For the horizontal and vertical combinations this could be the combinations this could be the formula:
(N-1) x N + (N-1) x N = All possible horizontal and vertical combinations.
Does it work? Testing the formula…
After working out the total number of arrangements for two more grids the 9x9 and 10x10, I put the formula to work by checking it against my diagram results.
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
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. This means you need one less line between them to join them all up (N-1). This means there are 8 combinations in one line across. Next you have to multiply the total combinations in one line across by the amount of lines across that have to give the total amount of combinations running across (horizontally) (N-1). So far in the formula we have: (N-1) x (N-1). This formula also works for the vertical combinations. That is why we repeat the first part of the equation. This causes the equation to then look like this: (N-1) x (N-1) + (N-1) x (N-1).
The next diagonal section of the formula (highlighted in red on the equation) works the same with the diagonal right as the diagonal left. It works in a similar way to the vertical/horizontal equation in that the number of combinations left and right need one less line to join them up this gives us (N-1) we then times this by the total number of rows N or 9 in this case. So far in the diagonal we have (N-1) x N. We then repeat to give us the other direction (left or right). Now the diagonal section of the equation is complete (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
- Combinations on a 4x4 grid = 42
- There are 30 more combinations on a 5x5 grid than on a 4x4 grid
- Other grids see page containing diagram on first page
- 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.
