may 8 20225 12x12 difficult

Cages start from 0, from top-down left-right, ordered by the cell containing the total & operator text.

Max difficulty: 4.2; total difficulty: 132.4

0: Filled in initial combinations & candidates
1: (0) Naked single: 8 in r2c11
2: (0) Naked single: 12 in r3c12
3: (0) Naked single: 2 in r5c7
4: (0) Naked single: 3 in r5c11
5: (0) Naked single: 7 in r7c9
6: (0) Naked single: 4 in r7c11
7: (0) Naked single: 7 in r8c11
8: (0) Naked single: 12 in r7c10
9: (0) Naked single: 5 in r8c4
10: (0) Naked single: 11 in r8c5
11: (0) Naked single: 3 in r9c2
12: (0) Naked single: 4 in r9c4
13: (0) Naked single: 6 in r10c4
14: (0) Naked single: 11 in r12c1
15: (0) Naked single: 3 in r12c8
16: (0) Naked single: 2 in r12c10
17: (0) Naked single: 9 in r12c12
18: (1.5) Pointing candidates: all combinations for cage 1 contain 8r1,12r1; eliminations in r1c7, r1c8, r1c12, r1c11
19: (1.5) Pointing candidates: all combinations for cage 2 contain 2r1,7r1; eliminations in r1c1, r1c2, r1c11, r1c12, r1c7, r1c8
20: (0) Naked single: 3 in r1c1
21: (0) Naked single: 4 in r6c1
22: (0) Naked single: 9 in r4c1
23: (0) Naked single: 4 in r4c2
24: (0) Naked single: 1 in r7c1
25: (0) Naked single: 5 in r1c2
26: (0) Naked single: 6 in r7c2
27: (0) Naked single: 1 in r8c2
28: (1) Hidden single: 6 in r5c1
29: (1.5) Claiming candidates: cage 55 must contain 5 in r10c1, r11c1: from 5 combinations to 4
30: (1.5) Claiming candidates: cage 55 must contain 7 in r10c1, r11c1: from 4 combinations to 2
31: (0) Naked single: 2 in r10c2
32: (1.5) Claiming candidates: cage 8 must contain 9 in r2c2, r3c2: from 3 combinations to 2
33: (0) Naked single: 4 in r3c3
34: (1.5) Claiming candidates: cage 60 must contain 10 in r11c2, r12c2: from 12 combinations to 8
35: (1.5) Claiming candidates: cage 60 must contain 12 in r11c2, r12c2: from 8 combinations to 4
36: (1.5) Pointing candidates: all combinations for cage 5 contain 1r1,10r1; eliminations in r1c7, r1c12, r1c8, r1c11
37: (1.5) Pointing candidates: all combinations for cage 9 contain 1c4; eliminations in r4c4, r12c4
38: (0) Naked single: 1 in r12c3
39: (0) Naked single: 7 in r12c4
40: (1.5) Pointing candidates: all combinations for cage 14 contain 1c7,5c7; eliminations in r6c7, r7c7, r10c7, r11c7
41: (1.5) Pointing candidates: all combinations for cage 25 contain 9c4,10c4; eliminations in r7c4, r11c4
42: (1.5) Pointing candidates: all combinations for cage 34 contain 1r6,5r6,7r6; eliminations in r6c11, r6c12, r6c3, r6c5, r6c6, r6c2, r6c7
43: (0) Naked single: 7 in r5c2
44: (0) Naked single: 8 in r6c2
45: (1) Hidden single: 12 in r5c3
46: (0) Naked single: 7 in r4c3
47: (0) Naked single: 12 in r5c3
48: (0) Naked single: 8 in r1c3
49: (0) Naked single: 12 in r1c4
50: (3.1) Region parity: Cols 11, 12 must sum to Even number; cage 53's cells within this region must sum to Even number; reduced from 32 combinations to 16
51: (3.1) Region parity: Cols 8, 9, 10 must sum to Even number; cage 38's cells within this region must sum to Even number; reduced from 13 combinations to 5
52: (2) X-Wing: 8 in c4, 8
53: (0) Naked single: 10 in r7c7
54: (1) Hidden single: 5 in r7c12
55: (0) Naked single: 6 in r8c12
56: (0) Naked single: 9 in r1c11
57: (1.5) Pointing candidates: all combinations for cage 46 contain 9c7,12c7; eliminations in r2c7, r6c7, r11c7, r10c7, r12c7
58: (0) Naked single: 6 in r1c7
59: (0) Naked single: 7 in r2c7
