JE Pattern Eliminations and Inferences Terms Member digit: A digit that appears as a candidate in the base cells

Base digit: A digit that is true in one of the base cells.

Spot Candidate: Any instance of a of a member digit that would prevent it from being able to occupy two S cells (similar to a fin for a fish pattern).

Target Cells in different boxes Note: The eliminations and inferences detailed here apply to the original JE specification which requires that the base and object cell pairs all should occupy different boxes in the JE band. However cases are possible where the two target cells can also occur in the same box. At the time of writing. (June 2013) these cases haven't been studied to investigate whether the pattern will produce any inferences that aren't subsumed by simpler methods.

Common To All Pattern Forms- Code: Select all
`*-----------*-----------*----------*`

| #ab #ab . | . . . | . . . | base cells = (ab)r1c12

| . . . | #a . . | Q . . | target 1 = (a)r2c4

| . . . | \ . . | Q ax ay | Q = object cells = r23c7

*-----------*-----------*----------* non-member digits = x,y

This pattern is common to all two target box forms of JE.

1) The JE theorem proves that a digit in a target cell must also occur in the base cells which eliminates instances of it in sight of either.

2) The theorem also proves that object cells pairs will contain different base digits

3) A digit that is true in a target cell will also be true in the base cells and one of the two non-object cells in the remaining unoccupied line in the third box.

Eliminations 1. Any digit that has been eliminated from either the base or target cell pair can be eliminated from the other cell pair.

2. A digit found to be true in a target cell can also be eliminated from the other target cell and the cells in sight of the base cells

3. Any member digit that isn't common to one target cell and the non-object cells in the diagonal mini-line in the other target box can be eliminated from these three cells.

Inferences1. A weak inference exists between instances of the same member digit in the object cell pairs* - it can't be true in both

2. For a true base digit, a conjugate inference exists between the two pairs of object cells* - it must be true in just one of them

3. An equivalence inference between a digit's truth in the base and the four object cells* - it can't be true in one and false in the other

4. An equivalence inference between instances of a member digit in a target cell and the non object-cells* in the diagonal mini-line in the other target box.

5. Weak inferences between a member digit in the base or target cells and all the spot candidates for the same digit.

* Usually the inference is confined to the target cells, but this covers cases when a target could be either object cell.

Other inferences are now specific to the different forms the JE pattern can take

The Diagonal JE form - Code: Select all
`*-----------*-----------*-----------*`

| #ab #ab . | . . . | . . . | base cells = (ab)r1c12

| . . . | #a bx bx | \ . . | target 1 = (a)r2c4

| . . . | \ . . | #b ay ay | target 2 = (b)r3c7

*-----------*----------*------------* non-base digits = x,y

Here the target cells are on diagonal mini-rows within JE band.

1) From the previous proof, each mini-line containing a target will also contain the true base digit that will occupy the other target.

2) The diagonal target mini-lines can only contain one non-member digit at most.

Inferences 1. If one non-member digit is locked in a target mini-line, all other non-member digits can be eliminated from that mini-line

The Collinear JE form- Code: Select all
`*-----------*-----------*------------*`

| #ab #ab . | . . . | . . . | base cells = (ab)r1c12

| . . . | #a . . | #b . . | target 1 = (a)r2c4

| . . . | \ bx bx | \ ay ay | target 2 = (b)r2c7

*-----------*-----------*------------* non-base digits = x,y

Now the target cells are on the same line.

1) The target and companion cell mini-lines will each contain one true base digit.

Inferences1. If two non-member digits are locked in a companion mini-line, all other non-member digits can be eliminated from that mini-line

The Twin JE Form- Code: Select all
`*-----------*-----------*--------------* base cells = (ab)r1c12`

| #ab #ab . | . . . | . . . | target 1 = (a)r2c4

| . . . | #a . . | #bz-x . . | target/companion 2 = (b)r23c7

| . . . | \ . . | #bz-y ax ay | locked non-base digit = (z)r23c7

*-----------*-----------*--------------* non-base digits = x,y

The twin JE form departs from the others in that a pair of object cells contains a locked non-member digit, and both object cells may contain a mix of member and non-member digits.

1) One cell must eventually hold the locked non-member digit leaving just one capable of holding a member digit, and so complies with the JE pattern requirements.

Eliminations 1. For a twinned object cell pair all non-member digits apart from the one that is locked can be eliminated from the object cells.

Note 1. The object cell pair can be seen as an Almost Hidden Pair consisting of the locked non-base digit and one base digit.

