The Relationship between the T-number and T-total

I chose three T-shapes from the table above, they are:

I am going to test the T-shape one, use letter n for T-number and use letter T for T-total. The result is showing below:

        

From the T-shape above we can work out the relationship between T-number and T-total in size nine by nine number-grid.

The steps are showing below:

So the relationship between T-total and T-number in size 9 by 9 number-grid is: T=5n – 63.

Then I am going to check the answer,

• n = 20
• T = 1 + 2 + 3 + 11 + 20 = 37
• 37 = 5 × 20 – 63

So the result shows this formula is right, but I have to find out if this formula can be used anywhere on a size 9 by 9 number-grid. After I tested this T-shape I tested the two other T-shapes above, the results are same as T-shape one.  

The result of T-shape two:

• n = 25
• T = 6 + 7 + 8 + 16 + 25 = 62
• 62 = 5 × 25 – 63

The result of T-shape three:

• n = 78
• T = 59 + 60 + 61 + 69 + 78 = 327
• 327 = 5 × 78 – 63

These two T-shapes are both correct; I used same formula as T-shape one.

So this means this relationship can be used for anywhere of a size 9 by 9 number-grid.

Key point – the formula of the size 9 number-grid is showing below:

Question two:

• Use grids of different sizes. Translate the T-shape to different position. Investigate relationships between the T-total, T-number and the grid size.

I chose two different sizes of number-grid to see the relationships between the T-number, T-total and the grid size.

The first one is size of 6 number-grid.

 First of all I chose three T-shapes are showing below:

 

I am still using letter n for T-number, letter T for T-number try to find the relationship between the T-number and T-total in size six number-grid.

The result is showing below:

Now I am going to work out the T-total of this T-shape.

I found the relationship between the T-total and T-number is:

T = 5n – 42

The next thing is check the answer see if it is really can used for these T-shapes.

The results are showing below:

In T-shape one:

• n = 14
• T = 1 + 2 + 3 + 8 + 14 = 28
• 28 = 5 × 14 – 42

In T-shape two:

• n = 32
• T = 19 + 20 + 21 + 26 + 32 = 118
• 118 = 5 × 32 – 42

In T-shape three:

• n = 35
• T = 22 + 23 + 24 + 29 + 35 = 133
• 133 = 5 × 35 – 42

This is means the relationship of the T-number and T-total in size six by six number-grid can be used for anywhere in this size grid.

Key point- the formula of this size grid is:

The second one is size of 12 number-grid:

Same as last size grid, I chose three T-shapes from table above, they are:

     

I am going to use letter n for T-number and use letter T for T-total to find the relationship between T-number and T-total in this grid size.

The result is showing below:

Now I am going to work out the T-total of this size grid:

Next step is check the answer to see if it can be used for this size grid:

In T-shape one:

• n = 26
• T = 1 + 2 + 3 + 14 + 26 = 46
• 46 = 5 × 26 – 84

In T-shape two:

• n = 78
• T = 53 + 54 + 55 + 66 + 78 = 306
• 306 = 5 × 78 – 84

In T-shape three:

• n = 143
• T = 118 + 119 + 120 + 131 + 143 = 631
• 631 = 5 × 143 – 84

So the formula below can be used anywhere is this size grid.

Now I already down the three different size grids I put the all the formula below to compare them what is the different:

1. Size 9 by 9 number-grid:

2. Size 6 by 6 number-grid:

3. Size 12 by 12 number-grid:

Now I try to use letter g for number of grid size.

The formula is showing below:

Now I am going to check the answer below:

1. T-shape one of size 9 by 9 number-grid.

        

2. T-shape one of size 6 by 6 number-grid.

        

3. T-shape one of size 12 by 12 number grid.

        

So the formula T = 5n – 7g can be used for anywhere in any grid size!

Key point – the relationship between the T-number and T-total in any grid size is showing below:

        

Now I am try to find out where is it not possible to place the T-number. First I want to try the size nine number-grid there is a table showing below:

From the table above I found out that we could not use some numbers next to the edge of number-grid, the reason for this is showing below:

The diagram above shows some numbers that I chose from the size nine number-grid, all these numbers could not be T-numbers. That is because we cannot use these numbers to form a complete T-shape. For example, I chose number one as a T-number, the meaning of T-number is the number at the bottom of the T-shape. But the problem is the number one is the top row of the whole number-grid so we cannot find any line above it to form a T-shape. Next we try the number ten as a T-number, still could not get a whole T-shape out, if we use ten the shape just like this:

Same problem as all the number in the last right hand-side column we cannot make a T-shape out use this numbers.

Now I found out that all the number in the first two rows across, the first column and last column cannot be a T-number.