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Two in a line

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Introduction

Two in a line

The following are allowed, as being next to each other, they are two in a line.

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But the following are not two in a line, since there are spaces between them..

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Show that there are 42 ways to put 2 counters in a line.

Investigate

We can draw ALL 42 grids and find all the different ways but this takes too long and towards the end gets more difficult as we get closer to the total of 42.

Lets look at it in a methodical way.

There are three types of patterns for two in a line,image00.pngimage00.png

  1. horizontal                2) vertical                3) diagonal             orimage00.pngimage00.pngimage00.png

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Horizontal

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With a grid width of 4,

There are three possible arrangements of two in a line (for one row).

...read more.

Middle

image20.pngimage19.png

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With a grid width of 4,

Diagonal patterns take up two rows and there are three possible arrangements of two in a line

There are three pairs of two rows, i.e., row 1&2, row 2&3 and row 3&4.

So there are 3 x 3 = 9 ways diagonally. (in this direction)

For the diagonal patterns there are two different directions.

  1. The direction in the squares above
  2. The direction in the squares below

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This means that if there are 9 in one direction then there are 2 x 9 = 18 in both directions

Therefore there are

12 (horizontally) + 12 (vertically) + 18 (diagonally) = 42

EXTEND INVESTIGATION

Based on the idea above we can investigate other size grids.

We could start with square grids but it is just as easy to look at rectangular grids.

Horizontal

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This grid is 3 x 5 (3 wide and 5 long)

There are 4 arrangements for each row (1 less than the length)

There are 3 possible rows with the same arrangements (same as the width) so 3 x 4 = 12

So for any rectangle

The number of horizontal arrangements will be equal to, 1 less than length (L-1)  multiplied by the width. I.e.,

                (L – 1) x W = W(L-1)

In the example above W = 3 and L = 5 giving,         3 x (5 – 1) = 12

Vertical arrangements follow a similar structure.

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...read more.

Conclusion

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For length 5

For 2 in a line we can make 4 arrangements                5 – (2 – 1) = 4

For 3 in a line we can make 3 arrangements                5 – (3 – 1) = 3

If L is the length of the rectangle and n the number in a line

We can make L – (n – 1) arrangements

For the total number of diagonal arrangements we need to multiply the number of diagonals for one row by the number of rows available.

For 2 in a line the number of rows available is         3 – (2 – 1) = 2

For 3 in a line the number of rows available is        3 – (3 – 1) = 1

The number of rows available is W – (n – 1)

Therefore, the total number for diagonals in one direction is

(L–(n–1))x(W–(n–1))        = (L–n+1)x(W–n+1)         = LW – Ln + L – nW + n2 – n + W – n + 1

                        = LW – Ln – Wn + L + W + n2 – 2n + 1

This needs to be multiplied by two because of the two different directions for diagonal arrangements. Thus

2 x (LW – Ln – Wn + L + W + n2 – 2n + 1) gives

2LW – 2Ln – 2Wn + 2L + 2W + 2n2 – 4n + 2

The general rule for any size rectangle L x W and with n in a line, we need to add all parts together.

(LW – Wn + W)  + (LW –Ln + L)  + (2LW – 2Ln – 2Wn + 2L + 2W + 2n2 – 4n + 2)

Collect terms.

4LW + 3L + 3W – 3Ln – 3Wn + 2n2 – 4n + 2

...read more.

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