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

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

Forms 5, 6 and 7 are enclosed by a 2×4 rectangle. The surface of 8 squares is smaller than the surface of 9 squares we encountered in the previous chapter, but nevertheless there seem to be more variations, because there are (1) 3 forms instead of 2 and (2) 8 orientations per form instead of 4.

As always we start with 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}
Forms 5, 6 and 7 have 8 orientations: k×{(8-n)×(8-m)-6}=8×{(8-2)×(8-4)-6}=8×{6×4-6}=8×18=144

In reality we find:

• Form 5 has 151 positions, so 7 extra compared to the minimum
• Forms 6 and 7 have 154 positions, 10 extras

Here is an overview of all orientations of forms 5, 6 and 7, grouped by their number of positions. 

6 orientations have the minimum 18 positions
5L1	5L2	5P1	5P2	6P1	6P2
xxx xxx xxx xxx xxx	xxx xxx xxx xxx xxx	xxx xxx xxx xxx xxx	xxx xxx xxx xxx xxx	xxx xxx xxx xxx xxx	xxx xxx xxx xxx xxx

11 orientations have an extra position, adding up to a total of 19 The extra position is shown in the table Notice the black box, which is part of the enclosing rectangle, but not of the board
5L3	5P3	6L1	6L2
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

Orientations 5 - 8 with 19 positions
6L3	7L1	7L2	7L3
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

Orientations 9 - 11 with 19 positions
7P1	7P2	7P3
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

5 orientations have 20 positions
5P4	6L4	6P3	7L4
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
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

Orientation 5 with 20 positions
7P4
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

2 orientations have 21 positions
5L4	6P4
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
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

Let's have a closer look the separate forms.

Form 5 has one blank corner, while the other two have 2 blank corners. That explains (at least partly) why form 5 has only 7 extra positions, while the other have 10.

There are 4 orientations without blank corners on the right and hence without extra's: 5L1, 5L2, 5P1 and 5P2.

Orientations 5L3 and 5P3 have 1 extra position.

Orientation 5P4 has 2 extra positions, where the enclosing rectangle covers either 1 or 2 extra fields in the top right hand corner.

Really special is 5L4: 3 extra positions covering 1, 2 or 3 extra fields in the lower right hand corner. This is only orientation covering 3 extra fields outside the board.

Form 6 has 2 blank corners, located on the same side of the long axis. This leads to big differences between landscape and portrait orientations.

Let's view landscape first: 6L1,  6L2 and 6L3 have an extra position, 6L4 has 2 extra positions in the lower right hand corner.

The portrait orientations 6P1 and 6P2 have no extra positions, because both blank corners are on the left.

6P3 and 6P4 are the only extra orientations with extra positions in both right hand corners; no other orientation of any form has this property. 

- 6P3 has 2 extra positions, 1 in each corners.

- 6P4 has 3 extra positions, 2 at the top and 1 at the bottom.

Form 7 has 2 blank opposing corners, which means that all 8 orientations have at least 1 extra position. Only 7L4 and 7P4 have 2 extra positions.

Summarising we can say that the 3 forms have big differences:

• Form 5 (with 1 blank corner) has extremes: 4 orientations without extra positions, but also the only orientation covering 3 extra fields outside the board.
• Form 7 (with 2 opposing blank corners) is very moderate: all 8 orientations cover 1 or 2 extra field.
• Form 6 (with 2 adjacent blank corners) is somewhere in the middle.

The maximum number of filtered positions per square

(You find the list for form 5 here; a link to the form 6 list is on the page. There you find a link to form 7.)

In the previous chapter we saw that the 3×3 forms are more complicated than 2×3 forms when it comes to minimum and maximum filtered positions. The 2×4 forms are even more complicated.

The theoretical maximum is 8 orientations × 5 covered fields = 40. However, there are no squares on the board that reach this maximum.

