12 + 13 + 14 + 22 + 31 = 92
80 x 5 – 63 = 337
61 + 62 + 63 + 71 + 80 =337
38 x 5 – 63 = 127
19 + 20 + 21 + 29 + 38 = 127
All the examples worked of the formula worked perfectly.
What happens when I turn the t-shape on its side and alter the direction slightly?
After working on the first t-shape I can use the same technique I used before for working out the formula but on this t-shape. The t-number is the highlighted number.
4 + 11 + 12 + 13 + 22 = 62
38 + 47 + 56 + 48 + 49 = 238
52 + 61 + 69 + 70 + 71 = 323
I’m going to begin with this t-shape.
This t-shape totals to 62. By moving it one place right I get a t-shape consisting of 5, 12, 13, 14 and 23.
I remember from the first t-shape that when I moved the t-shape one place right it increased by 5. Lets see if the same rule applies to this t-shape.
5 + 12 + 13 + 14 + 23 = 67 67 – 62 = 5
The rule we learnt from the first t-shape still applies even when the t-shape I rotated through 90 degrees.
If I move the t-shape one place back from the original I predict that the new t-total will be 62 – 5 = 57
Lets see if I am right, 3 + 10 + 11 + 12 + 21 = 57. My prediction is correct.
Now I am going to look at the rest of the t-shapes to see if this theory works for them too.
T-total = 238
If I take the t-shape to the right of it I have 39, 48, 49, 50 and 57. I think the t-total will be 238 + 5 = 243.
39 + 48 + 49 + 50 + 57 = 243. It worked again.
Now I can move it one place back and have 37, 46, 47, 48 and 55. I predict the total will be 238 – 5 = 233
37 + 46 + 47 + 48 + 55 = 233
For this t-shape all the original rules from the first t-shape applied to this changed t-shape.
T-total = 323
To move this one step right I would have a t-shape with the numbers 53, 62, 70, 71 and 72. I think the t-total is 323 + 5 = 328.
53 + 62 + 70 + 71 + 72 = 328. I was right.
To move this step one step left, I would have a t-shape consisting of 51, 60, 68, 69, and 70. I believe this t-total will be 323 – 5 = 318.
51 + 60 + 68 + 69 + 70 = 318.
I am sure that the rule of moving any rotated t-shape forward or backwards you just increase or decrease the t-total by 5 or more and it effectively works.
For the next section of working out the formula, I deducted the numbers, one by one, in the t-shape from the t-number.
4 + 11 + 12 + 13 + 22 = 62
This is the t-shape with the calculations shown in it to represent which box represents which calculation.
11 – 12 = -1
11 – 4 = 7
11 – 13 = -2
11 – 22 = -11
This shows the
t-shape with the
Results in it.
On the first t-shape I added the answers in the box up. The original answers were all minuses but on this t-shape we are left with pluses and minuses. I will still add these up I know that the answer will not be the same as before.
1 + 2 + -7 + 11 = 7
I remember that when I did turn the t-shape around and moved it left and right it remained its original rule of increasing or decreasing by five. I am going to still keep the 5 in the original equation but substitute the old –63 for the new 7.
n x 5 + 7 = t
Lets do some tests to see if this formula actually works:
5 x 33 + 7 = 172
33 + 34 + 26 + 35 + 44 = 172
The formula works.
5 x 65 + 7 = 332
65 + 66 + 67 + 58 + 76 = 332
The formula works.
5 x 20 + 7 = 107
20 + 21 + 22 + 13 + 31 = 107
The formula works.
All of the examples worked.
I am going to repeat the last stage with a different t-shape:
38 + 47 + 48 + 49 + 56 = 238
Now I take the numbers inside the t-shape away from the t-number.
49 – 38 = 11
49 – 47 = 2
49 – 56 = -7
49 – 48 = 1
When following the procedure I add these numbers up:
-11 + -2 + -1 + 7 = -7
It is the same as the last formula I did but on this one the +7 is now substituted for a –7.
With this being the change the formula now stands as:
n x 5 - 7 = t
Lets test this formula to see whether it works or not.
