T-totals. I am going to investigate the relationship between the t-total, T, and the t-number, n. The t-number is always the number at the bottom of the t-shape when it is orientated upright.

T-Totals

I am going to investigate the relationship between the t-total, T, and the t-number, n. The t-number is always the number at the bottom of the t-shape when it is orientated upright. Here the t-number would be 20. The t-total is the sum of the cells inside the t-shape. Here it would be 37 as 1+2+3+11+20 = 37

I will calculate the t-total for different t-numbers on 9×9 grid. Working algebraically, I will find a relationship that will express the t-total in terms of the t-number and the grid size. I will test this generalisation for t-shapes in an 8×8 and 10×10 grid. I will then transform the t-shape and investigate it’s affect on this relationship.

T-shapes in a 9×9 grid

The t-total increases by 135 when the t-number is increased by 27 and by 15 when the t-number increases by 3. It thus follows that for every unit increase in the t-number there will be an increase of 5 in the t-total.

The t-shapes in a 9×9 grid can be represented algebraically. The t-total, T, can therefore be written in terms of the t-number, n, as T= 5n - 63. Using similar reasoning we can express T in terms of n in an 8×8 and 10×10 grid.

T-shapes in an 8×8 grid

The t-total increases by 120 when the t-number is increased by 24 and by 15 when the t-number increases by 3. It thus follows that for every unit increase in the t-number there will be an increase of 5 in the t-total.

The t-shapes in an 8×8 grid can be represented algebraically. The t-total, T, can therefore be written in terms of the t-number, n, as T= 5n - 56.

T-shapes in a 10×10 grid

The t-total increases by 150 when the t-number is increased by 30 and by 15 when the t-number increases by 3. It thus follows that for every unit increase in the t-number there will be an increase of 5 in the t-total.

The t-shapes in a 10×10 grid can be represented algebraically. The t-total, T, can therefore be written in terms of the t-number, n, as T= 5 n - 70.

T-shapes in a g×g grid

We can express the t-shape in any grid size algebraically. Using g= grid size, the t-total can be written as T= 5n - 7g. Its validity can be checked by recalculating the t-total of some of some of the t-shapes that were used earlier in different grid sizes.

Validation

The t-total of any t-shape in any grid size can clearly be written as T = 5n – 7g. We can now perform transformations on the t-shape and investigate it’s affect on this relationship.

Translation by the vector

The t-number can be translated by the column vector where a is the horizontal distance (positive to the right and negative to the left) and b is the vertical distance (positive being upwards and negative being downwards).