T-Totals.I will investigate what makes up a T-total and what makes up a T-number. This will be done by drawing out a nine by nine grid first. Then looking at the grid I will try to see what kind of relationships there are

Adam May

T-Totals

What I will do:

I will investigate what makes up a T-total and what makes up a T-number. This will be done by drawing out a nine by nine grid first. Then looking at the grid I will try to see what kind of relationships there are. Then I will do the same again but with other sized grids e.g. a ten by ten grid. This will then show me if there any changes and if so what are they. I will look at these by drawing out the grids then, using a piece of paper I will cut out a T shape that will measure three squares across and three down. This will then allow me to clearly see the five numbers in question, with out getting them mixed up with other numbers. Then once I have taken a reasonable sample with the three square “T” I will increase its size to a five by five “T” shape and then this will show up any other possible connections between the T shape and the T number. I will also look at the “T” number, this is the number at the bottom of the “T” shape. It is always the largest number.

Example of what the cut out would look like;

Piece of paper

Holes cut out

in a “T” shape

Hypothesis:

I think the T-total will go up in up fives on a nine by nine grid, this is because each number will go up by one and five ones will make an extra five on to the T-total.

This will happen; no matter what size of the grid it could be a nine by nine, or a ten by ten the size would not matter because the grid numbers always the same. But this would change with the size of T-shape. If it increased to a five across five down the t total will go up by nine every time. This is because there are nine squares and each one would go up by nine thus adding nine on every time the “T” is moved along one space right.

T shape 1 T shape 2 T shape 3

I have labelled these equation’s 1, 2 and 3. The red numbers indicate the T shape. In this I used a three square “T” shape and a nine by nine grid.

This has shown me that for “T” shape 1, the T-total is 37 this is because 1+2+3+11+20=37. This tells me as well that the “T” number is 20.

In “T” shape 2 the “T” number is 50, the “T” total is 187 ...

This is a preview of the whole essay

T shape 1 T shape 2 T shape 3

I have labelled these equation’s 1, 2 and 3. The red numbers indicate the T shape. In this I used a three square “T” shape and a nine by nine grid.

This has shown me that for “T” shape 1, the T-total is 37 this is because 1+2+3+11+20=37. This tells me as well that the “T” number is 20.

In “T” shape 2 the “T” number is 50, the “T” total is 187 this is because 31+32+33+41+50=187.

In “T” shape 3 the “T” number is 71, the “T” total is 292 this is because 52+53+54+62+71=292

So far looking at this information I can not find any possible link between the “T” total and the “T” number so I will us the same size grid again but a five square “T” instead o see if this helps me.

T shape 4 T shape 5

Again I have labelled these equations 4 and 5.

In T shape 4 is has shown me that the “T” number is 39, then I added up all the numbers in red in t shape 4 to give a “T” total of 117. This is because 1+2+3+4+5+12+21+30+39=117.

In T shape 5, 61 is the “T” number and the “T” total is 315 this is because 23+24+25+26+27+34+43+52+61= 315

I have looked at both of these results very hard and come up with an idea. My idea is that If for example you look at T shape 1 the “T” number is 20, but if you replaced this number with “N” you get;

“T” number = N

Then the number in the “T” square, directly above the “T” number is then nine places back in the grid so it is N – 9. The number directly above that is then N-9-9 = N-18. The two remaining numbers in the “T” shape are N-18-1 and N-18+1.

This also works with T shape 4 but some alterations have to happen because it is a five by five “T” shape.

If we take the “T” number as 39 this then transforms in to N. This means the number directly above that N-9, the one above that is N-9-9= N-18, the one above that is N-9-9-9= N-27 and the one above that is N-9-9-9-9= N=36 the other number would be N-36-2, N-36-1,N-36+1 and N-36+2.

These formulas would help if only the “T” number was given and I was asked to find out what the “T” shape is, but looking in to it further I have realised it will help me to work out the “T” total.

Then as I was looking at this, I noticed a what was similarity between T shape 1,2 and 3 when I used the formula N+(N-9)+(N-18)+(N-18-1)+(N-18+1) = 5N-63 and doing this you end up with the “T” number.

