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

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In this chapter we will explain what filtered positions are and why they 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 the squares of the pair.

As we have only covered form 1 so far, the below examples only handle this form. This gives an indication on what to expect in other forms, but there may be differences.

We start at the corner squares: Jan Jun 7 28 29 31. These all filter out 2 positions, 1 portrait and 1 landscape position. Here are the 2 position for square Jan:

Landscape	Portrait
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

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. There are some squares in the middle that indeed filter out 12 positions: Here are the 12 positions that are filtered out by field 3:

Landscape

Portrait

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

Here is the complete overview:

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

Let's see how this works out in 2 pairs: January 28 and "double number" 3/12:
January 28 has 2 corners squares, which both filter 2 positions. Hence in total 4 of the 48 positions are filtered.
3/12 has 2 centre squares, which both filter 12 positions. However, there is one position, which is filtered by both squares, the landscape position covering 3, 4, 5, 10, 11 and 12. So in total 2×12-1=23 positions are filtered, almost half of the total of 48.

Why do we investigate filtered positions here? The background is the enormous differences we find in solutions per pair; see the page Statistics reverse sorted by number of solutions.
At this moment, with 42% of the pairs investigated, the number of solutions per pair ranges from 7 to 282, with an average of 68.

The number of filtered positions (of all 8 forms) ranges from 58 to 387 with an average of 208. 

We are investigating if there is a (linear) relation between the number of solutions and the number of filtered positions per pair. The idea is, that if there are less filtered positions (or rather: more unfiltered position) there are more solutions, simply because there are possibilities to match.

Let's start with a big spoiler: the effect is not (yet) found. Notice the "(yet)", which is the motivation to keep searching on. There are 2 things on the roadmap for this investigation:

1. Calculate all 903 pairs and see if that gives better results.
   Frankly my hopes are not high but you never know. Besides, I want to finish the investigation I started.
2. Summarise the pairs to square level (each pair adding to 2 squares) and see if anything comes out.

Let's have a look at some pairs:

• So far October 6 has the least solutions: 7. The number of filtered positions is 280, which is well above average, but not extreme.
• The date (but not the pair) with the most solutions in January 25: 216 solutions. There are 134 filtered positions, which is well below average, but again, not extreme.
• January 28, mentioned above, has 62 filtered positions, almost the minimum. The number of solutions is 145, above average, but less than January 25.
• "Double number" 3/12 has 387 filtered positions, the maximum so far. The number of solutions is 37, which is low but much more than October 6.

Conclusion: there may be some relation between number of filtered positions and number op solutions, but it is not so clear. 

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