GCSE: Number Stairs, Grids and Sequences
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Number Grid Investigation.
= x�+33x (x+3)(x+30) = x�+33x+90 (x�+33x+90)  (x�+33x) = 90 Therefore: (x+3)(x+30)  x(x+33) = 90 n x n Sections Size of section Difference 2x2 10 3x3 40 4x4 90 N x n 10(n1) � [image001.gif] n X x + (n1) x + 10(n1) x + 10(n1) + (n1) [image002.gif] n x*[x+10(n1)+(n1)] = x*[x+11(n1)] = x� + 11nx  11x [x+(n1)]*[x+10(n1)] = x�+10xn10x+10(n1)�+nxx = x� + 11xn  11x + 10(n1)� [x� + 11xn  11x + 10(n1)�]  [x� + 11nx  11x] = 10(n1)� Therefore: {[x+(n1)]*[x+10(n1)]}  {x*[x+10(n1)+(n1)]} = 10(n1)� The difference for any sized square on the number grid is 10(n1)� when n is the length of the side of the square.
 Word count: 1897

Number Stairs
To generate a sequence, I will need a starting value and a rule to find the next term. My starting value will be the Stair Total of the first 3step stair, and the rule to find the next term will be found by working through the Stair Total difference of first 3step stair and the second 3step stair, by subtracting the two totals. 12 11 12 1 2 3 1+2+3+11+12+12=50 The Stair Total of this 3Sstair is 50 22 12 13 2 3 4 2+3+4+12+13+22=56 The Stair Total of this 3Sstair is 23 13 14 3 4 5 3+4+5+13+14+23=62 The
 Word count: 4332

Number grids
2�2 55�66=3630 56�65=3640 36403630 = 10 89�100=8900 90�99=8910 89108900=10 22�33=726 23�32=736 736726=10 27�38=1026 28�37=1036 10361026=10 After multiplying the corners of the 2�2 squares I then took the lowest away from the highest. This number is always 10. In this section: a represents the number in the top left hand corner of the inset square. a a+1 a+10 a+11 (a+1)�(a+10) a�(a+11)Here I have multiplied the opposite corners of the grid [a�+11a+10][a�+11a] Here I have multiplied out the brackets and simplified the rule a�+11a+10  a�+11a Here I have subtracted the two sections to prove my overall rule.
 Word count: 2057

I have been asked to investigate the amount of squares protected by a queen in any position on a chessboard.
I will tabulate my results because this allows comparison and patternfinding much easier. Q = queen in this position protected squares protected squares 8x8 chessboard Q Q 22 22 22 22 22 22 22 22 22 24 24 24 24 24 24 22 22 24 26 26 26 26 24 22 22 24 26 28 28 26 24 22 22 24 26 28 28 26 24 22 22 24 26 26 26 26 24 22 22 24 24 24 24 24 24 22 22 22 22 22 22 22 22 22 The positions where the most squares are protected are the middle four, with concentric rings decreasing in twos.
 Word count: 2409

Number Grid.
In an effort to expand the investigation, I decided to carry out the same test but by varying the size of the grids within the 10 x 10 matrix. I first of all increased the size of the cell selection to a 3 x 3 area. My results were as follows: Example 1 (1 x 23)  (3 x 21) = 40 Example 2 (53 x 75)  (55 x 73) = 40 Example 3 (78 x 100)  (80 x 98)
 Word count: 1050

Number stairs
The overlapped numbers will be added up to create a total figure this is the number i am trying to work towards. The formula created will have to be formed to calculate the total from using the bottom left hand number on the stair shape.
 Word count: 218

Number Grids
N N+1 N+10 N+11 So how can we get around this? We will use algebra. This grid (right) represents a 2x2 grid. It can be used with the correct formulas to work out any difference for a 2x2 grid. So we start with a formula to find the two numbers multiplied together. n� (n+11), but we can improve on this buy tuning the to formulas into 1. In this second formula, we put both sums together. (n+1) (n+10) when we expand the brackets, it reveals the difference for any 2x2 grid.
 Word count: 2887

