Maths GCSE Investigation - T Numbers
Maths GCSE Investigation - T Numbers
Introduction
In a number grid, a T-shape can be drawn outlining five numbers. The sum of all the numbers within the T-shape is the T-total, and the number at the base of the T is the T-number.
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In this 5 by 5 grid example, the T-number is 12, and the T-total is 1+2+3+7+12, which is 25. My task is to investigate the relationship between the T-number and the T-total for T-shapes within the number grid. The factors I can change which might change the pattern of numbers are:
* Grid size
* Direction of T-shape (with the T pointing N/S/E/W)
* Translating the T-shape within a number grid to investigate patterns.
Investigation#1-upright T-shape
First of all, I am going to focus on the T-shape pointing S, the upright T:
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I first thought that any T-shape could be created if the T-number was known, as the rest of the numbers in the T-shape are related to the T-number. For example, in the above 6 by 6 square, a T-shape is selected:
T-number=21
T-total=21(T-number) + 21-6(15) + 21-12(9) + 21-12-1(8) + 21-12+1(10)
To create a formula, I will now substitute in X for the T-number:
T-total=X(T-number) + X-6(15) + X-12(9) + X-12-1(8) + X-12+1(10)
=5X - 42
I will now see if my formula works for another T-shape in the same grid:
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T-total = 5X-42
= 5*34-42
= 170-42
=128
21+22+23+28+34=128
My formula works. Now I will try it for a different sized grid:
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T-total = 5X-42
= 5*49-42
= 245-42
= 203
49+40+30+31+32=182
My first formula does not work for the 9 by 9 grid. Using the first method for finding the formula, the formula for the 9 by 9 grid should be:
T-total = 5X-63
I will now analyse my results so far:
Grid Size
Formula
6 by 6
5T-42
9 by 9
5T-63
I noticed that the number at the end of the formula is always seven times the grid width. From this assumption, the formula for an upright T-shape in any grid should be:
T-total = 5X-7Z
Where Z is the grid width.
I will now test my new formula on a 7 by 7 grid:
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This is a preview of the whole essay
T-total = 5X-7Z
Where Z is the grid width.
I will now test my new formula on a 7 by 7 grid:
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T-total = 5X-7Z
= 5*26-7*7
= 130-49
=81
1+12+13+19+26=81
My formula works for the 7 by 7 grid, and is also proved to work for both the 6 by 6 and the 9 by 9 grids, so I can confidently say that it works for any sized grid, as the grid size is included in the formula.
Investigation#2-Upside-down T-shape
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From the above diagram, I can derive the following formula:
T-total = X(7) + X+Z(18) + 3(X+2Z)-X+2Z, 29, is the middle of 3 consecutive numbers, so I can refer to them as 3(X+2Z), 3*29.
T-total = 5X+7Z
I will now test this for a 5 by 5 grid:
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T-total = 5X+7Z
= 5*14+7*5
= 70+35
= 105
4+19+23+24+25=105
My formula works for the 5 by 5 grid, so I can confidently say that it is the correct formula for an upside-down t-shape.
Investigation#3-Sideways T-shape (pointing E)
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From the above diagram, I can derive the following formula:
T-total = X(14) + X-1(13) + X-2(12) + X-2-Z(4) + X-2+Z(20)
T-total = 5X-7
Due to the fact that the grid sizes are not present in the formula, I need to test my formula for other grid sizes:
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T-total = 5X-7
= 5*36-7
= 173
24+34+44+35+36 = 173
My formula works for the 10 by 10-sized grid, so it is correct for a sideways T-shape pointing E.
Investigation#4-Sideways T-shape (pointing W)
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The upright T-shape and the upside-down T-shape had an opposite in their formulas: -7Z and +7Z. Due to this fact, I predict that the formula for a sideways T-shape pointing W would be 5X+7, opposite to the sideways T-shape pointing E's formula of 5X-7.
To test my prediction, I will apply it to the above 9 by 9 grid:
T-total = 5X+7
= 5*51+7
= 262
51+52+53+44+62 = 262
My formula works, so 5T+7 is the formula for a sideways T-shape pointing W.
