From 26 to 25 the difference is 1.
So therefore 10-9=1, this method works on any number square grid and anywhere on the grid.
E.g. 4 by 4 number square grid
Top right corner number stair, the total is always higher.
The total of the numbers inside the top right corner number stair shape is:
-
1st Line: 78+79+80
-
2nd Line: 88+89
-
3rd Line: 98
T=Total T=452
Bottom left hand corner number stairs, the total is always lower.
E.g.
The total of the numbers inside the bottom left hand corner number stair shape is:
-
1st Line: 1+2+3
-
2nd Line: 11+12
-
3rd Line: 21
T=Total T=50
Finding the formula
Hypothesis: I have found out that to find the formula for any number square you must firstly make one of the numbers in that stair pattern as ‘x’. I chose to make the bottom left number ‘x’, because I consider it as being the easiest number to make ‘x’ and also to find the formula.
E.g. 10 by 10 number square grid
Collect the entire ‘x’ values and add them up. For e.g. the above Grid there are 6 ‘x’, so add them all together it will make, ‘6x’. Now add up all the digits 20+11+10+2+1= 44, for that reason the formula would be ‘6x+44’.
Now to verify my formula, place ‘x’ in the formula which is 25, for this occasion.
6x25=150 150+44=194
To guarantee my answer even more I tried adding up all the numbers in that number stair.
The total of the numbers inside the stair shape is:
-
1st Line: 25+26+27
-
2nd Line: 35+36
-
3rd Line: 45
T=Total T=194
To sum up I have tried my formulas on other number square grids and they have worked successfully on all of them. I believe that through out my work I have found an easier way of finding out the total to any number stair and I have also found out the relationship between the stair total and the position of the stair shape on the grid.
Formulas
2 steps 3x+11
→ 33
3 steps 6x+44 → 33
→ 66
4 steps 10x+110 → 44
→ 110
5 steps 15x+220 → 55
→165
6 steps 21x+35
To get the first part of the equation, I realized that it is going up in triangle numbers, therefore for the 6 step I would add 6 and get 21x as the first part of my formula.
It is evident that a specific order occurs in the last part; therefore I would suppose that the next number in the pattern is 55. From here I would use the number on the five steps equation which is 220 and take 55 from it. Eventually my answer is 165.
To get the rest of the 6 step equation, I added 165 to 220 which is 385. Finally I got the formula for the 6 steps which is 21x + 385.
To test this formula I took the numbers from a 6 step stair on a 10x10 grid and summed them up. The answer was 406. I then placed the number on the first block of the 6 step stair, which was 1 in my formula. 21 x 1 = 21 + 385 = 406. Since the two numbers were equivalent, this shows that my prediction was correct and my formula is working.
9x9 grid – 2 step stair
From the above grid, I will explore and try to find a formula for a 2 step stair on any grid size.
e.g 23,24,32
23+24+32=79
x+x+1+x+9 = simplified 3x+10
What I have noticed is that the grid size also relates to how I will find my formula, through using the step stairs.
e.g. As 23 is x in my case, you will have to add 1 to it to get 24.
Then to get 32, the algebraic formula would be x + g. (x being 23 and the grid size being 9). So my answer would be 32. The final algebraic formula for a 2 step stair on any grid size would be 3x+1+g.
To guarantee that my formula is correct, I will have to test it by collecting the numbers from a 2 step stair and placing it in the formula.
e.g.
1 + 2 + 10=13
So I tested out my formula 3x + 1 + g by:
3 x 1 = 3 + 1 = 4 + 9 (g) = 13
The above example shows that the numbers in a 2 step stair add up to equalize the answer of my formula, which was 13. My formula is Correct.
9x9 grid – 3 step stair
For a 3 step stair I will have to go through the same procedure I did with a 2 step stair.
1 + 2 + 3 + 10 + 11 + 19 = 46
Algebraic = x + x + 1 + x + 2 + x + g + x + g + 1 + x + 2g =
Simplified 6x + 4g + 4
Again I tested out my formula by placing the digits in the grid and I have found that it is also correct.
6 x 1= 6
+ 4 x 9 = 36 + 4 = 46
So the formula for a 3 step stair is : 6x + 4g +4
Again I predict that this formula will work for any 3 step stair on any grid size. Below is how I found this out:
e.g. 4x4 grid Formula = 6x + 4g + 4
1 + 2 + 3 + 5 + 6 + 9 = 26
6x1=6
+ 4 x 4 = 16 + 4 = 26
9x9 grid – 4 step stair
1 + 2 + 3 + 4 + 10 + 11 + 12 + 19 + 20 + 28 = 110
Algebraic: x + x + 1 + x + 2 + x + 3 + x + g + x + g + 1 + x + g + 2 + x + 2g + x + 2g + 1 + x + 3g
Formula: 10x + 10 + 10g
I tested out this formula on any grid size I preferred (6 x 6 grid).
1 + 2 + 3 + 4 + 7 + 8 + 9 + 13 + 14 + 19 =80
10 x 1= 10 + 10=20 10 x 6 = 60
60 + 20 = 80 Correct.
9x9 grid – 5 step stair
Total = 215
Algeraic: x + x + 1 + x + 2 + x + 3 + x + 4 + x + g + x + g + 1 + x + g + 2 + x + g+ 3 + x + 2g + x + 2g + 1 + x + 2g + 2 + x + 3g + x + 3g + 1 + x + 4g = Formula: 15x + 20 + 20g
I will now test this formula on a 7 x 7 grid, but still staying with a 5 step stair.
Total = 175
So, 15 x 1 = 15 + 20 = 35 20 x 7 = 140
140 + 35 = 175
Again this shows that my 5 step stair formula for any grid size is correct.
Finding the algebraic formula
2 step stair = 3x + 1 + g
3 step stair = 6x + 4 + 4g
4 step stair = 10x + 10 + 10g
5 step stair = 15x + 20 + 20g
??? = 21x + 35 + 35g
I have noticed a certain pattern which occurs constantly through the formulas. In the first column it goes up in triangle numbers:
3x, 6x, 10x , 15x
I believe that the next number will be 21, because 15 add the next triangle number in the pattern which is 6 is 21. Also for the last part of the formula I had to find the difference from the numbers at the end of the formulas so that I could notice a pattern.
… g
→ 3
… 4g → 3
→ 6
… 10g → 4
→ 10
… 20g → 5
→ 15
21x + … + …?(35g)
Algebraic formulas for and grid size
2 step stair = 3x + 1 + g
3 step stair = 6x + 4 + 4g
4 step stair = 10x + 10 + 10g
5 step stair = 15x + 20 + 20g
6 step stair = 21x + 35 + 35g
