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How many arrangements for 2 counters in a line, in grids of various sizes ranging from 2x2 up to 8x8? (Basic data and information). I will also be finding out how many combinations on an NxN grid.

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Introduction

Some Questions

  • How many arrangements for 2 counters in a line, in grids of various sizes ranging from 2x2 up to 8x8? (Basic data and information).
  • I will also be finding out how many combinations on an NxN grid.

The Plan

First I am going to find out how many arrangements there are in different grid sizes.

(2x2, 3x3, 4x4, 5x5, 6x6, 7x7, 8x8). I will then put all these results into a table. I will then be able to analyse the data and look for patterns within the numbers. This will allow me to start calculating what a possible formula might be.

image00.png

Possible Formula

For the diagonal combinations this could be the formula

(N-1) x (N-1) + (N-1) x (N-1) = All possible diagonal combinations.

...read more.

Middle

9x9 diagonal (9-1) x (9-1) = 64 + (9-1) x (9-1) = 128

9x9 horizontal and vertical (9-1) x9 = 72 + (9-1) x9 = 144

128 + 144 = 272

Total combinations for 9x9 grid according to formula = 272

10x10 diagonal (10-1) x (10-1) = 81 + (10-1) x (10-1) = 162

10x10 horizontal and vertical (10-1) x10 = 90 + (10-1) x10 = 180

162 + 180 = 342

Total combinations for 10x10 grid according to formula = 342

This table is proof that the formula works

image01.png

Why does it work?

The formula:

(N-1) x (N-1) + (N-1) x (N-1) +(N-1) x N + (N-1) x N = Total Combinations is the vertical/horizontal equation (highlighted in blue) and the diagonal equation (high lighted in red) put together. This works because on the horizontal and vertical combinations that you have on a 9x9 grid you have 9 dots across.

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Conclusion

(N-1) x N + (N-1) x N. You then add all the results together to give a total number of combinations on your grid.

Some Answers

  1. Combinations on a 4x4 grid = 42
  2. There are 30 more combinations on a 5x5 grid than on a 4x4 grid
  3. Other grids see page containing diagram on first page
  4. Arrangements on a NxN grid =

(N-1) x (N-1) + (N-1) x (N-1) + (N-1) + (N-1) x N + (N-1) x N = arrangements

Conclusion

I conclude that my investigation was successful. I think this because the formula I have created is more effective than and just as accurate as the diagrams. This is because drawing lots of diagrams is longwinded and time consuming and working out this short formula is much quicker and easier.

...read more.

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