T-Shapes Coursework

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Maths Investigation 2

Higher Tier Task - “T Shapes

I was given a number grid, like Fig 1.1. On it, a “T Shape” was to be placed in any possible position on the grid, such that all of the shape remained within the boundaries. The task was to investigate the patterns connecting the numbers in the “T” themselves, and also patterns in combinations of these numbers.

Fig 1.1

Any number of investigations was possible, but I chose an investigation in which I would add up all the numbers in the “T”.

Section 1: 3x1 “T” on Width 8 Grid

1) Introduction

In this section, a 3x1 “T” will be used. This means that there are 3 squares along the top of the “T” (the “wing”, in yellow), and 1 on the bottom (the “tail”, in green). This is as shown originally in the task brief.

2) Method

To discover any patterns, I will calculate the sum of the numbers in the “T” for 5 sequential locations of the box within the grid. As it would be impractical to do all possible calculations, 5 should be enough to display any patterns that may lie therein. The “Middle Number” will be the number in the wing that is adjacent to the tail. It is highlighted in red on Fig 1.1.

3) Data Collection

Here are the results of the 5 calculations for 3x1 “T” on Width 8 Grid:

4) Data Analysis

From the table, it is possible to see a couple of useful patterns:

  1. The Sum of the Wing is always 3 times the Middle Number;
  2. The Sum of the Tail is always 8 more than the Middle Number;

5) Generalisation

It can be assumed that for all possible locations of the 3x1 “T” on the width 8 grid, these patterns will be true. Therefore, the following logic can be used to create a formula where:

n = the Middle Number

  1. Using Pattern 1 above, we can say that the Sum of the Wing = 3n
  2. Using Pattern 2 above, we can say that the Sum of the Tail = n + 8
  3. Because Total Sum = Sum of Wing + Sum of Tail, then Total Sum = 3n + n + 8
  4. This simplifies to give the formula:

Total Sum = 4n + 8


This means I predict that with a 3x1 “T” on a width 8 grid, the sum will always be 4n + 8.

6) Testing

My formula works as shown with the following, previously unused values:

1) Where n = 19

2) Where n = 38

7) Justification

The formula can be proven to work with algebra. The basic algebraic labelling of the 3x1 “T” on a width 8 grid is:

n is the Middle Number;

(n + 1) is the top-right number in the “T”, because it is always “1 more” than n;

(n - 1) is the top-left number in the “T”, because it is always “1 less” than n;

(n + 8) is the tail, because it is always “8 more” than n.

In the “4n + 8” formula, the “4” represents the number of  “n”s which are present within the “T”. Every square in the “T” is referred to in terms of n and so with 4 squares added together, there are 4n.

The “8” comes from the “Sum of the Tail”, which is 8 more than the Middle Number.

The “Sum of the Wing” is 3n because there are 3 “n”s present in the 3 squares of the wing. The “-1” and “+1” cancel out each other when added to together.

8) Conclusion

After this justification, it can now be said that for every possible 3x1 “T” on a Width 8 Grid, the Total Sum of all of the squares contained within it is 4n + 8.

9) Extension

Having done this, I saw that my formula would only work for a 3x1 “T” on a Width 8 grid. To improve the usefulness of my formula, I wondered what would happen to the Total Sum if I varied the width of the grid on which the 3x1 “T” was placed.

Section 2: 3x1 “T” on Width “g” Grid

1) Introduction

Throughout this section, the variable g will be used to represent the width of the grid i.e. the number of columns on the grid. The variable n will continue to be used for the Middle Number of the “T” i.e. the location of the “T” upon the grid.

2) Method

Varying values of g will be tested to give different widths of the grid. Grid sizes to be used for data collection will range from 9 to 14. On these grids, from 5 sequential positions of the 3x1 “T”, the Total Sum will be calculated. Again, I believe 5 calculations are enough to display any patterns.

3) Data Collection

Figs 2.1 to 2.5 are the grids used for the varying values of g. An example of the 3x1 “T” has been highlighted on each one.

(a) Here are the results of the 5 calculations for a 3x1 “T” on Width 9 Grid (Fig 2.1):


(b) Here are the results of the 5 calculations for a 3x1 “T” on Width 10 Grid (Fig 2.2):

(c) Here are the results of the 5 calculations for a 3x1 “T” on Width 11 Grid (Fig 2.3):

(d) Here are the results of the 5 calculations for a 3x1 “T” on Width 12 Grid (Fig 2.4):

(e) Here are the results of the 5 calculations for a 3x1 “T” on Width 13 Grid (Fig 2.5):

4) Data Analysis

From the tables (a)-(e), it is possible to see that when the grid width is varied, only the Sum of the Tail changes. In fact, if we take one constant Middle Number, 2, from each of the above tables, we get the following:

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From these tables, it is possible to see a useful pattern:

  1. The Sum of the Tail equals the Middle Number plus the Grid Width.

5) Generalisation

It can be assumed that for all possible locations of the 3x1 “T” on the width g grid, these patterns will be true. Therefore, the following logic can be used to create a formula where:

n = the Middle Number

g = the Grid Width

  1. Using Pattern 1 above, we can say that the Sum of the Tail = n + g
  2. Using the patterns from Section ...

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