x u ) )
For this position, in 4 step stairs, the formula would be:
n = 220 + ( ( s x r ) + ( 10s x u ) )
For this position, in 4 step stairs, the formula would be:
n = 230 + ( ( s x r ) + ( 10s x u ) )
For this position, in 4 step stairs, the formula would be:
n = 130 + ( ( s x r ) + ( 10s x u ) )
2 Step Stairs
This is a 3 step stair on a 10 x 10 number grid:
All 2 step stairs consist of 3 squares and its total is found by adding all the numbers inside the squares.
Total = (25+26+35)
Total = 86
This 2 step stair is in a certain position on the grid. This position is called P1 (see below)
This position is called P1
This position is called P2
This position is called P3
This position is called P4
Formulas
There are 4 different formulas to find the total of the 3 step stairs, each one depending on the position of the stairs.
Number added = (1+2+11)
Number added = 14
Number of squares = 3
Squares right = 0
Squares up = 0
For this position, in 2 step stairs, the formula would be:
n = 14 + ( ( s x r ) + ( 10s x u ) )
For this position, in 2 step stairs, the formula would be:
n = 24 + ( ( s x r ) + ( 10s x u ) )
For this position, in 2 step stairs, the formula would be:
n = 24 + ( ( s x r ) + ( 10s x u ) )
For this position, in 2 step stairs, the formula would be:
n = 24 + ( ( s x r ) + ( 10s x u ) )
Proof
This is a randomly selected 3 step stair
I can work out the total by using my formula:
n = 50 + ( ( s x r ) + ( 10s x u ) )
= 50 + ( ( 42 ) + ( 360 ) )
N = 452
Simple total = 68+69+70+78+79+88
= 452
This is a randomly selected 4 step stair
I can work out the total by using my formula:
n = 120 + ( ( s x r ) + ( 10s x u ) )
= 120 + ( ( 60 ) + ( 600 )
N = 780
Simple total = 67+68+69+70+77+78+79+87+88+97
= 780
This is a randomly selected 2 step stair
I can work out the total by using my formula:
n = 14 + ( ( s x r ) + ( 10s x u ) )
= 14 + ( ( 18 ) + ( 180 )
N = 212
Simple total = 67+68+77
= 212
Maths Investigation
Number Stairs
Alex O'Carroll 10a