For example look at “T” shape 2 and use the above formula:

N+(N-9)+(N-18)+(N-18-1)+(N-18+1)= 5N-63 which equals187 which is the “T” total for “T” shape 2.

So now I have found a formula for the five square “T” shape and checked it by making sure it works with “T” shape 2. I now I must change this formula To suit the nine square “T” shape.

I looked at this problem and worked out the change for the formula. This is the formula for a nine square “T” shape

N=(N-9)+(N-18)+(N-27)+(N-36)+(N-36-1-1)+(N-36-1)+

(N-36+1)+(N-36+1+1)=9N-234 then I checked this by using “T” shape five and the equation is correct because it gave me the answer of 315 which is the “T” total.

This has proved that I have now found the formula to work out the “T” total just from the “T” number, or the “T” number just from the “T” total.

I will now investigate a seven by seven “T” shape just to make sure that the formula works, I will also change the size of the grid to see if this affects the formula in any way. I will change the grid to a ten by ten grid.

T shape 6

Again I have labelled these “T” shape 6.

The “T” number for “T” shape 6 is 96 and the “T” total is 678

I have found that when you change the size of the grid you also have to change the first number in the formula. Where as it used to be take away nine on a nine by nine grid, now I have to take away ten because it is a ten by ten grid. So if we were to change the grid again for example to a six by six grid you would have to change the first number in the formula to six.

So to make sure I am 100% correct I will use the formula to check the seven by seven “T” shape labelled “T” shape 6.

N=(N-10)+(N-20)+(N-30)+(N-40)+(N-50)+(N-60)+(N-60-1-1-1)+(N-60-1-1)+(N-60-1)+(N-60+1)+(N-60+1+1)+(N-60+1+1+1)=13N-570

When I used this formula I got the answer 678 which is the correct answer as the “T” total is 678.

This has made me now 100% sure that my working is correct. I now am sure whatever size grid I was given and whatever size “T” shape I will be able to use my formula to work out the “T” number or “T” total.

Now I will investigate just by changing the size of the grid and using a three by three “T” shape, what changes I will have to make to the formula to account for the change in grid size.

“T” shape 7

I have labelled this “T” shape “T” shape 7, this is a three by three “T” shape and it Is placed on a six by six grid.

The formula to find out the “T” number for this shape will have to be N+(N-6)+(N-12)+(N-12-1)+(N-12+1)=5N-42

The formula has to change for example on a nine by nine grid the first number you take away is nine, then after that multiplies of nine. This would not work on a six by six, so you have to change the formula so the first number you take away is 6 and multiplies of six there after.

So to make sure I am correct in what I will do I will no check this theory twice to make sure I’m right.

“T” shape 8

Again I have labelled the “T” I labelled it “T” shape 8. It is a five by five “T” shape and it is on a five by five grid.

Iwill change the formula again, this time I will take away five and then multiples of five.

N+(N-5)+(N-10)+(N-10-1)+(N-10+1)=5N-35 this gives me the answer of 55 which is correct as this is the “T” total for “T” shape 8.

I will check my working one more time to make sure I’m correct.

“T” shape 9

I have labelled the “T” shape, “T” shape 9. I have put it on a eight by eight grid. It is a five by five “T” shape.

I will change the formula to incorporate the size of the grid.

N+(N-8)+(N-16)+(N-16-1)+(N-16+)= 5N-56

When I work out this equation I get the answer 119, which proves my theory, is correct because 119 is the “T” total.

Summery:

So in this I have shown that I have created a formula that will allow you to work out any “T” number or “T” total. This can simply be done by adjusting the formula to suit the problem. I have found this formula will solve any questions whatever the size of “T” shape or grid.

Evaluation:

I feel that I have worked extremely hard in this coursework to find out my formula. I am now confident that whatever size grid I am given I could adjust the formula to suite, because I have explored any of the possibilities. I am also confident whatever size “T” shape I am given I would also be able to solve this using my formula.

I liked this question because it tested me, and it was a challenge to find the answer.