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Srinivasa Ramanujan

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In my mathematics GCSE coursework I shall be investigating the total and difference in sets of stairs in different grid sizes. I will be investigating the relationship between stair totals on different grids. This will also include different stair sizes.

I will choose to display my findings using a mixture of graphs, tables and grids. I shall also try to work out the formula for each of the number of step sizes.

After I have finished investigating the different sized number stairs I will attempt to find out the different sized number grids.

At the end of my investigation I will conclude by demonstrating and explaining the relationship between all sizes of stairs.

Method

I shall be carrying out my stair challenge by finding out the correct information. I shall do this by experimenting with the size of the grid corresponding to the size of the stairs used in the grid.

I shall try to work out the formulas by recognising any similarities in the results obtained.

Part 1- Three Step grid

I have investigated the three step stair (some of my findings can be viewed on the graph paper in the back). During my investigation I believe that I have worked out the formula for the total inside the three step stairs on a 10 x 10 grid. Below is an example of just one of the three sets of stairs that I used to conduct my investigation:

I have worked out the formula for the total inside the three step stairs in a 10 x 10 grid

Key: x = corner number

t = total inside stairs

1. I worked out the common relationship with the numbers inside the stairs and to be accurate here are two examples:

x+x+1+x+2+x+10+x+11+x+20

= 6x+44

2. Noticing that this is the total I realised that it is not so hard to work out the formula for the total. Therefore due to the above diagram it is evident that the formula must be, 6x + 44. It is easy to work out because there is six 'x's then you have to times x by 6 then you just add the other numbers together to get 44.

3. Lastly I have noticed that this formula does in fact work for any stairs anywhere in this number grid.

I noticed something about the common stair total. It is evident that the larger the corner number is the larger the stair total is going to be. The reason for this is that the higher 'x' is then the more it is multiplied. Although the corner number affects the multiplication of x, it does not affect the number 4 which always remains the same no matter what the corner number is. So, the three step stairs could be located anywhere in the grid and the only part of the formula that will change is the amount of times that you multiply x. Here is a table simplifying this theory:

X

Total

25

194

63

422

58

394

Conclusion

In conclusion, in this part I have learnt that the formula for any three step total in a 10 x 10 grid is: 6x + 44. Furthermore I have noticed that the higher the corner number is then the higher the total inside the steps will be. Also I have found out that the size of the corner number does not affect the amount that 'g' is multiplied nor does it affect the number that is added on the end of the formula. Also I have found out that the value of 'g' is different in each grid depending on the size of the grid. In this investigation it has also been proved that wherever the stairs are in the grid the formula for their total still applies.

Part 2 - Other Steps on Other Number Grids

Now that I have finished the three step stair on just a 10 x10 number grid I shall go on to changing first the grid size and then the steps size.

First of all I decided to investigate using the same size steps on an 8 x 8 number grid. Here is what I found out:

x+x+1+x+2+x+8+x+9+x+16 = 6x+36

Therefore as you can see above the formulas for the three step stairs on different sized number grids do not seem to be exactly the same (other than each time the multiplication of 'x' has been the same). So I then went on to investigate the same step size on another different size of grid, this time a 6 x 6 size grid. Here is what I found:

When I investigated the different sized grid I noticed that each different grid size gave the stair total a different formula because each time a different number was added. Now that I have experimented with three different grids but with the same size of steps I have also noticed that the number that is always added vertically is always the number that is the same size of the grid. So for example for a 10 x 10 grid as the numbers increase upward from the corner number each time it is increased by ten. I have shown this below representing the grid size by using the letter 'g'.

x+x+1+x+2+x+g+x+g+1+x+g+g = 6x+4g+4

After checking this formula to see if it works for each size of grid it has been proven that it does work. So for the three step total on any size grid the formula is; 6x+4g+4

Different Stair Sizes

Next I shall investigate different stair sizes and the formulas for their totals. The first grid size I will choose to use is 10 x 10 and the size stairs will be a four step size. After finding the totals for each of the stairs that I will use I shall work out the formula by using the same method as before. Here is what I found out:

X+x+1+x+2+x+3+x+g+x+g+1+x+g+2+x+g+g+x+g+g+1+x+g+g+g = 10x+10g+10

Looking at the above information it is visible to know that the four step stair shape follows the same simple pattern as the three step stairs in the way that the letter g is added vertically and the numbers added horizontally. Therefore I have found out that that the formula for steps of this size is; 10x+10g+10. Next I shall attempt to find out about a one step stair. Here it is:

Total = x

As you can see above a one step stair only contains one block so this is the whole staircase in one. Therefore the formula for this size of step is; x. of course this was the same for all of the grids on which I investigated this stair size on. Next I went on to investigate the two step stair shape here is what I found out:

X+x+1+x+g = 3x+g+1

This formula was the same for the two steps on any of the number grids which I experimented on. Lastly I went on to learn about the fifth and final stair size; the five step stair. Once again here are the results:

X+x+1+x+2+x+3+x+4+x+g+x+g+1+x+g+2+x+g+3+x+g+g+x+g+g+1+x+g+g+2+x+g+g+g+x+g+g+g+1+x+g+g+g+g = 15x+20g+20

On all of the grids on which I tested this I found that this was constantly the formula. So the formula for the five step shape is; 20x+15g+20. Here are all of the above results in the form of a table:

Stair Shape

Total

1

T = x

2

T = 3x + g + 1

3

T = 6x + 4g + 4

4

T = 10x + 10g + 10

5

T = 15x + 20g + 20

Formula Linking All of the Formulas

After now finding out all of the formulas for the entire stair shapes I shall now try to figure out a way to link the all into on formula. This final formula is:

t = Tnx + ? Tg + ?T

My workings out for this formula can be found in the back of his booklet on a piece of paper.

Conclusion

In conclusion, after working out these formulas including the final formula I have noticed a number of things. Firstly all of the different sizes of stairs follow the same simple rule and that is; as you increase vertically you add the grid size number and then when you go along horizontally you always increase by one digit. The larger the number stairs are the more they have increased both horizontally and vertically therefore giving a larger total inside the stairs. The rule which I have just explained is applied in every grid and for any size number stairs.

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