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

Forms 2 and 3 are enclosed by a 3×3 rectangle. Within this rectangle 5 squares are covered by the form, giving the 4 other squares plenty of opportunities for blank corners. 

Both forms have 4 orientations; that means k=4 in the formula for the minimum number of orientations: 
k×{(8–n)×(8–m)–6}=4×{(8–3)×(8–3)–6}=4×{5×5–6}=4×19=76
In reality we find:
  • Form 2 has 80 positions
  • Form 3 has 82 positions

Here is an overview of all orientations of forms 2 and 3, grouped by their number of positions. We show the positions where the 3×3 rectangle overlaps square outside the board.
Notice the following:
  • The orientation ID does not contains an L of P, but a dash (-): since height and width are the same there is no portrait of landscape.
  • The 3×3 rectangles around the positions are shown with black square outside the board as we did with form 8. New are the shaded boxes with caption on the board.
Form 2 has 2 orientations with the minimum 19 positions.
Notice all blank corners are on the left side.
2-12-2
xxx xxx xxx
xxx
xxx
xxx
xxx
xxx xxx xxx

Form 3 has 2 orientations with the 20 positions.
Notice there is 1 blank corner are on the right side.
3-13-2
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

Form 2 & 3 both have 2 orientations with the 21 positions.
Notice there is 2 blank corners are on the right side.
2-32-43-33-4
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

As always we have a look at the filtered positions and as always we compare this with the forms already handled. In this case we are looking if there are fundamental differences between forms enclosed by a 2×3 rectangle and those enclosed by a 3×3 rectangle.

The maximum number of filtered positions per square

(You find the list of form 2 here; a link to the form 3 list is on the page.)
The maximum number of filtered positions per square is 4 orientations × 5 covered fields = 20. This is reached for squares 3, 4, 10 and 11. The 2×3 forms also had square 12 in their list of maximum values, but a 3×3 rectangle with the lower left hand corner on square 12, has its upper right hand corner just outside the board. Hence it cannot contain 2-2, 2-3, 3-1 and 3-4. However, with 18 filtered positions square 12 is still the best-of-the rest.

The maximum of filtered positions per pair

(You find the list of form 2 here; a link to the form 3 list is on the page.)
In 2×3 forms, the pair 3/12 was the top of the list. This changes now square 12 is no longer in the top group. There is a difference between forms 2 and 3:
  • Form 2 filters 38 positions using the squares 3/11 and 4/10. This means an overlap of 2×20-38=2 filtered positions.
    Next are squares 3/12 and 4/12  with 36 filtered positions. Notice that square 12 is back!
    All other 4 combinations of 2 top squares (3, 4, 10 and 11) filter only 32 positions, an overlap of  2×20-32=8 filtered positions caused by the fact that they are all neighbouring squares.
  • Form 3 has a top 4 of pairs filtering 36 position: 3/11, 3/12, 4/10 and 10/12. Notice again square 12 appears.
    Again the other 4 combinations of 2 top squares filter only 32 positions, an overlap of  2×20-32=8 filtered positions.
We are not going into all differences between forms 2 and 3, but the difference in overlapping positions needs some attention for the top 2: 3/11 and 4/10. We will show 3/11; 4/10 shows a comparable pattern:
Form 2 covers squares 3 and 11 in 2 ways
2-12-4
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 3 covers squares 3 and 11 in 4 ways
3-33-33-23-2
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 minimum number of filtered positions per square