26 x 5 – 7 = 123
26 + 25 + 24 + 15 + 33 = 123
It works.
67 x 5 – 7 = 328
67 + 66 + 65 + 56 + 74 = 328
It works.
13 x 5 – 7 = 58
13 + 12 + 11 + 20 + 2 = 58
It works.
The final t-shape format looks like this:
52 + 61 + 69 + 70 + 71 = 323
Now I will deduct the numbers inside the t-shape from the t-number.
52 – 61 = -9
52 – 69 = -17
52 – 70 = -18
52 – 71 = -19
These results seem very similar to the first t-shape. They are all the same numbers but this time they are all pluses. I will add them all up and then put them into the formula. 9 + 17 + 18 + 19 =63. The formula is now:
n x 5 + 63 = t
Now I will show some examples to prove my formula.
5 x 11 + 63 = 118
11 + 20 + 28 + 29 + 30 = 118
It works.
5 x 42 + 63 = 273
42 + 51 + 59 + 60 + 61 = 273
It works.
5 x 39 + 63 = 258
39 + 48 + 56 + 57 + 58 = 258
It works.
While I have been looking at these new formulas I have noticed how that the t-shapes with a t-number either below or above. E.g.
had the same formulas with the exception that one of the
numbers was the opposite i.e. A plus number becoming a minus
number. When the t-shapes with their t-numbers left or right e.g.
their formula was the same with the exception of one number
that was the opposite to the last formula. These numbers were
7 and 63. I wanted to find a connection between the two.
Eventually I found it. 7 x 9 = 63. I wondered what did the 9 have to do with anything but of course, 9 was the grid, 9x9. With this information I felt like I might be able to predict a formula for a shape with a 8x8 grid. I believed that the 5 in the formula represented the 5 boxes for the t-shape so I believed that would stay the same. If I were to predict a formula for a regular t-shape but with an 8x8 grid I would have to consider all factors of my formula as it stands now.
n x 5 - 63 = t
If I was to keep the ‘5 x …’ it would seem correct as there is only going to be 5 boxes in the t-shape. For the 63 I think I will change it to 7 x 8 being the 8 to represent the 8x8 grid, 7 x 8 = 56. With all this together we get a formula that is predicted to be for an 8x8 grid:
N x 5 - 56 = t
Now I am going to draw an 8x8 grid and run though the procedures as before skipping out the unnecessary areas.
Before I move on to the working out, I am going to just check whether my old formula does work on an 8x8 grid.
5 x 61 – 63 = 242
44 + 45 + 46 + 53 + 61 = 249
It does not work.
10 + 11 + 12 + 19 + 27 = 79
I am going to start with the procedure of taking away the numbers in the t-shape away from the t-number.
27 – 10 = 17
27 – 11 = 16
27 – 12 = 15
27 – 19 = 8
Lets see if my prediction for this formula is correct by adding these numbers in the t-shape up.
-17 + -16 + -15 + -8 = -56
I will exchange this number for the original 63 in the formula.
n x 5 - 56 = t
If this is the formula, it means that my prediction is correct. To find out if it is, I will try it out and see if it works or not.
Here are some examples.
5 x 61 – 56 = 249
44 + 45 + 46 + 53 + 61 = 249
It works.
5 x 20 – 56 = 44
3 + 4 + 5 + 12 + 20 = 44
It works.
5 x 43 – 56 = 159
26 + 27 + 28 + 35 + 43 = 159
It works.
All these tests of my predicted formula worked perfectly.
Now I need to find a formula for all the other t-shapes in a 8x8 grid.
First I will try the old formula just incase it works or can help me.
5 x 52 – 7 = 253
42 + 50 + 58 + 51 + 52 = 253
The old formula still works with an 8x8 grid.
Here are some tests to make sure that it defiantly works.
5 x 27 – 7 = 128
17 + 25 + 26 + 27 + 33 = 128
It works.
5 x 46 – 7 = 223
36 + 44 + 45 + 46 + 52 = 223
It works.
5 x 16 – 7 = 73
6 + 14 + 15 + 16 + 22 = 73
It works.
Lets now try to find out the next t-shape formula.
First I will try the old formula.
