GCSE: Number Stairs, Grids and Sequences
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Number Stairs.
Below is the stair shape one square to the right of stair1. This stair is called stair2 as the number in the bottom left hand corner is 1. Stair2 is a translation of stair1 one square to the right. 22 12 13 2 3 4 The stair total for this stair shape is 2 + 3+ 4 + 12 + 13 + 22 = 56. Now we are going to find the step total for stair3 (a translation of stair2 one step to the right).
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For 3step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid. Investigate further the relationship between the stair totals and other step stairs on other number grids.
PART 2 Investigate further the relationship between the stair totals and other step stairs on other number grids. PART 1 The total of the squares inside the stair is all the squares added together. The stair number is the number in the bottom left hand corner of the stair. We can call this number n. In order to see a pattern between the totals of the shapes, we can arrange information in a table. n 6n +44 1 6 50 2 12 54 3 18 64 4 24 68 25 150 194 As you can see from the above table we can come to a formula of: T = 6n + 44 T is the stair total, n the stair number and 44 is the remaining number.
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Stepstair Investigation.
By investigating the formula above you will find that it is the formula for all 3step stairs on a size 10 grid. I worked this out by adding together all the numbers in the 3step stair and then using the formula to see if the formula comes up with the total of all the numbers in the 3step stair. In the diagram coloured red above; 15+16+17+25+26+35=134. The formula 6X + 44 comes up with the answer 134 as well proving that this formula works.
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Number Grids.
I will need a new way to get around this problem so I try to introduce algebra. N N+1 N+10 N+11 This grid (above) represents a 2x2 grid; it can be used with the correct formula to work out any difference for a 2x2 grid. I then tried the formula to fin the two numbers multiplied together. N?(n+11), but could this be improved? I tried tuning the formula into one formula. (n+1)(n+10) when I expanded the brackets, it reveals the difference for any 2x2 grid.
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Number stairs.
Therefore it can be written as: n+n+1+n+2+n+10+n+11+n+20= 6n+ 44 To describe this formula in words, we need to multiply 6 by the stair number of the stair i.e. the first bottom left number. Then if we add 44 to it we will get the stair total. This formula can also be reversed. We can also find the stair number by the same formula. That can be done if we subtract 44 by the stair total and then divide it by 6 which will then give us the stair number.
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Number grid.
and then found the product of the top left number: 12 x 23 = 276 I then repeated this with the top right number and the bottom left number: 13 x 22 =286 On completion of this I found the difference: 286  276 = 10 I repeated this process and recorded the results in a table: Box size: 2 x 2 Box numbers difference 1 12 x 23 = 276 13 x 22 = 286 286  276 = 10 2 27 x 38 = 1026 28 x 37 = 1036 1036  1026 = 10 Box numbers difference
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Open Box Problem.
Notice that the maximum value, 4 is 24/6 (24 being the length/width of the square). The scatter diagram below shows the volume of the open box against its square cut (x) for a square with a 24cm by 24cm measurement. 30cm by 30cm Square Cut x L (302x) W (302x) Volume 1 28 28 784 2 26 26 1352 3 24 24 1728 4 22 22 1936 5 20 20 2000 6 18 18 1944 7 16 16 1792 8 14 14 1568 9 12 12 1296 10 10 10 1000 11 8 8 704 12 6 6 432 13 4 4 208 14 2 2 56 15 0 0 0 Volume = x(302x)(302x)
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He analysis of number patterns on various types of number grids.
The Rule that would have to be applied to the above numbers is as follows: A box is drawn around the four or more numbers as shown in the above diagram. Find the product of the top left number and the bottom right number in the box. Do the same with the top right and the bottom left numbers, Calculate the difference between these products, making sure the number is a positive number. Note that in the above grid of 10 x 10, we have the following boxes and the specific numbers in the boxes.
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As part of my G.C.S.E Maths we had to do a piece of coursework on connect four winning lines.
V Box The Vertical box on the grid has the same pattern as the h box. D Box The first value will always be two, the second value will be the squared number from the first value in the horizontal/vertical box The algebra term Grid size h v d total 4x4 1x4 1x4 2x1 10 5x5 2x5 2x5 2x4 18 6x6 3x6 3x6 2x9 54 NxN (N3)xN (N3)xN 2x(N3) H+V+D = total amount of winning lines Rectangle grids for connect four on next page.
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Investigate the difference of the product of the diagonally opposite corners of a certain shape, drawn on a 10x10 grid with the individual squares numbered off 1 to 100.
46 x 55 = 2,530 45 x 56 = 2,520 Diff = 10 It seems so. I now predict that all 2x2 squares will have a difference of 10. I shall test this prediction: 70 x 79 = 5,530 69 x 80 = 5,520 Diff = 10 I am correct; all 2x2 squares have a difference of 10 This brings up the obvious question: * Is the difference different when the square changes size? In answer to this I will do a test for a 3x3 square and a 4x4 square 3x3 square 1 2 3 4 5 6 7
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Step Stairs  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
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Number Stairs investigation.
 Try and explain any links between figures and anything that has been found. 10x 10 grid 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 71 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 31 32 33 34 35 36 37 38 39 40 21 22 23 24 25 26 27 28 29 30 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7
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Number Grid Investigation.
Taking these results into account, I predict for any 2x2 square the result will always be 10. Below is a table of results for 2x2 squares that were randomly chosen from the 10x10 grid. Table 1. 2x2 results. 1st No. multiplication 2nd No. multiplication Difference 5 x 16 = 80 6 x 15 = 90 10 17 x 28 = 476 18 x 27 = 486 10 32 x 43 = 1376 33 x 42 = 1386 10 68 x 79 = 5372 69 x 78 = 5382 10 The results from T.1 show that the difference is a constant 10, so my prediction was correct.
