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

Under construction

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

In the previous chapters we have introduced some concept using form 1 as an example.
Form 1 differs from all other forms in serveral 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 rectangle that encloses it. This gives us 3 groups:

Forms enclosed by a 2×3 rectangle: 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.

Let start with 4 orientations:

xx xx xx xx xx

xx xx xx xx xx

xx xx xx xx xx

xx xx xx xx xx

 

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 show the same pattern as form 1; the number of filtered positions has more or less doubled. Don't forget the restriction: form 1 has a maximum of 2×6=12 times, since k=2 and the rectangle covers 6 squares. All other forms cover only 5 squares. So for form 4 the maximum is 4×5=20.

• 4 filtered positions: Jan Jun 7 28 29 31
• 5 filtered positions: 30
• 7 filtered positions: Feb May Jul Dec 14 21 22 27
• 9 filtered positions: Mar Apr 1 8 15 25 26
• 10 filtered positions: 23
• 12 filtered positions: Aug Nov 6 20 24
• 14 filtered positions: 13
• 16 filtered positions: Sep Oct 2 9 16 18 19
• 18 filtered positions: 5 17
• 20 filtered positions: 3 4 10 11 12

• The following corner squares filter a minimum of 4 (=k, number of orientations) positions: Jan Jun 7 28 29 31. With form 1 the exact same squared where filtered twice, since k=2.
• The following centre squares filter a maximum of 20 (=4×5=k×(number of squares cover by this form): 3 4 10 11 12. 

Form 8

Now there is a form with extra positions. k=8, but the number of positions in NOT 8×24=192, but 4×24+4×25=196:

Orientations with 24 positions
xx xx xx xx xx	xx xx xx xx xx	xx xx xx xx xx	xx xx xx xx xx
Orientations with 25 positions
xx xx xx xx xx	xx xx xx xx xx	xx xx xx xx xx	xx xx xx xx xx

Form 8 (and all other forms to follow) have something that forms 1 and 4 have not: blank corners. This gives the situation that orientation fit on a the board, where the enclosing rectangle does not. Notice that blank corners on the left side give no extra positions and blank corners on the right side do.

Here are the extra positions. The rectangle is completed with a black square, which covers a location not on the board: 

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

###

• 6 filtered positions: Jan Jun 28 29
• 8 filtered positions: 7 31
• 11 filtered positions: 30
• 13 filtered positions: Feb May Jul 21 22 27
• 14 filtered positions: 14
• 15 filtered positions: Dec
• 16 filtered positions: Mar Apr 1 8 15 26
• 18 filtered positions: 25
• 25 filtered positions: 6 24
• 26 filtered positions: 23
• 28 filtered positions: Aug 20
• 29 filtered positions: Nov
• 32 filtered positions: 13
• 34 filtered positions: Sep Oct 2 9 16 19
• 35 filtered positions: 18
• 38 filtered positions: 5 17
• 40 filtered positions: 3 4 10 11 12

###

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