THE TRIDAGON ELIMINATION RULE

Some forms of this rule have been known under the names of trivalue-oddagon or Thor's Hammer.

As I can't see anything related to oddagons, neither in the pattern nor in the way the rule can be proven, and "Thor's Hammer" does not sound very serious, I've found this short name, but I'm ready to change it.

For references or some history, see the next post.

Let there be four blocks forming a rectangle in two bands and two stacks:

b11 b12

b21 b22

Let there be three digits (the target digits), say 1 2 3, such that:

- in each of the four blocks, there are three cells in different rows and different columns such that:

--- each of these 4x3 cells contains the three digits,

--- eleven of these cells (the 123-cells) do not contain any other digit,

--- the twelfth cell (the target cell) contains at least one more digit,

- some additional conditions to be found below are satisfied.

Then the 3 target digits can be eliminated from the target cell.

The purpose of this thread is:

- to find the necessary and sufficient additional conditions for the rule to be valid;

- to prove the rule in these conditions;

- to analyse the possibility of missing candidates in the 11 cells;

- to list references to the first mentions of (some versions of) this pattern

- to discuss examples;

...

My first goal here is to find the additional conditions, i.e. all the possible patterns of cells in b22.

I propose here a T&E-ish direct proof, assuming the basic condition that in each block there is only one of the 3 cells per row and per column.

It will be very inelegant, but it will not use any uniqueness argument.

First, notice that in such conditions, it is easy to:

- make permutations of stacks and bands such that the target cell is in block b11

- make permutations of rows in the first band such that the target cell is in row r1

- make permutations of the columns in each of the 2 stacks and then permutation of the rows in the band of b21 and b22, such that we have the following pattern, with the target cell in r1c1 and with the 123-cells of the first 3 blocks in their anti-diagonal.

- Code: Select all
`+-------------------------------+-------------------------------+`

! 123456789 . . ! 123 . . !

! . 123 . ! . 123 . !

! . . 123 ! . . 123 !

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

! 123 . . ! * * * !

! . 123 . ! * * * !

! . . 123 ! * * * !

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

Blocks not concerned by the pattern are not represented.

456789 means that each of these candidates may be in the cell and at least one must be in it.

. means a cell that does not participate in the pattern or its proof

* means a cell for which we are trying to decide whether it can be part of the pattern

At this point, there is nothing we can say about where the 3 cells of interest in b22 are. We only know that, given the conditions at the start, they can form only 6 non-isomorphic patterns of cells in b22.

Notice that, at this point, we have exhausted all the possibilities of applying isomorphisms.

By symmetry of the conditions wrt 1, 2, 3, we can always assume that r1c1 = 1, which gives:

- Code: Select all
`+-------------------------------+-------------------------------+`

! 1 . . ! 23 . . !

! . 23 . ! . 123 . !

! . . 23 ! . . 123 !

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

! 23 . . ! * * * !

! . 123 . ! * * * !

! . . 123 ! * * * !

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

Again, by symmetry of this situation wrt digits 2 and 3, we can always assume that r2c2 = 2, which gives:

- Code: Select all
`+-------------------------------+-------------------------------+`

! 1 . . ! 23 . . !

! . 2 . ! . 13 . !

! . . 3 ! . . 12 !

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

! 23 . . ! * * * !

! . 13 . ! * * * !

! . . 12 ! * * * !

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

At this point, we can have either r4c1 = 2 or r4c1 = 3

First suppose that r4c1 = 2

- Code: Select all
`+-------------------------------+-------------------------------+`

! 1 . . ! 23 . . !

! . 2 . ! . 13 . !

! . . 3 ! . . 12 !

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

! 2 . . ! * * * !

! . 3 . ! * * * !

! . . 1 ! * * * !

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

We can now have either r1c4 = 2 or r1c4 = 3

if r1c4 = 2

- Code: Select all
`+-------------------------------+-------------------------------+`

! 1 . . ! 2 . . !

! . 2 . ! . 3 . !

! . . 3 ! . . 1 !

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

! 2 . . ! 13* 1* 3* !

! . 3 . ! 1* 12* 2* !

! . . 1 ! 3* 2* 23* !

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

At this point, we can check how a contradiction could arise in b22, depending on where the 3 cells with no other candidates are placed.

There are 3 cases:

cells 357 in b22 (anti-diagonal): 3 appears twice in b22

cells 249 in b22: 1 appears twice in b22

cells 168 in b22: 2 appears twice in b22

if r1c4 = 3:

- Code: Select all
`+-------------------------------+-------------------------------+`

! 1 . . ! 3 . . !

! . 2 . ! . 1 . !

! . . 3 ! . . 2 !

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

! 2 . . ! 1* 3* 13* !

! . 3 . ! 12* 2* 1* !

! . . 1 ! 2* 23* 3* !

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

At this point, we can check how a contradiction could arise in b22, depending on where the 3 cells are placed.

There are 3 cases:

cells 357 in b22 (anti-diagonal): 2 appears twice in b22

cells 168 in b22: 1 appears twice in b22

cells 249 in b22: 3 appears twice in b22

Same possibilities as before

At the point where we could have either r4c1 = 2 or r4c1 = 3

suppose r4c1 = 3

- Code: Select all
`+-------------------------------+-------------------------------+`

! 1 . . ! 23 . . !

! . 2 . ! . 13 . !

! . . 3 ! . . 12 !

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

! 3 . . ! * * * !

! . 1 . ! * * * !

! . . 2 ! * * * !

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

We can now have either r1c4 = 2 or r1c4 = 3

if r1c4 = 2:

- Code: Select all
`+-------------------------------+-------------------------------+`

! 1 . . ! 2 . . !

! . 2 . ! . 3 . !

! . . 3 ! . . 1 !

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

! 3 . . ! 1* 12* 2* !

! . 1 . ! 3* 2* 23* !

! . . 2 ! 13* 1* 3* !

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

At this point, we can check how a contradiction could arise in b22, depending on where the 3 cells are placed.

There are 3 cases:

cells 357 in b22 (anti-diagonal): 2 appears twice in b22

cells 168 in b22: 1 appears twice in b22

cells 249 in b22: 3 appears twice in b22

Same possibilities as before

if r1c4 = 3:

- Code: Select all
`+-------------------------------+-------------------------------+`

! 1 . . ! 3 . . !

! . 2 . ! . 1 . !

! . . 3 ! . . 2 !

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

! 3 . . ! 12* 2* 1* !

! . 1 . ! 2* 23* 3* !

! . . 2 ! 1* 3* 13* !

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

At this point, we can check how a contradiction could arise in b22, depending on where the 3 cells are placed.

There are 3 cases:

cells 357 in b22 (anti-diagonal): 1 appears twice in b22

cells 249 in b22: 2 appears twice in b22

cells 168 in b22: 3 appears twice in b22

Same possibilities as before

Conclusion: when the patterns of cells in the first 3 blocks are fixed as at the start (anti-diagonal, with the target cell in r1c1), there are three and only three possible patterns of 123-cells for the fourth block that make the rule valid:

- Code: Select all
`+-------+`

! . . * !

! . * . !

! * . . !

+-------+

+-------+

! . * . !

! * . . !

! . . * !

+-------+

+-------+

! * . . !

! . . * !

! . * . !

+-------+

Notice that, when cell r1c1 is fixed, the first two are non-isomorphic, but the last two are isomorphic, via r2->r3, c2->c3, r5->r6, c5->c6.

This is half of the possible patterns satisfying the conditions at the start.

.

[edit]: for non ambiguity, explicitly repeated the condition on target cell being in r1c1 in the conclusion.

[Edit 2]: take into account 999_spring remark.