# &#147;C&#148; Totals Investigation.

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Introduction

## “C” TOTALS INVESTIGATION

WHAT IS THE INVESTIGATION ABOUT?

In this investigation, I have tried to experiment using different grids but the same C shape.

WHAT IS THE PROBLEM?

I am going to start with a 9x9 grid with 81 counting numbers in it and write them down. I will then draw a as shown below in my mini diagram:

The bottom right number is going to be my C number (n) and the total of all the numbers in the C is going to be my C total, (I am going to call this “t”).

Finally, I am going to see if there is a relationship between “n” & “t”.

MY 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 |

Below I have drawn some random C’s, which I took from the table.

41 | 42 |

50 | |

59 | 60 |

N=60

T=60+50+59+41+42=

252

8 | 9 |

17 | |

26 | 27 |

N=27

T=8+9+17+26+27=

87

56 | 57 |

65 | |

74 | 75 |

N=75

T=56+57+65+74+75=

56

57+

113

65+

178

149+

327

3 | 4 |

12 | |

21 | 22 |

N=22

T=3+4+12+21+22=

62

From above I can see a pattern forming; the T numbers are ending in a 2 or a 7:

252, 87, 327, and 62.

Middle

22

23

N=23

T=4+5+13+22+23=

67

5 | 6 |

14 | |

23 | 24 |

N=24

T=5+6+14+23+24=

72

6 | 7 |

15 | |

24 | 25 |

N=25

T=6+7+15+24+25=

77

### A TABLE SHOWING MY RESULTS…

## N | 20 | 21 | 22 | 23 | 24 | 25 |

T | 52 | 57 | 62 | 67 | 72 | 77 |

+5 +5 +5 +5 +5

I can see that my prediction was right because all the numbers go up in 5’s.

### THE FORMULA…

I have already established from the table that the formula will have 5 in it somewhere, this is because the numbers as shown in the table go up in 5’s.

The start of my formula is going to be like this:

T=5n-?? ~(?? =The 0th term)

Now I have to find the 0th term.

To do this I am going to multiply 20 by 5 giving us 100 then I am going to subtract 52-the 20th term from 100 giving me the 0th term, which is: -48

0th | 20th |

-48 | 52 |

So now, I can complete my formula:

T=5n-48

### EXPERIMENTATION WITH C’S…

Now I am going to translate the C downwards.

1 | 2 |

10 | |

19 | 20 |

N=20

T=1+2+10+19+20=

52

Translating the C downward by one square…

10 | 11 |

19 | |

28 | 29 |

N=29

T=10+11+19+28+29=

97

All the numbers seem to be in relation with the number 9:

~97-52=45

45\5=9

~RED=

10-1=9

~YELLOW=

11-2=9

~GREEN=

19-10=9

~BLUE=

28-19=9

~PINK=

29-20=9

I think that the number 9 keeps appearing is because the grid, which, I have drawn, is a 9x9 grid.

Below is a C like the other one’s but with n terms in it instead of the numbers from the grid.

N-19 | N-18 |

N-10 | |

N-1 | N |

From this I discovered that wherever in the 9x9 grid you go the N terms will always be the same. Though I do not think the C can be rotated etc.

Below I am going to draw a 9x9 grid again but this time I am going to alter the way I draw the C’s. This time I am going to put the C on its side like shown below:

Conclusion

N=3

3*5=15+22

=T=37

###### EXPERIMENTING WITH GRIDS OF DIFFERENT SIZES…

Below I am going to draw a 4x4 grid, in which, like the 9x9 grid I am going to experiment with.

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

Now I am going to take some C’s and experiment with them

1 | 2 |

5 | |

9 | 10 |

N=10

T=1+2+5+9+10=

27

2 | 3 |

6 | |

10 | 11 |

N=11

T=2+3+6+10+11=

32

3 | 4 |

7 | |

11 | 12 |

N=12

T=3+4+7+11+12=

37

5 | 6 |

9 | |

13 | 14 |

N=14

T=5+6+9+13+14=

47

Because the grid is too small for me to see what the T number is going to be when the N number is 13. However, I assume it is going to be 42.

A table with my results…

N | 10 | 11 | 12 | 13 | 14 |

T | 27 | 32 | 37 | 42 | 47 |

The Formula is going to have the number 5 in it, because the T numbers go up in 5’s:

T=5n????

10*5=50

27-50=-23

Therefore the formula is:

T=5n-23

To check this:

N=10

5*10=50-23

=T=27

1 | 2 |

5 | |

9 | 10 |

N=10

T=10+9+1+2+5=

27

5 | 6 |

9 | |

13 | 14 |

N=14

T=5+6+9+13+14=

47

All the numbers, like the 9x9 grid, seem to be associated with the number 4.

~47-27=20

20/5=4

~ 5-1=4

~ 6-2=4

~ 9-5=4

~ 13-9=4

~ 14-10=4

I think that the number 4 keeps appearing is because the grid, which, I have drawn, is a 4x4 grid.

Mehjabeen Iqbal 10NDM Page 5/10/2007

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