Thor's Hammer (TH) is new pattern that has received much discussion in the "hardest patterns" thread in the New Sudoku Player's Forum over the last year, and in a new thread started by
Denis_berthier recently March '22).
The simplist TH pattern is the figure below, which has been called "Thor's Hammer", due to its obviously similar appearance. Each asterix represents the same 3 digits. In the
threads mentioned above, it has been shown that without one or more extra digits, this constitutes a deadly pattern. So there must be one or more additional digits, or guardians. The
asterixes are rising (clockwise) in top left box, and falling (or anticlockwise) in the other 3. Aternatively, they can be rising in 3, and falling in one. To complicate matters,
there are 3 arrangements of asterixes which are rising, and 3 which are falling. So TH patterns can have any one of the rising patterns and any 3 of the falling ones, or
vise versa. When there is only one of the twelve pattern cells with extra digit(s), the situation is similar to a Type 1 Unique Rectangle, allowing the 3 pattern digits to be
removed. When 2 cells have an extra digit, it requires that at least one of the two extra digits must be true. The extra digits are also referred to as guardians
Thor's hammer
. . * | * . . | . . * | | * . . | | . * . | | * . . | | . . * | | . * . |
. * . | . * . | . * . | | . . * | | * . . | | . * . | | * . . | | . . * |
* . . | . . * | * . . | | . * . | | . . * | | . . * | | . * . | | * . . |
------+------
* . . | * . . Rising patterns Falling patterns
. * . | . * .
. . * | . . *
TH type 1 examples
Many puzzles published by Mith in the "hardest patterns" thread have the TH pattern after removal of basics. The following are examples of these
........1......234.....5.6.....7......72.8.96.2956...8.96.52...2.86.7...75.98....
........1.12.34....561.7..........85..8...6.9.65.8.12..89....165.1...9.262.9..85.
..............1.23....45.67.27....8986....37.9.38..2.6.39......28.......7.6.9..32
........1.....2.34....56....17......2.8......65..27..8.2678.1..17..65...8.52.17..
........1.....2.3.....45..6.1457.8..5.28.17..78..241...51..8...42..57..88.7......
........1....12.3....456782....2596....6......561.92...19.6...45.2.94..664...1...
.......12.....13.4..1.2356.....14.53.1.36.42..4.2.51.61.......57...3....8.9.56...
........1.....2.3..45...267....48.....83.5.9..5492..83.895.43..4.328....52..39...
The following puzzle (Regis, by Shye) has 2 consecutive type 1 TH's:
.12.3456.6.52.13.443.65....7..1.........2.........3..8....16.451.45.26.3.5634.21.
THs where one or more of the pattern cells contain only 2 of the 3 digits are also implemented.
12..345.65.32.6....6415..3..12.4365.4.56.13..63..2.......47..6....3..8.9.......7.
.......12.....13.4..1.2356.....14.53.1736.42..4.2.51.61.......57...3..8.8.9.56...
....4...1......234.....5.6.....7......72.8.96.2956...8.96.52...2.86.7...75.98....
67..8.1..8.4..9....5946......5.47..89.75...6..8.6..5.......48......7..93...9...2.
With some puzzles, some basic eliminations need to be executed to expose the TH. In the following example, after basics, a potential TH can be seen in boxes 1278, but with many guardians. However, after the simple ALS move ((2=9)r1c249 - (9=2)r1234c7 => -2 r1c8) and follow-on basics, all but one of the guardians are removed, leaving a type 1 TH.
..3.56....571.9.3..6.37..5.....1..956.....8.27.....4.33.........7....3.1.96.....7
TH type 2 examples
With 2 of the TH cells having an extra digit (guardian), at least one must be true to avoid a deadly pattern. If one is postulated to be false, then the other must be
true, creating a strong link that can be used in AICs. When the 2 extra digits are the same, then that digit can be eliminated from any cell which can see them both.
The first 3 puzzles below are examples.
