Forms enclosed by a 2×3 rectangle: 4 and 8

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Previous chapter: Explaining filtered positions using form 1, the 2×3 rectangle

In the previous chapter we have introduced some concepts using form 1 as an example.
Form 1 differs from all other forms in several ways:

1. It covers 6 squares, while the others cover only 5;
2. It is a rectangle, which is easy to manage;
3. It has 2 orientations while forms 2 to 4 have 4 orientations, and forms 5 to 8 have 8 orientations.

About point 2: for forms 2 to 8 we still use rectangles, but now the smallest enclosing rectangle. This gives us 3 groups:

Forms enclosed by a 2×3 rectangle: 1, 4 and 8

Forms enclosed by a 3×3 rectangle: 2 and 3

Forms enclosed by a 2×4 rectangle: 5, 6 and 7

For rectangles we found the following formula:
The total number of positions for an n×m rectangle with k orientations is k×{(8-n)×(8-m)-6}

For other forms we can state:

The minimum number of positions for a form with k orientations, that can be enclosed by a n×m rectangle, is k×{(8-n)×(8-m)-6}

We will explain later where the extra positions come from.

 

Now we continue with the forms fitting in a 2×3 rectangle. The formula gives k×24. 

Form 4

With respect to the formula, we ask a little patience from our readers: form 4 has no extras, so minimum and total number of positions are the same. There are 4 orientations:

4L1	4L2	4P1	4P2
xxx xxx xxx xxx xxx	xxx xxx xxx xxx xxx	xxx xxx xxx xxx xxx	xxx xxx xxx xxx xxx

Each of the orientations has 24 solutions, giving a total of 4×24=96 solutions. This is simply form 1 doubled.

Also the filtered positions per square (click here for the list) show the same pattern as form 1. 

• The same centre squares have the most filtered positions: 3 4 10 11 12
  This matches the number of orientations times the number of squares per form: 2×6=12 for form 1 and 4×5=20 for form 4.
• The same 6 corner squares have the least filtered positions: Jan Jun 7 28 29 31
  This matches the number of orientations: 2 for form 1 and 4 for form 4.

It comes as no surprise, that filtered positions per pair (click here for the list) also have similarities with form 1. 

• Pair 3/12 filters the most positions: 38. Both squares filter 20, so there are 2 overlapping: 4L1 and 4L2 with square 3 in the top left hand and square 12 in the lower right hand corner.
• The 3 pairs containing squares 29, 30 and 31 filter the least positions: 6.

Form 8

Back to the formula: The minimum number of positions for a form with k orientations, that can be enclosed by a n×m rectangle, is k×{(8-n)×(8-m)-6}
For forms 1 and 4 this minimum is the actual value, but that is not the case for the other forms. The reason is that form 8 and all other forms to follow, have something that forms 1 and 4 have not: blank corners. This gives the situation that an orientation fits on a the board, where the enclosing rectangle does not.
Form 8 has k=8 orientations but the number of positions in NOT 8×24=192, but 4×24+4×25=196.

Notice the following in the below tables:

• Blank corners on the left side yield no extra positions and blank corners on the right side do.
• The extra position are also shown in the below table. The rectangle is completed with a black square: 

Orientations with 24 positions
8L1	8P1	8L2	8P2
Orientations with 24 positions
xxx xxx xxx xxx xxx	xxx xxx xxx xxx xxx	xxx xxx xxx xxx xxx	xxx xxx xxx xxx xxx

Orientations with 25 positions Also shown: positions where enclosing rectangles do NOT fit
8L3	8P3	8L4	8P4
Orientations with 25 positions
xxx xxx xxx xxx xxx	xxx xxx xxx xxx xxx	xxx xxx xxx xxx xxx	xxx 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

Form 8 shows similarities with forms 1 and 4 when it comes to filtered positions, but the blank corners create differences as well.

First the filtered positions per square (click here for the list): 

• The same centre squares have the most filtered positions: 3 4 10 11 12
  This time 8×5=40 positions are filtered.
• There are only 4 corner squares have the least filtered positions: Jan Jun 28 29
  The number filtered positions is not 8 (the number of orientations) but only 6. This too is due to blank corners in the form. Take for example January: all 8 orientations can be placed in the top left hand corner, but 8L1 and 8P1 do not cover the Jan square.
• The corner squares 7 and 31 filter 8 positions, so 2 extra positions each, which you find in the above table: 
• The extra positions of 8L3 and 8P3 cover square 7.
• The extra positions of 8L4 and 8P4 cover square 31.

Next the filtered positions per pair (click here for the list):

• Again pair 3/12 filters the most positions: 78. Both squares filter 40, so there are 2 overlapping: 8L2 and 8L3, placed with square 3 in the top left hand and square 12 in the lower right hand corner. Notice that due to blank corners 8L1 and 8L4 cannot be placed this way.
• There are 8 pairs filtering 12 positions. These are all 4×3÷2=6 combinations of the 4 corner squares. The other pairs are 29/30 and 29/31.
• Notice 30/31 is not mentioned this time, but it appears next with 13 filtered positions.

Next chapter: Forms enclosed by a 3×3 rectangle: 2 and 3
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