Number Stairs
In doing so, I will use different size grids and use different size stairs. Again, investigating relationships and discovering formulae for each problem I encounter. I am going to start with a 10 by 10 grid with a 3step stair. 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53 54 55 56 57 58 59 60 41 42 43 44 45 46 47 48 49 50
 Word count: 3711

e.(mc)2
87 88 89 90 91 92 93 94 95 96 97 98 99 100 35 36 45 46 * Analysis 10 by 10 Grids 13 14 23 24 66 67 76 77 18 19 28 29 x y Difference D1 D2 1 1 0 10 2 2 10 20 30 3 3 40 20 50 4 4 90 20 70 5 5 160 20 8 by 8 Grids 57 58 65 66 11 by 11 Grids 101 102 112 113 12 by 12 Grids 42 43 54 55 13 by 13 Grids 140 141 153 154 14 by 14 Grids
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In this project I hope to achieve the ability to explain how and why certain formulae can be associated with this particular problem. I also hope to extend the investigation in order to find interesting patterns, which can be relevant to the task.
Therefore there are 6 ? 5 ? 4 ? 3 = 360 combinations With 5 colours: Once the red, blue, yellow and green squares have been fixed (360 possible combinations), there are 2 remaining places in which the grey can be placed. Therefore there are 6 ? 5 ? 4 ? 3 ? 2= 720 combinations With 6 colours: There is one remaining place in which the brown can be placed after the other 5 are fixed. Therefore there are 6 ?
 Word count: 2718

The Open Box Problem
AIM 1 First I will be looking at aim 1 which uses a square sheet of card. A square is being cut from each corner. Aim 1 is to find out what fraction of the whole original square needs to be cut from each corner to make the largest possible volume. This chart shows the volume compared to the 3 different sizes of original card (10cm�, 20cm� and 30cm�) and the size of the squares cut out. For each size of A (original piece of card)
 Word count: 1063

the Open Box Problem
To get that, I need to get rid of the yellow square's lengths. These together will give me the area of the base. Next, I time the base area by x, which is the height to get the volume of the box. Example: x = 2 20X20 Square v = x(l2x)(w2x) v = 2(204)(204) v = 2*16*16 v = 512cm� Now I will do a trail run with a 20X20 square. The results will be given on the table below: Small Square Length (cm)
 Word count: 1334

Number stairs
However, on larger or smaller number grids some blocks may not exist or more blocks may exist. The formulae in the furthest right hand side column are always in the furthest right hand side column on every size of grid as these define what the number is at the end of each row. However, the height and width of the number grid may be more than, or less than, five. Additional blocks on a larger grid will follow the same pattern as that which can be seen above and if on a smaller number grid there will just be less
 Word count: 2084

Investigate the difference between the products of the numbers in the opposite corners of any rectangles that can be drawn on a 100 square.
I will also do one test at a random site on the grid to make sure that results are as close to perfect as possible. I predict that the difference will increase as the shape moves across or down the grid. 1 2 11 12 1 x 12 = 12 2 x 11 = 22 22  12 = 10 5 6 15 16 5 x 16 = 80 5 x 15 = 90 90  80 = 10 9 10 19 20 9 x 20 = 180 10 x 19 = 190 190  180 = 10 From this
 Word count: 5216

Boxes made in the shape of a cube are easy to stack to make displays in supermarkets. Investigate
2x5=10 As you can see my formula was correct! 2d Triangles I shall try another shape: I will find a formula for this pattern by, first of all, drawing a difference table: Number of Layers (N) Number Of Boxes (B) 1st difference 2nd Difference 1 1 2 2 3 1 3 3 6 1 4 4 10 1 5 5 15 As this table goes into a second difference this means it is a quadratic equation, implying the highest power of N is N2.
 Word count: 5257

Opposite Corners Investigation
What about a 3x3 number square? X X + 2 X + 6 X + 8 (X + 2) (X + 6) = X2 + 2X + 6X +12 = X2 + 8X +12 X (X + 8) = X2 + 8X Difference = 12 So, the difference between the products of the opposite corner numbers in a 3x3 number square is 10. What about Other squares? X This investigation does not work with a square size of 1x1, as the square does not have four corners. X X + 1 X + 2 X + 3 (X + 1)
 Word count: 2282

