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GCSE: Number Stairs, Grids and Sequences

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  1. Investigating the relationship between the total of a three-step 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
  2. Number Stairs.

    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). 23 13 14 3 4 5 The stair total for this stair shape is 3 + 4 + 5 + 13 + 14 + 23 = 62.

    • Word count: 2779
  3. Number Stairs - For part one, I am investigating the relationship between the stair total and the position of the stair shape on the grid.

    I have studied these answers and concluded that they are all even. To investigate further, I looked at what would happen if I moved the 3-step stairs to the left and right and up and down. I predict that if I move the 3-step stair upwards, the stair total will be greater and if I was to move the 3-step stair downwards, the total will be less. If I moved the 3-step stair to the left, it would be less than moving it to the right.

    • Word count: 2784
  4. An investigation into Number Grids.

    Due to the fact that the difference is always 10 we have decided to show this in a table algebraically. If we multiply the bottom left digit by the top right digit and the top left digit by the bottom right digit we can see that the N's disappear to leave just 10 which is the difference. As is shown below. N(n+1)=n2+1 =see below N(n+21)=n2+21 =N2+21-N2+1 =10 For a two by two grid we can see that there is an algebraic table This is shown below also: N N+1 N+10 N+11 N2+1 N2+21 When subtracted the n cancels out leaving the number 10 which is actually the sized number grid I am working with.

    • Word count: 2630
  5. Investigation into Number Grids.

    For 5 x 5. 24 25 26 27 28 34 35 36 37 38 44 45 46 47 48 54 55 56 57 58 64 65 66 67 68 So (24 x 68)-(28 x 64) 1632-1792 = -160 Just like the normal 5 x 5 grid the answer is the same; -160. This shows that no matter what the numbers are, the answer, always, for a square will always be the same, as long as there is the same number of columns, and therefore rows.

    • Word count: 2622
  6. Number grids

    2�2- 55�66=3630 56�65=3640 3640-3630 = 10 89�100=8900 90�99=8910 8910-8900=10 22�33=726 23�32=736 736-726=10 27�38=1026 28�37=1036 1036-1026=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
  7. 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 pattern-finding 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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
  13. The Open Box Problem

    This shows that the maximum volume is obtained when x is between 1 and 1.5, I will now make another three spreadsheets, to find the maximum volume correct to three decimal places, as I believe that this first spreadsheet does not give an accurate result. These show, that for a square of width 6cm, that the maximum volume is obtained when x=1.000. I will now repeat the spreadsheets above for a square of width 12cm, to find the maximum volume for this size of card, and to attempt to find any patterns in maximum volumes.

    • Word count: 2831
  14. China in the 20th Century Sources Questions

    In the past they have just stood back and watched as dissidents' wives stimulated publicity." I think that this is an advantage to the spouses, as the women would just make a fool of themselves by stimulating publicity. However, they would have had an effective response by stimulating publicity because then other citizens would know what is going on. "In a similar case, the dissident Wang Dan, a leader of the 1989 Tiananmen Square movement, has gone into hiding after receiving death threats police who stalked him for more than a year and a half.

    • Word count: 2643
  15. To Find the Diagonal Difference Taken From Small Nxn Grids, Like 3x3.

    The investigation I will start my investigation with the 2x2 grids. As there are many values for the smaller grids like 2x2 I will take my values from the larger grid in a distinct fashion. That is starting from the lowest value, top left, then moving down and right towards the bottom right. With grids like the 7x7 once I have taken the values from one angle I will then take them from the opposite angle. Like illustrated below. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

    • Word count: 2494

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