First I will draw a 3 stair shape at the bottom left with the number 1. Then I am going to add up the numbers in the stairs to get the total. Then I am going to move the stairs one step right and find the total of the particular stair shape. Then I am going to repeat this once more moving it one step right and finding its total. Finally I am going to find the relationship between these stair shapes.
Grid: 10 Steps: 3 @1 1+2+3+11+12+21 = 50
Grid: 10 Steps: 3 @4 4+5+6+14+15+24 = 68
Grid: 10 Steps: 3 @7 7+8+9+17+18+27 = 86
The total of all three numbers stairs equals 50, 68, 86 the common difference is 18 you get 18 if you minus two stair shape totals i.e. 50-68 = 18 meaning the pattern is 18 if I want to find the total of the next stair shape 68 +18 = 86 hence the pattern is 18.
Since I found the pattern going side ways now I am going to prove that if I move one square any direction I will get 18 the difference in other words patterns I am going to move it up one squared upwards to see if there is a pattern.
Grid: 10 Steps: 3 @31 31+32+33+41+42+51 = 230
Grid: 10 Steps: 3 @1 1+2+3+11+12+21 = 50
Grid: 10 Steps: 3 @31 31+32+33+41+42+51 = 230
Grid: 10 Steps: 3 @61 61+62+63+71+72+81 = 410
I have found a different pattern proving my hypothesis wrong of 18 being the pattern. The pattern I got is by subtracting the total of each 3 step stair shape i.e. 230-410 = 180
being the pattern.
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The difference when moving one square upwards
18 The difference when moving one square side ways
I have also found a formula to get the total of each stairs: 6n + 44
I got this formula via:
The 6n comes from the six numbers in the stair shape i.e.:
The Formula = 6n +44, how I get the 6n is mentioned above now I am going to show how I get the 44.
This shows the six n and numbers. 20+10+11+2+1= 44
This is how I get 6n +44 which will work on finding any 3 step stairs at least that’s what I think.
This is how I worked out my formula:
T = n + (n+1) n + 2 + n + 10 + n +11+ n + 20 = 6n+44
6n + 44 will work with any number in this 10 × 10 grid and with any 3 step stair shape.
Now for part 2 I am going to investigate the further relationship between the stair totals and other step stairs on an other number grid. For this I am going to use 9× 9 and a 3 step stair shape.
Grid: 9 Steps: 3 @1 1+2+3+10+11+19 = 46
Grid: 9 Steps: 3 @4 4+5+6+13+14+22 = 64
Grid: 9 Steps: 3 @7 7+8+9+16+17+25 = 82
The total of all three numbers stairs equals 46, 64, 82 the common difference is 18 you get 18 if you minus two stair shape totals i.e. 46-64 = 18 meaning the pattern is 18 if I want to find the total of the next stair shape 64 +18 = 82 hence the pattern is 18.
Since I found the pattern going side ways now I am going to prove that if I move one square any direction I will get 18 the difference in other words patterns I am going to move it up one squared upwards to see if there is a pattern
Grid: 9 Steps: 3 @1 1+2+3+10+11+19 = 46
Grid: 9 Steps: 3 @28 28+29+30+37+38+46 = 208
Grid: 9 Steps: 3 @55 55+56+57+64+65+73 = 370
I have found a different pattern proving my hypothesis wrong of 18 being the pattern. The pattern I got is by subtracting the total of each 3 step stair shape i.e. 208-370 = 162
being the pattern.
162 The difference when moving one square upwards
18 The difference when moving one square side ways
I have also found a formula to get the total of each stairs: 6n + 40
I got this formula via:
The 6n comes from the six numbers in the stair shape i.e.:
The Formula = 6n +40, how I get the 6n is mentioned above now I am going to show how I get the 40.
This shows the six n and numbers. 18+9+10+1+2= 40
This is how I get 6n +40 which will work on finding any 3 step stairs at least that’s what I think.
This is how I worked out my formula:
T = n + (n+1) n + 2 + n + 9 + n +10 + n + 18 = 6n+40
6n + 40 will work with any number in this 9 × 9 grid and with any 3 step stair shape.
The difference between a 10× 10 grid and a 9 × 9 is that to find a total of 3 step stair in a 10× 10 grid is that the formula is different 6n +44 from a 9 × 9 grid which is 6n + 40 this is the result of my investigation.
I am going to investigate the further relationship between the stair totals and other step stairs on an other number grid. For this I am going to use 8× 8 and a 3 step stair shape.
Grid: 8 Steps: 3 @41 41+42+43+49+50+57 = 282
Grid: 8 Steps: 3 @44 44+45+46+52+53+60 = 300
