Formulae
C= 4
S = (A – 1) 2
N = 4(A – 1)
I have also discovered that the spacer (S) is increasing in square numbers.
I must also state that the letter (A) represents the arrangements. However when this letter is in formulae, you must take the first number of the arrangement in order to work out either (S) or (N).
I have discovered that (N) increases by 4 when the arrangements are in order.
By using the formulae I am able to find out the following arrangements shown in the table and the diagrams.
Result Table 2:
The drawing below is drawn to express that my formulae is correct. I must state that this formula is used so that the person will not have to draw large diagrams.
Using the results from the table, I want to see if my formulae work. I decide to use the 6 by 6 arrangement as shown below,
This is what I discovered,
By referring back to Result Table 2, I can conclude that my formulae are accurate and do work.
This can be proven below,
C = 4 = (This is correct)
S = (A – 1) 2 = 6 – 1 = 5 = 25 (This is correct)
N = 4(A – 1) = 6 – 1 = 5 × 4 = 20 (This is correct)
Knowing that my formulae is correct, the builder if given a square arrangement will know how many different spacers he will need, Below is a table to demonstrate this,
To check if these calculations for the different spacers for the different arrangements are right we can do a simple mathematical workout.
We take the arrangement 499 by 499,
- We know that S = 248004 (if we square root this number we get a total of 498, we plus 1 which gives 499.
- If we take the take 1 away from the first arrangement number, like this 499 - 1 = 498, we then use the formulae to work out N,
4(498 – 1) = 1988. The difference between this number (1988) and the answer to the arrangement of 499 by 499, which is (1992) is 4. I started that (N) increases by 4 earlier on in the investigation.
- We know that C always = 4 because so there aren’t any ways of seeing that this is accurate.
Extension
I’m going to extend this investigation by investigating into arrangements, which are rectangular.
I began by drawing diagrams as shown below
1 by 2
2 by 3
3 by 4
I put the results from the diagrams into a table,
Table of Results:
I started to notice patterns. Obviously we know that (C) = 4. However (N) is increasing by 4 every time.
I carried on drawing the rectangular tile arrangement, as shown on the rough diagrams sheet.
These are the results that I discovered
I have discovered the formula to work out S
I must state like I did earlier on in the investigation that the letter (A) represents the arrangements. However when this letter is in formulae, you must take the first number of the arrangement in order to work out.
A - A = S
However this formula only applies in certain arrangements, for example, if the first number in the arrangement is 12 it has to by 13, one added on to the first number.
Using this formula I can know how many spacers I need to fill between the tiles.
