Walkthrough of Puzzle 39 courtesy of udosuk. It's really a marvellous X-puzzle! The Killer part is not too hard, but the cages are laid pretty nicely too... Smile Too bad you can't make them perfectly symmetrical though... Neutral Here is my walkthrough (for the Vanilla-X part): After the killer part: 278 123789 13589 | 6 4 57 | 3589 123789 125 2357 1279 6 | 39 8 57 | 3459 *1249 123457 3578 4 89 | 39 2 1 | 59 3789 6 -------------------------+-------------------------+------------------------- 9 6 2 | 5 1 8 | 7 #34 #34 4 5 7 | 2 3 9 | 1 6 8 1 38 38 | 7 6 4 | 2 5 9 -------------------------+-------------------------+------------------------- 357 137 *145 | 8 9 2 | 6 -1347 -13457 2358 129 134589 | 14 7 6 | 34589 1289 12345 6 12789 1489 | 14 5 3 | 489 124789 127 Crossover: Either r2c8 or r7c3 must be 4 on d/ => r7c8 cannot be 4 Turbot fish: Either r2c8 or r7c3 must be 4 on d/ Either r4c8 or r4c9 must be 4 on r4,n6 => r7c9 cannot be 4 => Hidden single on r7: r7c3=4 278 123789 #13589 | 6 4 57 | 3589 123789 125 2357 1279 6 | 39 8 57 | 3459 129 123457 *3578 4 89 | 39 2 1 |*59 3789 6 -------------------------+-------------------------+------------------------- 9 6 2 | 5 1 8 | 7 34 34 4 5 7 | 2 3 9 | 1 6 8 1 38 38 | 7 6 4 | 2 5 9 -------------------------+-------------------------+------------------------- 357 137 4 | 8 9 2 | 6 137 1357 2358 129 #13589 | 14 7 6 |-34589 1289 12345 6 12789 189 | 14 5 3 | 489 124789 127 Turbot fish: Either r3c1 or r3c7 must be 5 on r3 Either r1c3 or r8c3 must be 5 on c3 => r8c7 cannot be 5 => 5 in c7,n3 locked in r123c7 => Hidden single on d/: r3c7=5 278 123789 13589 | 6 4 57 | 389 123789 12 2357 -1279 6 | 39 8 57 | 349 *129 12347 378 4 89 | 39 2 1 | 5 3789 6 -------------------------+-------------------------+------------------------- 9 6 2 | 5 1 8 | 7 34 34 4 5 7 | 2 3 9 | 1 6 8 1 38 38 | 7 6 4 | 2 5 9 -------------------------+-------------------------+------------------------- 357 137 4 | 8 9 2 | 6 137 1357 2358 *129 13589 | 14 7 6 | 3489 -1289 12345 6 12789 189 | 14 5 3 | 489 124789 127 Crossover: Either r2c8 or r8c2 must be 9 on d/ => r2c2 and r8c8 cannot be 9 => Hidden single on d\: r3c3=9 => r23c4=[93] => r1c9+r2c8={12} (naked pair in n3) *278 12378 1358 | 6 4 57 | 389 3789 12 2357 *127 6 | 9 8 57 | 34 12 347 #78 4 9 | 3 2 1 | 5 #78 6 -------------------------+-------------------------+------------------------- 9 6 2 | 5 1 8 | 7 34 34 4 5 7 | 2 3 9 | 1 6 8 1 38 38 | 7 6 4 | 2 5 9 -------------------------+-------------------------+------------------------- 357 137 4 | 8 9 2 | 6 -137 1357 2358 129 1358 | 14 7 6 | 3489 128 12345 6 12789 18 | 14 5 3 | 489 -124789 *127 Grouped turbot fish: Either r1c1+r2c2 or r9c9 must have 7 on d\ Either r3c1 or r3c8 must have 7 on r3 => r79c8 cannot be 7 Crossover: Either r9c2 or r9c9 must be 7 on r9 => r2c2 cannot be 7 => r2c2=1|2 => r2c28={12} (naked pair on r2) => r8c8 cannot be 1|2 => r8c8=8 => r3c18=[87] => r2c79={34} (naked pair on r2) => 3 in c1,n7 locked in r78c1 *27 -1237 135 | 6 4 57 | 389 39 12 57 *12 6 | 9 8 57 | 34 12 34 8 4 9 | 3 2 1 | 5 7 6 ----------------------+----------------------+---------------------- 9 6 2 | 5 1 8 | 7 34 34 4 5 7 | 2 3 9 | 1 6 8 1 38 38 | 7 6 4 | 2 5 9 ----------------------+----------------------+---------------------- -357 *17 4 | 8 9 2 | 6 13 1357 235 129 15 | 14 7 6 | 349 8 12345 6 12789 18 | 14 5 3 | 49 1249 127 XY-Wing: r2c2=1|2, r7c2=1|7, r1c1=2|7 => r1c2, r7c1 cannot be 7 27 123 135 | 6 4 57 | 389 39 12 57 12 6 | 9 8 57 | 34 12 34 8 4 9 | 3 2 1 | 5 7 6 ----------------------+----------------------+---------------------- 9 6 2 | 5 1 8 | 7 34 34 4 5 7 | 2 3 9 | 1 6 8 1 38 38 | 7 6 4 | 2 5 9 ----------------------+----------------------+---------------------- *35 -17 4 | 8 9 2 | 6 *13 1357 235 129 *15 | 14 7 6 | 349 8 -12345 6 12789 18 | 14 5 3 | 49 1249 127 XY-Wing: r7c1=3|5, r7c8=1|3, r8c3=1|5 => r7c2, r8c9 cannot be 1 => r7c2=7 => Hidden single in n9 => r9c9=7 => r1c1=2, and all naked singles from here. Did someone find a shorter solution (for example, one XY-Wing instead of two)? Question