60: (1) Hidden single: 8 in r12c7
61: (0) Naked single: 6 in r12c6
62: (0) Naked single: 8 in r12c7
63: (1) Hidden single: 11 in r6c7
64: (2) X-Wing: 12 in c1, 7
65: (4) Cage comparison: cages 5, 47 interact such that cage 47 has incongruous combinations; reduced from 15 to 14 combinations
66: (3.5) Region sum: Cols 3, 4 must sum to 156; cages 43, 21, 36's cells within this region must sum to 22; cages 21, 36 reduced from 5, 3 combinations to 3, 2
67: (4) Cage comparison: cages 5, 59 interact such that cage 59 has incongruous combinations; reduced from 1014 to 908 combinations
68: (4) Cage comparison: cages 6, 4 interact such that cage 4 has incongruous combinations; reduced from 8 to 6 combinations
69: (4) Cage comparison: cages 37, 47 interact such that cage 47 has incongruous combinations; reduced from 14 to 13 combinations
70: (4) Cage comparison: cages 38, 4 interact such that cage 4 has incongruous combinations; reduced from 6 to 5 combinations
71: (4) Cage comparison: cages 38, 47 interact such that cage 47 has incongruous combinations; reduced from 13 to 11 combinations
72: (4) Cage comparison: cages 55, 59 interact such that cage 59 has incongruous combinations; reduced from 908 to 833 combinations
73: (4) Cage comparison: cages 58, 59 interact such that cage 59 has incongruous combinations; reduced from 833 to 790 combinations
74: (4) Cage comparison: cages 60, 59 interact such that cage 59 has incongruous combinations; reduced from 790 to 725 combinations
75: (4) Cage comparison: cages 43, 47 interact such that cage 47 has incongruous combinations; reduced from 11 to 10 combinations
76: (4) Cage comparison: cages 52, 59 interact such that cage 59 has incongruous combinations; reduced from 725 to 713 combinations
77: (4) Cage comparison: cages 28, 59 interact such that cage 59 has incongruous combinations; reduced from 713 to 710 combinations
78: (4) Cage comparison: cages 30, 54 interact such that cage 30 has incongruous combinations; reduced from 4 to 3 combinations
79: (0) Naked single: 6 in r6c11
80: (1) Hidden single: 12 in r6c6
81: (3.4) Region sum: Col 6 must sum to 78; cages 51, 2, 61's cells within this region must sum to 24; cages 61 reduced from 31 combinations to 19
82: (4) Cage comparison: cages 56, 59 interact such that cage 59 has incongruous combinations; reduced from 710 to 660 combinations
83: (4) Cage comparison: cages 4, 59 interact such that cage 59 has incongruous combinations; reduced from 660 to 631 combinations
84: (4) Cage comparison: cages 62, 17 interact such that cage 17 has incongruous combinations; reduced from 11 to 9 combinations
85: (4) Cage comparison: cages 62, 61 interact such that cage 61 has incongruous combinations; reduced from 19 to 18 combinations
86: (4) Cage comparison: cages 62, 59 interact such that cage 59 has incongruous combinations; reduced from 631 to 621 combinations
87: (4) Cage comparison: cages 34, 59 interact such that cage 59 has incongruous combinations; reduced from 621 to 611 combinations
88: (4.2) Cage comparison: cages 5, 28, 59 interact such that cage 59 has incongruous combinations; reduced from 611 to 609 combinations
89: (3.6) Region sum: Col 11 must sum to 78; cages 17, 53, 54, 62's cells within this region must sum to 41; cages 17, 62 reduced from 9, 6 combinations to 6, 5
90: (1.5) Claiming candidates: cage 54 must contain 7 in r9c12, r10c12: from 5 combinations to 2
