Mathematics - Investigating Stair Shapes
Mathematics - 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
+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 ...
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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.
X
Total for 3 Step Stairs (left + right)
+2+3+11+12+21 = 50
2
2+3+4+12+13+22 = 56
3
3+4+5+13+14+23 = 62
4
Prediction = 68 - Correct
4
4+5+6+14+15+24 = 68
My prediction was correct. If this was to go backward, then obviously the numbers would be decreasing by 6, and not increasing.
X
Total for 3 Step Stairs ( up + down)
+2+3+11+12+21 = 50
1
1+12+13+21+22+31 = 110
21
21+22+23+31+32+41 = 170
41
Prediction = 230 - Correct
41
31+32+33+41+42+51 = 230
My Prediction was correct. If this was to go backward, then obviously the numbers would be decreasing by 60, and not decreasing.
Here is what we can conclude from this:
= x + 44
= 6 numbers = 6 x + 44
Test
If x = 1... 1+2+3+11+12+21 = 50 (6x1 add 44 = 50)
If x = 2... 2+3+4+12+13+22 = 56 (6x2 = 12 add 44 = 56)
It is important to test these kind of formulas, as simple mistakes could be found, which could alter the answer drastically, although they could be rectified very easily, so it is best to check your answers at an early stage.
Part 2
Here I will test different sized stair shapes, like 4, 5, and 6 stair shapes.
When a 4 stair shape is moved to the left once, its stair total is decreased by 10. Similarly, when it is moved once to the right, its stair total is increased by 10.
Stair number: 1
+2+3+4+11+12+13+21+22+31 = 120
Stair number: 2
2+3+4+5+12+13+14+22+23+32 = 130
When a 4 stair number is moved down once, its stair total is decreased by 100. Similarly, when it is moved up once, its stair total is increased by 100.
Stair number: 1
+2+3+4+11+12+13+21+22+31 = 120
Stair number: 11
1+12+13+14+21+22+23+31+32+41 = 220
Because in this case, instead of 6 numbers, there are ten, it is therefore easy to understand why the stair total increases by ten every time it moves once to the right. It increases by one for every number, and there are ten numbers, so
0x1 = 10.
And as for the sideways movement, because there are again ten numbers, it increases by ten for every number, and there are ten numbers, meaning ten times ten equals one hundred.
X
Table for 4 Stair Shapes (left + right)
+2+3+4+11+12+13+21+22+31 = 120
2
2+3+4+5+12+13+14+22+23+32 = 130
3
3+4+5+6+13+14+15+23+24+33 = 140
4
Prediction = 150 Correct
4
4+5+6+7+14+15+16+24+25+34 = 150
My prediction was correct. If it was going backwards, obviously it would be minus 10 each time.
X
Table for 4 Stair Shapes (up + down)
+2+3+4+11+12+13+21+22+31 = 120
1
1+12+13+14+21+22+23+31+32+41 = 220
21
21+22+23+24+31+32+33+41+42+51 = 320
31
Prediction = 420 Correct
31
31+32+33+34+41+42+43+51+52+61 = 420
My prediction was correct. If it was going backwards, obviously it would be minus 100 each time.
Here is what we can conclude from this:
= X = 110
= 10 numbers = 10x + 110
Test
If x = 1... 1+2+3+4+11+12+13+21+22+31 = 120
If x = 2... 2+3+4+5+11+12+13+22+23+32 = 130
(Because 10x2 = 20, add 110.)
Following this same pattern, I found out other formulas for a few other stair shapes. I have compiled them into a table, shown below:
Stair Shape
Formula
3 Stair Shape
6x + 44
4 Stair Shape
0x + 110
5 Stair Shape
5x + 220
6 Stair Shape
21x + 385
7 Stair Shape
28x + 616
8 Stair Shape
36x + 924
Using these formulas together, I managed to work out a general formula, which also takes into account the fact that there can be any grid size.
We all started off with the formula:
an3+bn2+cn+d
A = a 6th of the third difference, which is 11
D = the 0th term, which is 0
I finally worked out the formula:
This takes into account everything.
Evaluation
Because I found out the final formula, I believe that I did well to get that far. However, I do believe that if I had had enough time, I would have got to grips with the whole investigation a lot better, and would have then had the time and understanding to then expand on my work and hopefully get a better mark on it.