If you split the orientations in portrait and landscape, then you find the following:

Only square 9 filters 20 portrait orientations for all 3 forms Here are the extremes
7P4	5P3	5P2	7P3
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

Only squares 11 and 18 filters 20 landscape orientations for all 3 forms Here are the extremes
5L1	7L2	7L4	5L3
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

As you see, all examples are either form 5 or form 7. Form 6 has no opposing non-blank corners and that creates some extra solutions (to be checked by the readers):

• Square 10 filters 20 portrait positions
• Square 4 filters 20 landscape positions

Let's summarise the special squares:

• The 2×3 and 3×3 forms have there maximum in squares 3, 4, 10 and 11
  Runner-up is square 12
• The 2×4 have their portrait maximum in square 9
  Form 6 also has a maximum in square 10
• The 2×4 have their landscape maximum in squares 11 and 18
  Form 6 also has a maximum in square 4

There are differences, but we do see a top region on the board. A closer look per form:

• Form 5
• Square 11 filters 35 positions. We know this square filters all 20 landscape positions, so it filters only 15 portrait positions.
• Squares 4 and 10 filter 32 positions. Notice the decrease compared to their neighbour 11.
• Squares 9 and 18 filter 30 positions. We know they both filter 20 positions in one orientation, so the means just 10 in the other orientation.
• Form 6
• Squares 4, 10 and 11 filter 36 positions. Since the all filter 20 positions in one orientation, they filter 16 in the other orientation.
• Next on the list are squares 3 and 12 with 32 filtered positions. These squares are not among the maximum of either orientation. There seems to be a relative optimum here.
• Other maxima in 1 orientation are lower on the list:
• Square 18 filters 31 positions.
• Square 9 filters 30 positions
• Form 7
• Square 11 filters 38 positions: 20 landscape positions and 18 portrait positions. Let's take a closer look at this champion, which is only missing 2 filtered positions.
  Notice this square is in the middle of the 7×7 board. It filters all 20 landscape positions because it 3 squares away for both the left and the right border. Notice it is also 3 squares away from the top.
  That means the only blocking border is the bottom. But here too things look fairly OK: have a look at the above table with the caption "Orientations 9 - 11 with 19 positions". Here you see that orientation 7P1 and 7P2 can have their top in square 11 and their bottom in square 31. 
  This means that only orientations 7P3 and 7P4 are filtered 4 times instead of 5.
• The runners-up are squares 4 and 10, both filtering 34 positions.

Conclusion: we see a landslide victory for square 11, followed by 4 and 10.

The maximum of filtered positions per pair

(You find the list for form 5 here; a link to the form 6 list is on the page. If are at the form 6 list, there is a link to form 7.)

Since square 11 is so much ahead of the rest, followed by 4 and 10  you may expect that pairs 4/11 and 10/11 are winning pairs. However, this is not the case, because these squares are next to each other and hence have a lot of overlap. On the other hand, pair 4/10 turns out to be a winner

• Form 5
• Pairs 4/9 and 4/10 both filter 60 positions
• Next on the list are 3/11 and 10/18, both filtering 59 positions
• For the record: pair 4/11 filters 51 positions and pair 10/11 filters 52 positions.
• Form 6
• Pairs 4/17 and 5/10 both filter 65 positions.
• Pairs 4/9 and 4/10 both filter 64 positions.
• Notice squares 4 and 10 are both well represented in this top 4 and even pair 4/10 is there. However, co-winner square 11 occurs for the first time in 5 pairs filtering 60 positions.
• For the record: pairs 4/11 and 10/11 filter 56 positions.
• Form 7
• Pair 9/11 filters 64 positions.
• There are 5 pairs filtering 62 positions: Oct/11, 3/11, 4/18, 9/12 and 10/12. Notice square 11 is well represented, and also square 4 and 10 occur once.
• For the record: pair 4/10 filters 60 positions and pairs 4/11 and 10/11 filter 56 positions.

The minimum number of filtered positions per square

(Click here for the list form 6 and scroll up a little for the bottom of the form 5 list. That is easier than scrolling down from top of square 5.
A link on the page leads you to the square 7, where you can scroll up again.

Last there is a link to form 8 where you can scroll up again.)