91: (0) Naked single: 12 in r10c11
92: (0) Naked single: 8 in r4c12
93: (1.5) Pointing candidates: all combinations for cage 54 contain 11c12; eliminations in r1c12, r5c12
94: (0) Naked single: 4 in r1c12
95: (0) Naked single: 11 in r1c8
96: (0) Naked single: 10 in r2c12
97: (0) Naked single: 2 in r2c1
98: (0) Naked single: 10 in r3c1
99: (0) Naked single: 1 in r5c12
100: (0) Naked single: 3 in r6c12
101: (0) Naked single: 1 in r11c11
102: (0) Naked single: 2 in r11c12
103: (0) Naked single: 5 in r12c11
104: (1) Hidden single: 12 in r2c8
105: (0) Naked single: 3 in r2c9
106: (0) Naked single: 1 in r2c4
107: (1) Hidden single: 11 in r2c2
108: (0) Naked single: 9 in r3c2
109: (1) Hidden single: 11 in r3c11
110: (0) Naked single: 2 in r4c11
111: (0) Naked single: 11 in r4c4
112: (0) Naked single: 8 in r11c4
113: (0) Naked single: 2 in r7c4
114: (0) Naked single: 8 in r7c8
115: (0) Naked single: 4 in r8c8
116: (0) Naked single: 5 in r2c3
117: (0) Naked single: 4 in r2c6
118: (0) Naked single: 2 in r3c6
119: (0) Naked single: 8 in r3c9
120: (0) Naked single: 9 in r2c5
121: (0) Naked single: 6 in r2c10
122: (0) Naked single: 5 in r5c5
123: (0) Naked single: 9 in r5c8
124: (0) Naked single: 10 in r5c4
125: (0) Naked single: 3 in r4c6
126: (0) Naked single: 8 in r5c6
127: (0) Naked single: 9 in r7c6
128: (0) Naked single: 10 in r8c6
129: (0) Naked single: 9 in r6c4
130: (0) Naked single: 2 in r9c8
131: (0) Naked single: 10 in r10c8
132: (0) Naked single: 9 in r10c3
133: (0) Naked single: 10 in r9c11
134: (0) Naked single: 3 in r8c3
135: (0) Naked single: 2 in r8c9
136: (0) Naked single: 8 in r8c10
137: (0) Naked single: 12 in r8c1
138: (0) Naked single: 9 in r8c7
139: (0) Naked single: 12 in r9c7
140: (0) Naked single: 8 in r9c1
141: (0) Naked single: 1 in r9c5
142: (0) Naked single: 8 in r10c5
143: (0) Naked single: 2 in r6c3
144: (0) Naked single: 11 in r7c3
145: (0) Naked single: 6 in r9c3
146: (0) Naked single: 10 in r11c3
147: (0) Naked single: 12 in r11c2
148: (0) Naked single: 10 in r12c2
149: (0) Naked single: 4 in r5c9
150: (0) Naked single: 10 in r4c10
151: (0) Naked single: 12 in r4c5
152: (0) Naked single: 6 in r11c5
153: (0) Naked single: 5 in r10c9
154: (0) Naked single: 7 in r10c1
155: (0) Naked single: 7 in r9c12
156: (0) Naked single: 5 in r9c6
157: (0) Naked single: 1 in r10c6
158: (0) Naked single: 9 in r9c10
159: (0) Naked single: 3 in r10c10
160: (0) Naked single: 4 in r10c7
161: (0) Naked single: 3 in r11c7
162: (0) Naked single: 11 in r10c12
163: (0) Naked single: 5 in r11c1
164: (0) Naked single: 1 in r6c9
165: (0) Naked single: 10 in r1c9
166: (0) Naked single: 1 in r1c10
167: (0) Naked single: 11 in r9c9
168: (0) Naked single: 7 in r11c8
169: (0) Naked single: 5 in r6c8
170: (0) Naked single: 7 in r6c10
171: (0) Naked single: 5 in r3c10
172: (0) Naked single: 1 in r3c7
173: (0) Naked single: 6 in r3c8
174: (0) Naked single: 1 in r4c8
175: (0) Naked single: 5 in r4c7
176: (0) Naked single: 9 in r11c9
177: (0) Naked single: 4 in r11c10
178: (0) Naked single: 12 in r12c9
179: (0) Naked single: 11 in r11c6
180: (0) Naked single: 4 in r12c5
181: (0) Naked single: 11 in r5c10
182: (0) Naked single: 10 in r6c5
183: (0) Naked single: 7 in r3c5
184: (0) Naked single: 2 in r1c5
185: (0) Naked single: 7 in r1c6
186: (0) Naked single: 6 in r4c9
187: (0) Naked single: 3 in r3c4
188: (0) Naked single: 3 in r7c5