Same digit
........1.....2.3...4.15.......346...47..8.1.6.31.78...78...1.63...8..744.67.138.
.......12.....13....4.....5.3..647...6837...14.71.8....43.16...68.74.1.37..8.3.6.
...........1..2.34..2.1356..17.86..3.6.2.1.8.2.873.6...7..28...12.36..7.8.61.73..
Different digits
In the next examples, the digits are not equal, so further techniques are needed to achieve eliminations. I've set it up to find simple, discontinuous and continuous AICs, AIC/AHP, and forcing chains.
Again, all the examples are from Mith's hardest puzzles:
........1.....234...5..36.2....784.....49..36..46......125...6.4.31..52.56....1.. Simple chain x 2, stte
...........1.23.45..214.3.6......45..5...46.1.6..51.32.4.......7....2...8.9.36... Continuous loop
........1.....2.3.....4.56......7.....481.2..19..248...89..1..742..78..97.1.9.... Simple chain, btte
........1......234.....5.6.....78.....72.9.86.2856..79.76.52...2.98.....85.79.... Simple chains x2
........1.......2.....13.45...3.56...371..5..6.5.78....785.1...3.1.86...56.73...8 Simple chains x 3, continuous loop, forcing chain
........1.....2.34..2.1356.....24..3...1.6.45.4..5.12.2........7.8.65...96..31... Simple chains x2
.......12..1..34.5.5..4136.....7.6.....68925.56.1.4....15....4332.4....64......2. Simple x 2, continuous, forcing x2
........1....23.45....467.8..126..9..693.4.1.23..19....93..1...12.......6.4.321.. Simple chains x 3
........1.....2.34..2...56...578.....276.9...98..2.....6859.7..25.......7.92.86.. Simple chain
.......12..1..34.5.5..4136.....7.6.....68925.56.1.4....159...4.3..4...2.42......6 Simple chain x 2, continuous loop, forcing chains x2
9876........54..........3..89.2...737.6...92...3.....63........26...7.89.7...2... Simple chain, continuous loop
..3.56..9...1.9..3...37..1.........7..5...94....7.5.2.3.65..1...1........79.3.65. Simple chain
TH type 3 examples
When 3 of the TH cells have an extra digit, eliminations may be possible with chains eminating from the 3 digits.
Same digit
.......12..3.1456.16..524.3...2.513.3.1.78...6..391....3.....5141.5..62.5..1..3.4
Different digits
In the following examples, the 3 extra digits are different. By repeatedly applying the technique, multiple eliminations are achieved. The first actually gives a btte solution in
one step, the second after 6 consecutive type 3's.
........1.....234..35..4.62....7.4.6....89.3...36......125..6..3.41..25.56.....1. Forcing chain x1, btte
........1....23.....4..5.....6...47..78...1.99...7..86.197..6.4.8....7..6..1...98 Forcing chains x5, btte
........1.....2.3.....4.562.....7.....481.2..19..248...89.....742..78..97.1.9.... Forcing chain x1
........1...234.....2..5.....6.....7.2....18.78..9.6...671..9.82.8.....691....72. Forcing chains x5, btte
.......12.....13.4..1..256.....26..3..378.....46.93....3425....51....2..6.2.1..35 Forcing chains x5, then type 2 x2
.......12....1345...1.4.3.6.....75.....58926.56.13.....453...2.1.6....3.23......5 Forcing chains x5
.......12..3.1456.16..524.3...2.513.3.1.78...6..391....3.....5141.5..62.5..1..3.4 Forcing chains x2
1.3..6..9.5.1.9..3...37............7......941...7.5.2.3.65..1..51........79.3.6.. Forcing chains x5
Type 3 TH's with all 3 guardians in same box seem rarer. The following is one where there are 2 possible TH's but no eliminations:
74..589..9.86.75...6594.7...87..5...6.98.....45..9....8.6...4..........3....8.6.1