Here is a summary of the previously found squares filtering the least positions:

• Corner squares Jan, 28 and 29 never filter more positions of a form than the number of orientations. It may be less, because some orientations have an empty corner.
• Also corner square Jun never filters more positions of a form than the number of orientations. However, it may be covered by a position with a blank top right hand corner AND a blank square below that.
• Next in the list are corner squares 7 and 31. Again the number of filtered positions will not exceed the number orientation, but there may some extras because only any blank space will cause an extra filtered position.
• More complicated is the bottom row 29, 30 and 31.
  Already with the 2×3 forms we found, that square 30 can filter only 1 position of landscape orientation with a non-blank lower right hand corner. For 3×3 forms the same is true. (For a standard next-to-corner square this would be 2 positions per orientation.)
  For 2×4 forms we go a step further: landscape orientations without a blank bottom right hand corner simply do not fit on the bottom line. As a result square 30 may filter less positions than all other next-to-corner squares. Also square 29 may do better than Jan and 30, and 31 better than 7.

Let's see what we find per form:

• Form 5 has only 1 landscape orientation that fits the bottom row: 5L4. 3 positions fit, each covering exactly 1 bottom row square. You find these positions in left column of the table with the caption "2 orientations have 21 positions".
• Square 29 filters 4 positions: 3 portrait orientations and 5L4.
• Square 31 filters 5 positions: all 4 portrait orientations and 5L4.
• Squares Jan and 28 filter 6 positions. This is standard corner behaviour: 1 portrait and 1 landscape orientation are not filtered, because the blank corner does not cover the corner square.
• Squares Jun, 7 and 30 filter 7 positions, but in quite different ways
• Square Jun is a corner square filtering 1 extra orientation compared to Jan and 28. This is 5P4, as you can see in the bottom left most image of the table labelled "5 orientations have 20 positions".
  The only orientation not filtered by Jun is 5L3; see the left lower image of the table with the caption "11 orientations have an extra position, adding up to a total of 19".
  
• Square 7 also leaves 1 orientation unfiltered: 5P4; ; see the left lower image of the table with the caption "5 orientations have 20 positions".
• Square 30 is a completely different story. We have seen it filters 1 landscape position of 5L4. In other words: it filters 6 portrait positions, 1 for 5P1 and 5P3, and 2 for 5P2 and 5P4.
• Form 6 has 3 landscape positions that fit the bottom row:.
  - In the table with the caption "11 orientations have an extra position, adding up to a total of 19" you see that 6L1 covering square 31.
  - In the table with the caption "5 orientations have 20 positions" you see that 6L4 is covering both square 30 and square 31.
• Square 29 is the absolute winner: only 2 positions are filtered. We know these are both portrait positions.
• Corner squares Jan and 28 filter 4 positions: 2 landscape and 2 portrait.
• 2 squares filter 5 positions in 2 different ways:
• Square Jun is also a corner, but filters an extra 6P4, which you find in the table "2 orientations have 21 positions", the middle one of the 3 right side images.
• Square 30 filters 1 position of 5 different orientations: 6L4 (as we just saw) and all 4 portrait orientations.
• Square 31 filters 6 position. That is 1 more than 30, because (as we just saw) it filters an extra 6L1 position.
• Square 21 filters 7 positions, while squares 7 and 14 filter 8. It may be surprising that 21 performs slightly better than semi-corner 7, so we will have a closer look at these 3 squares on the right side.
  Squares 7, 14, 21 and 28 form a column of 4 on the right. We have seen that square 28 filters 2 portrait and 2 landscape positions.
  Before we discuss the other squares we have another look at form 6. There is a "backbone" of 4 squares + a 1 square bulge.
• Squares 7, 14 and 21 all filter 1 position of all 4 landscape orientations.
• Orientations 6P1 and 6P2 both have a position with a backbone covering the squares 7, 14, 21 and 28.
• Orientation 6P4 is listed in the table "2 orientations have 21 positions". You see the positions that cover either square 7 or square 14. You can move this orientation 1 square down to create a position covering square 21.
• Orientation 6P3 is listed in the table "5 orientations have 20 positions". You see the position covering square 7. You can move the position 1 square down to create a position covering square 14. BUT there is no position covering square 21.
• CONCLUSION: Squares 7 and 14 filter 8 positions, 1 for each orientation. Square 21 filters 7 positions, because orientation 6P3 is missing.
• Form 7 has 3 landscape positions (of 2 landscape orientations) that fit the bottom row. That sounds the same as form 6 (and also like form 5), but there is a big difference: the form 7 positions may cover 2 or even 3 bottom row squares, whereas the form 5 and 6 positions cover only 1 bottom row square.
  - The table "Orientations 5 - 8 with 19 positions" show 7L2 covering squares 29, 30 and 31.
  - The table "5 orientations have 20 positions" shows 2 positions of 7L4, covering 29 and 31 once, and by 30 twice.
• Squares Jan, 28 and 29, the corner squares, filters 4 positions, 2 portrait and 2 portrait.
• Square Jun filters 5 positions. The extra one is shown in the table with "Orientation 5 with 20 positions", the bottom image.
• Square 31 filters 6 positions. We know about the 2 landscape positions. 4 portrait positions represent all 4 portrait orientations.
• Squares 30 and 22 filter 7 positions.
• Square 30 is like 31, but with an extra landscape position.
• Square 22 is left to the reader