(Click here for the list of form 3 and scroll up a little for the bottom of the form 2 list. That is easier than scrolling down from top of form 2. A link on the page leads you to the form 4, where you can scroll up again.)
We have seen that we should search in the corners for the minima. But how do we define a corner?
For forms 1 and 4 we saw 6 corner squares: Jan, Jun, 7, 28, 29 and 31.
In form 8 we saw squares 7 and 31 (partly) lost corner status.
Now square Jun is in danger of losing its corner status
See the difference between form 2 and 3		Take a look at square 30
2-43-32-1
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 2 has a minimum of 3 filtered positions for the squares Jan, Jun, 28, 29 and 30.
    The number 3 is no surprise for corner positions: there are 4 orientations and each have only 1 blank corner, so 4-1=3 positions are filtered. This is true for Jan, 28, 29; for the other ones it is more complicated.
    • Square Jun is not covered by 2-4 as you can see in the above left most picture. It is "saved by the bell" because of the 2×2 blank corner of form 2.
    • Square 30 is a special case, because it is not a corner. We have seen this square before in the low range of filtered positions. The reason is that the bottom squares 29, 30 and 31 form a very narrow range where not many orientations fit. 2-2 and 2-4, the orientations with a solid bottom line, can be placed on these 3 bottom squares in one way.
      Also 2-1 can be placed in one way; see the right most image above. Notice that it covers 31, but not 29 and 30.
      Finally 2-3 can be placed at its lowest position in 3 ways, each covering exactly 1 bottom square.

    • Because of 2-1 (as shown in the above right most image), square 31 filters 1 extra position, making the total 4.
    • Also square 7 filters 4 positions.
    • There are 6 squares filtering 5 positions. They are situated next to corners and are part of the 2×2 square of the blank corners. The list:
      • Next to Jan: Feb and Jul
      • Next to Jun: Dec but NOT May; see left image above
      • Next to 28: 21 and 27
      • Next to 29: 22 (and as we saw NOT 30)
  • Form 3 has 2 blank corners. 
    • Hence the minimum number of filtered positions is 4-2=2. This number is only achieved by 3 "real" corner squares: Jan, 28 and 29.
    • Runner-up is square Jun with 3 filtered positions.
    • Next is a list of 8 with 4 filtered positions. We recognise some "almost corners", but also some fields next to corners: Feb Jul 7 21 22 27 30 31

The minimum of filtered positions per pair

(Click here for the list of form 3 and scroll up a little for the bottom of the form 2 list. That is easier than scrolling down from top of form 2. A link on the page leads you to the form 4, where you can scroll up again.)
  • Form 2 has its minimum on the bottom row. Also the next ones are found in that region. Going up the list we see other corners and fields next to it.
    • Pair 29/30 filters only 4 positions
      • 0 × 2-1 as we saw
      • 1 × 2-2 which covers the complete bottom row
      • 1 × 2-4 which covers the complete bottom row
      • 2 × 2-3 positions, 1 with its "foot" in square 29 and 1 with its "foot" in square 30.
    • 5 positions cover the pairs 29/31 and 30/31. Its is quite similar to the previous one, except for an extra 2-1 with its foot in square 31, which we showed before.
    • There are 15 pairs filtering 6 positions:
      • The 5 squares filtering 3 positions can be combined in 5×4×÷2=10 pairs. We already found pair 29/30, which leaves us which 9 pairs of squares that are far apart. In other words: there are no overlapping filtered positions and hence the sum of filtered positions is 2×3=6.
      • The other 6 pairs are corner squares (filtering 3) and 6 almost corner squares. The latter are mentioned above; they filter 5 positions, 2 of which overlap with the corner square: Jan/Feb Jan/Jul Jun/Dec 21/28 27/28 22/29
  • We saw that form 3 has 3 corner squares filtering 2 positions.
    • The minimum numbers of filtered positions per pair is 4. This is obtained by the 3×2÷2=3 combinations of the 3 corners: Jan/28 Jan/29 28/29
    • There are 9 pairs filtering 5 positions. There are 2 categories
      • We saw that square Jun filters 3 positions. Hence the combination with any corner square results in 5 positions: Jan/Jun Jun/28 Jun/29
      • We found 8 "almost corners" filtering 4 positions. 6 of these can be combined with a corner square giving a pair filtering 2+4-1=5 positions. In other words: there is 1 overlapping position for: Jan/Feb Jan/Jul 21/28 27/28 22/29 29/30

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