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The Magic of Vedic Mathematics.
Suppose you want to find the square of a number which is one less than the number whose square is known, you can use the following method: Square of 79 will be given as, (80)^2  (80 + 79) = 6400  159 = 6241. Finding the square of a number near 50. Now, if you want to find the square of 51, the formula will be, (5)^2+1/ (1)^2 = 25 + 1/ 01 = 2601. This is what you do: The LHS of the answer is given as (5)^2 + 1 and the RHS is given as the square of the difference of the number from 50.
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Number Grid Maths Coursework.
I can also say that for all 3x3 squares that the difference is 40 and for all 4x4 squares the difference is 90 in 10x10 grids. From this I can therefore say that all squares of the same size have the same difference when the top right and bottom left corner are multiplied and the answer is then subtracted from the product of the top left corner and bottom right corners. An LxL square in a 10x10 grid Since the difference results of any square is the same I will now make the size of the square variable and as it is variable and not a set number it will be known as L for the length of the side.
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Number Grids Investigation.
n n+1 n+10 n+11 (n+1)(n+10)  n(n+11) n�+11n+10  n�11n = 10 By working using algebra, I can see that I will always get an answer of 10 on a 2x2 grid. I will now test a 3x3 grid and carry out the same methods as before for my investigation. 1 x 23 = 23 3 x 21 = 63 the difference between the top number and the bottom number = 40 I will again do the same for another 3x3 grid.
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Diagonal Difference.
I will also do an extension by doing a rectangle and a square and then find the diagonal difference and the formula for this. I am going to find the formula by finding the diagonal difference of all the sizes within the 10 by 10 grids, and then try to find any patterns, which would help me in finding the formula by drawing a grid. This method would be good because it will show me the diagonal difference of any square of any size, because I would just have to insert the value of n in the formula to find
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Number grid.
5 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 Example One 12 13 22 23 12 13 DIFFERENCE x 23 x 22 286 = 276 = 286  276 = 10 I would like to show one more example of this to see if the difference will always be 10. 14 15 24 25 14 15 Difference x 25 x 24 360 350 360  350 10 I am now going to use
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Investigate the products of 2x2 number squares within a large 10x10 number grid.
57 58 67 68 * I will now find the products of the top left & bottom right numbers, and the top right & bottom left numbers. 57 x 68 = 3876 58 x 67 = 3886 * I will then write down what I notice I have noticed that the difference is 10 & that they are both even numbers I will now do this again for 2 more sets of 2x2 number squares. 12 13 22 23 12 x 23 = 276 I have noticed that the difference 13 x 22 = 286 between these two values is 10, and they are both even numbers.
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Random Sample.
Number Of Shape Calculations Total 1 1+2+3+11+12+21 50 2 2+3+4+12+13+22 56 3 3+4+5+13+14+23 62 4 4+5+6+14+15+24 68 5 5+6+7+15+16+25 74 Comment: There is clearly a pattern here; the totals are increasing by 6, we call this a linear sequence. From this information I predict that the next total will be 80 and the 8th will be 92. 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
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Investigate the number of winning lines in the game Connect 4.
These are the steps I will take to complete the set task: 1. I will draw a range of differently sized grids and count the total amount of possible winning solutions each has on it, this will aid me in creating a formula. 2. After I have gathered all my results I will record them in tables. 3. Using the table I will look for patterns and possible links. 4. Using the data I have collected I will attempt to calculate a Connect 4 general formula. 5. After I have established this formula I will try to extend the investigation perhaps for a new game of Connect 5.
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The Open Box Problem.
(All measurements will be measured in centimetres) Question One This is an example of what I will do. 10x10 cut out size 2cm 20x20 cut out size 3cm 30x30 cut out of 4 cm Question Two Some examples on how cut out will look like in question two 10x20 cut out size 3 10x30 cut out size 4 20x30 cut out size 5 Conclusion All the formulas I found were based on the results and graphs shown.
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Number Grid.
I am going to try one more: 17 18 27 28 17 x 28 = 476 18 x 27 = 486 The difference is 10 again! I am now going to try and work out a formula, and try another one using the formula: n n+1 n +10 n +11 If this formula is correct I should be able to pick any number from the grid and use the formula to get the right answer If n =77 then: (n) (N +11) = n2 + 11n. & (n + 1) (n + 10) = n2 + 11n +10. (77) (77+11)
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Investigate the diagonal difference of a 2 by 2 grid inside a 10 by 10 grid
I will also do an extension by doing a rectangle and a square and then find the diagonal difference and the formula for this. I am going to find the formula by finding the diagonal difference of all the sizes within the 10 by 10 grids, and then try to find any patterns, which would help me in finding the formula by drawing a grid.
 Word count: 1769

Investigating the relationship between the total of a threestep stair on a number grid.
46+36+26+37+27+28=200 I would then continue moving it to the right until I am unable to anymore. Part 1: I will start of in the bottom left hand corner and work myself to the right of the grid. Position 1 21 11 12 1 2 3 The total for this stair is: 21+11+1+12+2+3=50 Position 2 22 12 13 2 3 4 The total for this stair is: 22+12+2+13+3+4=56 Position 3 23 13 14 3 4 5 The total for this stair is: 23+13+3+14+4+5=62 I will now tabulate these results and others. Position Number (Lowest number in grid)
 Word count: 2509