The minimum of filtered positions per pair

(Click here for the list square 6 and scroll up a little for the bottom of the square 5 list. That is easier than scrolling down from top of square 5. 

A link on the page leads you to the square 7, where you can scroll up again.

Last there is a link to form 8 where you can scroll up again.)

Most results are derived easily from the sum of the minimum number of filtered positions per square, but there some overlapping positions.

• Form 5:
• Pairs 29/31 and 29/30  filter 9 positions, but notice the differences:
• Pair 29/31: square 29 filters 4 positions and square 31 filters 5 positions. So there are no overlapping position. That is surprising for 2 squares so close.
• Pair 29/30: square 30 filters 7 positions. So there are 2 overlapping positions.
• 4 pairs filter 10 positions. Again there are differences between number of overlapping positions:
• Pairs Jan/29 and 28/29: These "double corner pairs" have no overlaps; Jan and 28 filter 6 positions.
• Pair 22/29: Square 22 filters 9 positions; that is fairly low. But what really matters are the 4+9-10=3 overlapping positions.
• Overlaps are also important for the last bottom row pair 30/31: there are 7+5-10=2 overlapping positions.
• 6 pairs filter 11 positions. We see the same mix of pairs with adjacent squares vs. pairs with faraway (semi-)corner squares.
• Form 6: Pair 29, filtering only 2 positions, is available in quite a lot of pairs at the bottom of the list
• Pairs Jan/29 and 28/29 filter only 6 positions. The corner squares Jan and 28 filter 4 positions, so no overlaps here.
• 7 filtered positions are found for 2 pairs, in the well known categories, but now there is a strange twist:
• Pair Jun/29 has 2 faraway (semi-)corner squares. Jun filters 5 positions.
• Pair 29/30 contains 2 bottom row adjacent pairs. Square 30 filters 5 positions, so the number of overlapping pairs is 2+5-7=0. That is really remarkable for 2 adjacent squares.
• 8 filtered positions are found for 2 solutions in the well known categories and again with the same strange twist:
• Pair Jan/28 combines the 2 corner squares filtering 4 positions. Note this is the first pair with square 29.
• Pair 29/31 contains 2 bottom row adjacent pairs. Like with pair 29/30 there are no overlaps.
• Form 7:
• 3 pairs filter 8 positions, all combinations of 3 corners squares filtering the minimum of 4 positions: Jan/28, Jan/29 and 28/29.
• 6 pairs filter 9 positions. There are 2 groups
• Semi-corner square Jun filters 5 positions. This yields pairs Jan/Jun, Jun28 and Jun/29.
• 3 pairs contain square 29 and a square nearby: 22/29, 29/30 and 29/31.
  The adjacent squares 22 and 30 filter 7 positions, so there is an overlap of 4+7-9=2 squares with square 29.
  Square 31 filters 6 positions, so the overlap with 29 is just 4+6-9=1.

Next Chapter: Overview of forms, orientations and positions

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