Stair Investigation

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Stair Investigation

45
35 36
25 26 27

This is a 3-step stair. The total of numbers inside the stair shape is:

25+26+27+35+36+45 = 194.
The 3 stairs total for this step is 194.

The smallest stair is in the bottom left corner of the grid:

21
11 12
1    2  3

The total of this grid is 1+2+3+11+12+21=50. The 3 stairs total for this stair is 50. Three step stairs can also be placed in other locations on the grid, in different places and positions. I will start in the bottom left corner, and move to the right by one square each time, because this is logical and methodical; the numbers will be sequenced.

1+2+3+11+12+21 = 50
2+3+4+12+12+22 = 56
3+4+5+13+14+23 = 62
4+5+6+14+15+24 = 66
5+6+7+15+16+25 = 72
6+7+8+16+17+26 = 78

this can be shown in a table:


There is a trend in these results; every time the stair is moved to the right by one square, the total increases by six. Using this idea we can make a formula rule. If we use the bottom left corner as n, then n+1 is 50. The total increases by 6, so it is 6n. Therefore 6n+44 is the rule. This can be shown in previous results; 44+(6x1)=50, 44+(6x2)=56, 44+(6x3)=62, and so on. This is a consecutive stair.

This formula can be checked by trying a different area of the grid, e.g. 25 to 28:

25+26+27+35+36+45 = 194
26+27+28+36+37+46 = 200
27+28+29+37+38+46 = 206
28+29+30+38+39+48 = 212

as these results show, the rule 6n+44 still works. This rule can be shown in algebra square:

n+10
 +10
n+10 n+10
          +1

 n       n+1 n+2

When the stair is moved up, I believe the same rule will apply:

1+2+3+11+12+21 = 50
11+12+13+21+22+31 = 110
21+22+23+31+32+41 = 170
31+32+33+41+42+51 = 230
41+42+43+51+52+61 = 290
51+52+53+61+62+71 = 350

The pattern that emerges is that for every row the stair is moved up, the total increases by 60. If the bottom left square is n, then 6n+44 is the total of the square; therefore the rule still applies.

This can be proved by checking another area of the grid:

14+15+16+24+25+34 = 128
24+25+26+34+35+44 = 188
34+35+36+44+45+54 = 268
44+45+46+54+55+64 = 328
54+55+56+64+65+66 = 368

34
24 25
14 15 16

This shows the formula works no matter which way the square moves, as long as it is at that angle.

If we rotate the stair 90 degrees and then try to work out the total, we find differences:

91 92 93
81 82
71

=510

The formula for a 3-stair rotated 90 degrees is 6n-36; this can be shown by trying it in another area of the grid:

24+25+26+14+15+4 = 108
(24x6) – 36 = 108

25+26+27+15+16+5 = 114
(25x6) – 36 = 114

26+27+28+16+17+6 = 120
(26x6) – 36 = 120

As we can see, the formula of 6n-36 works, and also when the square is moved to the right the total increases by six. From this we can also work out that the square is moved left, the total will decrease by 6. When the square is moved up, the total increases by 60, and when the square is moved down the total decreases by 60. This rule applies with all 3-stairs, only the formula for working out their total changes.

When the square is rotated 180 degrees, I predict (from previous results) that the formula will be 6n-44.

98 99 100                n-2    n-1       n
    89   99                         n-10  n-10
           80                             -1  n-10

                                                     -10

=556                                    =6n-44

(100x6)-44=556

I was correct. I worked out it would be this because when there are squares adjacent on the right side on the n square, it adds (and also when squares are above n), and when the squares adjacent to n are beneath it, it is subtracted.

Therefore, from this we can work out that when the square is rotated 270 degrees the formula will be 6n+36.




               30                             n+10            
          19 20                               +10
       8   9 10                  n+10   n+10
                                          -1
                            n-2       n-1         n

Join now!

           =96                          =6n-36

This was also correct, and I know this from my earlier explanation of squares above or on the right of n being positive, and squares beneath or left of n being negative.

The formula for working out the original 3 stair is 6n+44.

n+10
 +10
n+10  n+10
           +1
n        n+1   n+2

The +10’s represent the next row up on the grid. These could also be called g, so the formula could also be 6n+4g+4.

We could ...

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