Opposite Corners
2 x 3 Rectangles 1 2 3 11 12 13 To keep things simple I have started with rectangles with a width of 2 squares. I kept the width to two squares and increased the length by one square. (see results table above). I discovered that the width increases by 10 every time the length increases by 1. The difference can be worked out for all rectangles with a width of 2 squares by using several formulas: 1.
 Word count: 516

Number Stairs Maths Investigation
However, on larger or smaller number grids some blocks may not exist or more blocks may exist. The formulae in the furthest right hand side column are always in the furthest right hand side column on every size of grid as these define what the number is at the end of each row. However, the height and width of the number grid may be more than, or less than, five. Additional blocks on a larger grid will follow the same pattern as that which can be seen above and if on a smaller number grid there will just be less
 Word count: 2089

My task is to investigate a 2x2 box on a 100 square
94= 6768 82 83 84 74 x 92= 6808 92 93 94 6808  6768 = 40 DIFFERENCE = 40 Test 3 1 2 3 1 x 23 = 23 11 12 13 3 x 21 = 63 21 22 23 63  23 = 40 DIFFERENCE = 40 Prediction I predict that in a 3 x 3 square the difference will always be 40 Proof 26 27 28 26 x 48 = 1248 36 37 38 28 x 46 = 1288 46 47 48 1288  1248 = 40 DIFFERENCE = 40 Algebra I will assign a letter to the first number in the 3x3square, n.
 Word count: 4357

The problem is to investigate the differences of corner numbers on a multiplication grid.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 From the squares I can find the difference for a 2x2 grid.
 Word count: 1974

Number Grid.
What I will do is pick out random numbers in a square this is a two by two pattern. 12 13 22 23 I am going to multiply the top right and bottom left numbers together and the top left and bottom right numbers together. 12 x 23 = 276 13 x 22 = 286 I am going to then work out the difference between these two numbers. 286  276 10 By looking at this I can see I have come to the difference of 10.
 Word count: 3243

For my investigation I will be finding out patterns and differences in a number grid.
Once I have found out the differences I will then write up a statement and the predictions for each of my 6x6 and 7x7 squares. After I have done some data calculation to prove my differences are correct I will then move on to working out differences for rectangular shapes for instance 2x3, 2x4, 2x5, 3x4, 3x5, 4x5 after doing this I will then do another set of statements and predictions to work out 2x6, 2x7, 3x6, 3x7, 4x6, 4x7, and 5x6, 5x7 this would be the second part complete of my investigation.
 Word count: 9480

My Investigation is called 'Number Stairs'
Method To start with I will work out the stairs going across. This, after working a few stairs a pattern occurs; this can be worked out by subtracting on stair total (e.g. 3) from the next stair total down (e.g. 2). This would be Square 3=62 & Square 2=56 that would be 6256=6. This is my method plan. Once this is done I will show my results using a straightline graph.
 Word count: 1945

Number stairs.
+44 Stair 3: 21 + 22 + 23 + 31 + 32 + 41 = 170 Y+Y+1+Y+2+Y+10+Y+10+1+Y+20= 6Y+44=170 6Y=17044 6Y=126�21=6 Y=21 6(21) +44 Stair 4: 25 + 26 + 27 + 35 + 36 + 45 = 194 L+L+1+L+2+L+10+L+10+1+L+20= 6L+44=194 6L=19444=150 6L=150�25=6 L=25 6(25) +44 Conclusion: Based on all of the 4 3 step stairs that I have investigated, I can now say that I have found a formula which works with all of the other 3 step stairs. The formula which I have found is 6A+44 which just simply means 6AX4X10+4.
 Word count: 4730

For other 3step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid. To start the investigation a 10x10numbered grid square is used as illustrated below in table 1:
 10 x  11 X x + 1 x + 2 By adding the values from the equation (20  10  9 + 1 + 2) = [36] Thus we can use the algebra equation 6x  36 = [the total value of squares in a 10x10 square] Using the above logic and method we can use it in other grids, such as an 11x11 and a 12x12 numbered grid square. Using the same 3step stair approach we can use the theory for the 11x11 and 12x12numbered square to find a pattern.
 Word count: 6878