5 x 21 + 7 = 112
21 + 22 + 23 + 15 + 31 = 112
This original formula worked too.
Here are some tests to make sure that it defiantly works.
5 x 34 + 7 = 177
34 + 35 + 36 + 28 + 44 = 177
It works.
5 x 62 + 7 = 317
62 + 63 + 64 + 56 + 72 = 317
It works.
5 x 10 + 7 = 57
10 + 11 + 12 + 20 + 4 = 57
It works.
There is only one more t-shape left to do now.
First I will try the old formula.
5 x 47 + 63 = 298
47 + 55 + 62 + 63 + 64 = 291
This old formula does not work.
47 – 55 = -8
47 – 62 = -15
47 – 63 = -16
47 – 64 = -17
Now I will add these numbers up to get an answer to use in my formula.
8 + 15 + 16 + 17 = 56. The formula must then be.
n x 5 + 56 = t
This formula follows a pattern the same as a 9x9 grid. The two t-shapes that have their t-numbers at the top or bottom of the t-shape e.g.
Their numbers in the formula are the same but one of
the numbers in their formula is the opposite i.e. a plus
and a minus. This pattern is followed in an 8x8 grid.
Here are some examples to test if this formula is right.
5 x 5 + 56 = 81
5 + 13 + 20 + 21 + 22 = 87
It works.
34 x 5 + 56 = 226
34 + 42 + 49 + 50 + 51 = 226
It works.
54 x 5 + 56 = 326
54 + 62 + 69 + 70 + 71 = 326
It works.
I have worked out the formulas for all the t-shapes in an 8x8 grid. I found that the t-shapes with their t-numbers on the left and right e.g.
have the same formula as they did for the 9x9. This could
mean that I have already found out their ‘general formula’
(formula for any grid).For the other type of t-shapes e.g.
have different formulas from the original. The 9x9 grid
formula consisted of n x 5 –(+) 63. The 8x8 grid
formula consisted of n x 5 –(+) 56.
I set my targets to finding a connection between the 63 and 56 and it was just numbers in the 7 times table:
7, 14, 21, 28, 35,
42, 49, 56, 63 and 70
Now it may be possible for me to work out a general formula for these t-shapes. I already have a general formula for 2 t-shapes but I haven’t tested it fully yet. If I knew the grid size and t-shape, I would be able to make a formula for it. E.g. a grid size of 6x6 and a normal upward t-shape, the formula is n x 5 + 42.
I worked that out by using the information above me.
I thought for a while and came up with an idea to incorporate the grid size into the formula. It is:
(n x 5) -(+) (7 x g) = t
The g stands for grid and for grid you would look at whatever grid it is e.g. 3x3
and use the 3 as the g. Note: Begin as normal through the formula but when you encounter the brackets work them out separately and the answer from the brackets will tell you what to add or minus it by.
The 4 general formulas are:
(t x 5) - (7 x g) = t
(t x 5) + (7 x g) = t
t x 5 + 7 = t
t x 5 - 7 = t
Here they are demonstrated with a random grid. I
I randomly chose a 4x4 grid.
(10 x 5) - (7 x 4) = 22
First I multiplied 10 by 5 which gave me 50,
then separately I worked out 7 x 4, that is
28. Finally I took 28 away from 50 which
gave me the answer.
1 + 2 + 3 + 6 + 10 = 22
(15 x 5) - (7 x 4) = 75 - 28 = 47
6 + 7 + 8 + 11 + 15 = 47
(2 x 5) + (7 x 4) = 10 + 28 = 38
6 + 10 + 11 + 9 + 2 = 28
(6 x 5) + (7 x 4) = 30 + 28 = 58
6 + 10 + 13 + 14 + 15 = 58
6 x 5 + 7 = 37
6 + 7 + 8 + 4 + 12 = 37
9 x 5 + 7 = 52
9 + 10 + 11 + 7 + 15 = 52
7 x 5 - 7 = 28
1 + 5 + 6 + 7 + 9 = 28
12 x 5 - 7 = 53
6 + 10 + 11 + 12 + 14 = 53
I have gained all possible formulas that can link you from a t-number to a t-total in any grid.