Investigating three step stairs.

Aim

I am set the task of investigating three step stairs (which I will go into some more detail on later on) and how their position on the number grid and the step total corresponds to the step number.

Prediction

My basic prediction is that…

Definitions

Step total – the total of all the numbers in the stair added up

Stair number – this is the lowest number in the stair and is found in the bottom left hand corner of them

Number grid – this is a ten by ten square with the numbers from one to one hundred in it as ten time ten is 100. The ten by ten grid is the grid we will be using in the first part of the investigation but in the second stage of the investigation other smaller and larger number grids will be tested.

Step number – this is the number of steps the stair is comprised of. As I have stated the step number I will be using for the main part for this investigation will be a three step stair, with a three step number there will be six numbers within the stair.

Establishing a connection

In order to establish a link between n (n=stair number) and the st (st=step total) I must conduct a series of tests on various different three step stairs in different positions on the ten by ten grid. I will, whilst examining the results I gain from different stairs, attempt to find a formula that can be used to predict the st of any three step stair on the grid. As an example of what I am going to do:

E.g.

→

In this example I show the three step stair for stair number one (as can be seen by the number in the lower left corner) so n=1. Also in this example the step total is fifty so st=50. Now that I have gained that information I will begin working on a formula. But how can I start off? I can’t say that all three step stairs are equal to fifty this can’t be so as the step total is dependant on the values of the numbers within the stair. I have to look for a common thing that ...

This is a preview of the whole essay

→

While all three step stairs do not contain the same numbers the size of the numbers in relation to the stair number does remain the same. Allow me to explain what I mean by this. In the grid above (stair number one) the numbers in the stair follow a pattern, this pattern:

←This is an underlying pattern that exists within all three step stairs. If some one gave me a number now and I was asked to take that number as the stair number and find the rest of the numbers within the three step stair without the aid of a number grid I could now theoretically do it.

Lets say I am given the task of finding out the rest of a three step stair where the n=5. In order to do this I would like at the pattern that all three step stairs follow (see left) all that is required for me to do is to fill in n as 5 and do the appropriate calculations for each box in the three step stair. My finished three step stair with all the calculations done looks like this:

By using the pattern grid I have accurately found the missing numbers. However this is not exactly a formula and whilst it may be useful for a question like the one I just answered its uses are limited. A formula to calculate the step total from just the stair number would be ideal and so I began to experiment and using trial and error.

To start with I looked at the number of numbers in a three step stair and began to look back on the pattern that was found to be behind all three step stairs. I looked at what was actually done to n and how much overall was added to n in a three step stair.

E.g.

n+(n+1) +(n+2) +(n+11) +(n+10) +(n+20)= n+44

Using this I came up with some very basic and at times some ludicrously stupid equations. To name but a few:
nx(44)=st?
n+(44)=st?

Of course both of these are indeed wrong. However it was while thinking about this that I saw something I had missed, I had been only using half of the necessary factors in the previous lot of formulas. I had without realising it left out one of the most important things that should have been in the formula. As I sat back and thought about what was going wrong I began to notice something. In any of the three step stairs I tried this formula on there was always a large bit missing for example:

n=32

n+44=st?
32+44=76

The step total for n=32 is not 76 but 193. This was puzzling me, as the amount of the numbers missing was not constant. Instead as the stair number became larger so did the amount missing and as the stair number got smaller so did the amount that was missing. The fact that it was not a set amount that was missing was nagging at the back of my min. seeing as the amount missing was not constant throughout the other three step stairs I found it increasingly hard to test any theory I had as to what exactly was going on. I was finding solutions to individual three step stairs but they didn’t apply to the others. Seeing as the problem was enlarging quite dramatically with the higher numbers I began to work my way down where hopefully I could find the route of the problem and the problem would be easier to find. I based my theory that the source of the problem would be easier to spot with lower stair numbers on that with lower numbers those that are unnecessary would be removed and I would be able to look and spot any problem more easily dealing with a stair number of one. Much like if I had to draw a cross on the exact centre of a circle it would be easier for me to do it holding the pencil at a close distance away from the circle rather than doing it with a metre long pencil and holding it by the end and at arms length. Things have a tendency to get exaggerated the further they are from the actual problem and the problem is often hidden better when surrounded by a larger problem than by a small problem. In order to stop a problem it must be stopped at the root or source (where it most vulnerable and exposed). Anyway, I went back and looked at stair number one and looked how far off I was from the stair total; I was six off the stair total. I then moved along to stair number two and saw how far off I was from its stair total; I was