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

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

In this chapter we will explain why filtered positions are interesting.
Filtered positions came up when creating the program to find all solutions per pair. This program first finds all positions for the 8 forms, filtering out the positions that will cover one of (and sometimes both) the squares of the pair.
We are especially interested in squares and pairs with either the minimum or the maximum number of filtered positions.

Elsewhere on this site you find 2 lists of filtered positions: the list of filtered positions per square and the list of filtered positions per pair. Comparing the lists leads to some nice insights.

How to read the lists: All 8 forms are on the webpage, but at the top there is sum: all filtered positions of all forms. The above links point to the start of form 1. That is nice for the maximum number, since the list is sorted from maximum to minimum. To find the minimum, you need to get to the bottom of the list; there are 2 ways:

• Scroll all the way down
• (Preferred way) At the top op the list for form 1, you find a link to form 2. Click that link and then scroll slightly up.

The minimum number of filtered positions per square

The corner squares Jan, Jun, 7, 28, 29 and 31  filter out only 2 positions, 1 portrait and 1 landscape position.

2 positions filtered by square Jan
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

The maximum number of filtered positions per square

On the other side of the spectrum there are squares that may be covered by each of the 6 parts of form 1 in all orientations. As there are 2 orientations and the form covers 6 squares, there are 2×6=12 possibilities for the 5 squares meeting this condition: 3 4 10 11 12.
Here are 3's 12 filtered positions:

12 positions filtered by square 3
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	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

The maximum number of filtered positions per pair

There are 5 squares with 12 filtered positions, but there is no pair with 24 filtered positions. The reason is that all these squares are in the middle of the board with an overlap with respect to the positions they filter.

In the list of filtered positions per pair we see that the pair 3/12 filters 23 positions, meaning the 2 squares have only 1 overlapping position. You find this position in the bottom row of the above table. Notice the portrait position on the left side covers both squares 3 and 12.

The minimum number of filtered positions per pair

We found that 6 corners squares filtering only 2 positions. There are 6×5÷2=15 pairs, which are combinations of 2 corners squares. One would expect these pairs (covering 2×2=4 squares) at the bottom of the list.

But there is something special going on in the bottom row. The corner squares 29 and 31 filter the exact same position of 1L1 (covering squares 22, 23, 24, 29, 30 and 31). As a result pair 29/31 filters only 3 positions.

And there is more: these 3 positions are also the only ones covering square 30 and hence also by pairs 29/30 and 30/31 filter only 3 positions. So there are 3 pairs filtering 3 positions.

Moving up in the list we find 22 pairs filtering 4 pairs. As we mentioned before, 14 of this are combinations of corner pairs.
The other 8 are combinations of a corner pair and an adjacent square. An example is Jan/Feb. The square Feb filters 4 positions, including the 2 filtered by Jan. Hence the pair filters 4 positions.

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