Explaining forms, orientations and positions using form 1, the 2×3 rectangle

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All forms have a sequence number from 1 to 8.  

The numbering is functional: a lower number refers to a form with less positions on the board. 
Number 1 is the 2×3 square, which has 2 orientations. We name these orientations after the standard paper orientations, landscape and portrait. In order to give orientations a unique id, we take the number of form, the first letter of the orientation and a sequence number (not so useful for this form, but that's the price for standards).

1L1	1P1
xxx xxx xxx xxx xx xx	xxx xxx xxx xxx xxx xxx

 

Per orientation we will now investigate the number of positions. Starting point is the fact that the board is a rectangle of 7x7 squares. 

All positions in the left column
1L1	1P1
Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx	Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx
Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx	Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx
Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx	Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx
Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx	Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx
Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx	Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx
Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx

All positions start on the third row
1L1	1P1
Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx	Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx
Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx	Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx
Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx	Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx
Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx	Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx
Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx	Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx
	Jan Feb Mar Apr May Jun xxx Jul Aug Sep Oct Nov Dec xxx 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 xxx xxx xxx xxx

Conclusion:

• In the left column 1L1 has 6 positions, and 1P1 only 5.
• In the third row 1L1 has 5 positions and 1P1 6

Combining horizontal and vertical we get (for both portrait and landscape) 5×6=30 positions for a complete 7×7 board. But we do not have a 7×7 board; there are 6 squares missing. Each of these squares blocks a position, so in total there are 30-6=24 positions per orientation. Since form 1 has 2 orientations, it has a total of 2×24=48 positions.

Let's repeat in more general terms:

An n×m rectangle with k orientations is placed in an p×p board, where q fields are missing at the side. The total number of positions is k×{(p+1-n)×(p+1-m)-q}

 

This is a bit too general, because for A-Puzzle-A-Day 2 values are fixed: p=7 and q=6. So:

The total number of positions for an n×m rectangle with k orientations is k×{(8-n)×(8-m)-6}

Sneak preview: since all the other forms are not rectangles, we will use this in a slightly adapted form in the next chapters. 

Next Chapter: Explaining filtered positions using form 1, the 2×3 rectangle

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