Note 2 There are other AHS sub-patterns that will satisfy the "one member, one non-member" JE pattern requirement, but they are comparatively rare and difficult to spot, and aren't covered here.

Double JEs with 3 Common Cross-Lines Double JEs occur when two sets of base cells with corresponding target cells in the same band of boxes. All instances seen have the use the same three cross-lines.

- Code: Select all
`*-------*-------*-------* *-------*-------*-------* B = base cell 1st JE `

| B B \ | . t . | . . . | | B B \ | . \ . | . . . | T = target cells 1st JE

| . . t | . \ . | T . . | | . . t | . # . | T . . | b = base cells 2nd JE

| . . . | . T . | \ b b | | . . . | . \ . | \ b b | t = target cells 2nd JE

*-------*-------*-------* *-------*-------*-------* # = common target cell

4 Target Cells 3 Target Cells

Two JEs can co-reside in the same band, and sharing the same three cross lines as these diagrams illustrate. On the left there are 4 target cells but on the right one target cell is common to both patterns and there are only 3. In the two base boxes the target and companion cells may be in either order so any mix of diagonal and collinear forms is possible. Twin forms are also possible provided that any shared target cell is not involved.

1) As each component JE can taken independently, the target cells must hold true base digits in their respective base cell pairs.

2) The target cells in the same boxes as one base set must hold a true base digit in the other base set.

3) The second true member in each base set must therefore be true in a target cell in the third box.

4) If there are two target cells in the third box they must hold different true digits in their respective base cells, otherwise a single target cell must hold a digit that is true in both pairs of base cells.

5) Each target cell must therefore hold a different digit, and together these cells will hold the same combination of digits as the full set of base cells.

Eliminations 1.Any member digit in sight of both base pairs

2 Any member digit in sight of all the target cells

Note 1: As soon it can be shown that a member digit must be true in one or other base set, the spot candidates for that digit can be eliminated. This usually happens as soon as a double JE is found.

Double JEs With 4 or more Cross lines ??? I'm not sure that we have any examples.

Spot Cell EliminationsTo check how many instances of a digit can occur in the S cells the minimum number of rows and/or columns that are necessary to cover all the S cells where it occurs as a candidate. To comply with the pattern requirements this must be no more than two.

External spot cells are those cells in the covering lines that are not S cells, and when one of the covering houses is a cross-line, these will lie in the JE band itself.

Internal spot cells are any S-cell at the intersection of two covering lines.

- Code: Select all
`B B . | . x . | . . . B = Base cells `

. . . | . # . | R . . R = Object cells

. . . | . # . | R . . # = Spot object cells

------+-------+------

. . / | . O . | / . .

x x O | x X x | O x x <

. . / | . O . | / . . O = Occupied (maximum)

------+-------+------ / = Excluded S-cell

. . / | . O . | / . . x = External spot cells

. . / | . O . | / . . X = Internal spot cell

. . / | . O . | / . .

Example with covering lines row 5 & column 5.

1) The two target cells within each pair of object cells are the only cells available in the cross lines capable of holding a true base digit in the JE band.

2) When both true base digits are confined to two instances in the S cells, within the 3 cross lines, each digit must occur once in the JE band and twice in the other bands, in the S cells.

3) A member digit confined to one instance in the S cells will therefore invalidate the pattern if it's true in the base cells or reduce the set of members if it's false.

4) If a spot cell for a digit were true, the S cells would be reduced to holding one instance and the JE band would have to hold two, at least one of which would be in sight of the base cells so it could not also be true in the base cells.

5) When a spot cells for a digit is also an object cell, then if it were true it would create a contradiction as it would also have to be a true base digit but there would be insufficient cells in the cross lines to hold the its required 3 instances.

6) If two covering houses intersect, their intersection cell (r5c5 above) is also an internal spot cell as if it were true no other S cell could also be true.

Eliminations 1, Any member digit that has a spot cell covering a target cell can be eliminated from that target cell

Inferences (repeated)

1. Weak inferences between a member digit in the base or target cells and all the spot candidates for the same digit.

2. Any member digit that is restricted to one instance in the S cells won't invalidate the pattern provided it can be shown to be false in the base cells by other means. Until this can be proved the pattern must be reduced to an 'Almost' status.

Note: With the targets in different boxes, it is only necessary to count the minimum number of rows and columns needed to cover the instances of a digit in the S cells. Should the targets lie in the same box, this check would need to be extended to include box covers as well. It may then be possible to select two covering houses in more than one way when more than one internal spot cell may be identifiable.[/quote]

Edit 15th June 2013: Spot Cell Eliminations section: Theory point 3) and corresponding changes under Eliminations & Inferences