Summary
From the above four investigations, I can summarise that:
* The formula for an upright T-shape in ANY grid size is:
T-total = 5X-7Z
* The formula for an upside-down T-shape in ANY grid size is:
T-total = 5X+7Z
* The formula for a sideways T-shape pointing E in ANY grid size is:
T-total = 5X-7
* The formula for a sideways T-shape pointing W in ANY grid size is:
T-total = 5X+7
Further investigations
There are several different variables which I can change to extend this investigation:
* Investigate in 3 dimensions
* Translation of T-shape
* Rotation of T-shape
* Reflection of T-shape
* Enlargement of T-shape
3D investigations
A basic 3D cube can be made from small cubes, each representing one number. However, T-shapes within the 3D cube would be very difficult to identify, and the order of placement of the cubes can vary. This leads to complications but the differences in pattern would still be related to the width of the cube, so the pattern would be similar to the first four investigations. This investigation does not vary any factors apart from stretching another dimension. This extra dimension would produce the same patterns as the original investigations, so I will not pursue this investigation.
Translation of T-shape
I will carry out this investigation. This is because that the variable factors, in this case the vectors of the T-number, can be varied and can produce interesting results.
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In the above grid, the T-number, 16, has been translated on a vector of (2, -3), to produce a new T-shape. I will need to repeat the translation and produce a third T-shape to find a pattern.
Translation 1 (2, -3):
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T-total
Original T-shape
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Translated 1 (red)
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Difference
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90
Translation 2 (2, -3):
T-number
T-total
Translated 1 (red)
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Translated 2 (blue)
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426
Difference
38
90
From the above two tables, I cannot distinguish a clear pattern, so I will repeat the investigation with another vector value:
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In the above grid, the translation of the T-number is (3, -1), and has been repeated on the green T-shape to form a third purple T-shape.
Translation 1 (3, -1):
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T-total
Original T-shape
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Translated 1 (green)
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Difference
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75
Translation 2 (3, -1):
T-number
T-total
Translated 1 (green)
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Translated 2 (purple)
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96
Difference
5
75
The above tables show differences of 15 and 75. When I compare these results to the first differences of 38 and 190, I notice that the difference in the T-total is 5 times the difference in the T-number. To confirm this, I will now experiment with a new translation with different vectors in a different size grid. This will tell me if my prediction is correct and will also tell me if the grid size is related to this pattern.
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In the above 9 by 9 grid, the vector of translation for the T-shape is (3, -3)
Translation (3, -3):
T-number
T-total
Original T-shape
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Translated 1 (brown)
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Difference
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The above table confirms my prediction that the difference in T-total for an upright T-shape is always 5 times the difference in the T-number no matter what the grid size or translation vector is. This pattern can be written as:
For this investigation, the variables are the vectors, so when I substitute the vector values into the formula, replacing the ?T-number, the formula becomes:
Where X and Y are the vector values and Z is the grid size. X + ZY can be used to replace ?T-number.
Rotation of T-shape
The first four basic investigations have already explored the basic rotations through 90°, 180° and 270°. Now I will investigate rotations through 45°.
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Applying the rules previously, the T-number in the above shape should be 35. The T-total is 105. The formula is:
T-total = X(35) + X-Z+1(25) + X-2Z+2(15) + X-3Z+1(3) + X-Z+3(27)
= 5X-7Z+7
= 5*35 - 7*11 + 7
= 105
3+15+25+27+35 = 105
From the derivation of my formula, I have proved that the T-shape and its number are in a fixed position, so it will work wherever it is placed.
Test formula for 6 by 6 grid:
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T-total = 5X-7Z+7
= 5*20-7*6+7
= 65
3+10+15+17+20 = 65
My formula works for the 6 by 6 grid. So the formula for a T-shape rotated through 45° is:
T-total = 5X-7Z+7
N.B. Since this investigation is based on patterns derived from a diagram, ALL the formulas derived only apply in cases which the T-shape itself fits into the grid.
Qiming Liu 10SM 09/05/2007
