logical thinking 
gadget1903 KwonTom Addict Puzzles: 325 Best Total: 23m 39s  Posted  2007.04.30 03:24:22
Quote: Originally Posted by chairman You could call this a pattern as well: 
I get the following too: 
thomastheman KwonTom Obsessive Puzzles: 844 Best Total: 18m 22s  Posted  2007.05.11 01:13:10 I've used a lot off time to try to understand how the patterns pictures you post work! Can anyone try to explain me what the picture below means?
for example:
4 lines, 5 x's:

Jankonyex KwonTom Obsessive Puzzles: 4781 Best Total: 9m 35s  Posted  2007.05.11 01:18:41
Quote: Originally Posted by thomastheman I...] 
That means in a given situation as shown, M line(s) and N x'(s) can be deduced. 
chairman KwonTom Obsessive Puzzles: 1397 Best Total: 17m 32s  Posted  2007.05.29 18:37:39
Another highlander pattern, spotted in naivoj's catch 22 puzzle. The three squares north west of the 2 must be blank, the others can be anything. Three x's are to be found. You can make variants by changing the 1 into a 2 or a 3 (a 0 leads to a contradiction).
Edit: Actually, I missed the crux myself. Not three, but five x's can be found. The core pattern is one without any numbers
The corner and the two adjacent squares must be blank, the question mark can be anything. Two x's can be deduced.
Last edited by chairman  2007.05.30 09:01:45 
Jankonyex KwonTom Obsessive Puzzles: 4781 Best Total: 9m 35s  Posted  2007.05.30 18:58:45
given 4 blanks, deduce 3 x's, if later deduce ? = x, 3 more x's then. (also true for chairman's case)
Edit (2007.07.04 15:31:36): ... I miss the wall. XP
Edit (2007.07.07 04:15:25): actually my explanation for this starts with an assumption, better use chairman's case as an example:
¡Ï¡@¡Ï¡@¡Ï¡@¡Ï ¡@¡@£A¡D£A¡@¡@ ¡Ï£A¡Ï¡@¡Ï¡@¡Ï ¡@¡D¡@¡D¡@¡D¡@ ¡Ï£A¡Ï¡@£C¡@¡Ï ¡@¡@¡@¡D¡@¡@¡@ ¡Ï¡@¡Ï¡@¡Ï¡@¡Ï
z should be e and I possibly split it into 2 cases:
case 1: (further deduce ? = x)
case 2: (further deduce ? = x)
but this solution only be accepted if all ?'s(if any) in the puzzle are zero, so it's usually negligible.
therefore 2 x's can be deduced:
if ? = x, back to case 1, further deduce 4 x's. if ? = line, back to case 2, further deduce nothing.
Edit (2007.07.07 18:09:42): I use a very simple example to explain why 4 x's appears.
¡Ï¡@¡Ï¢é¡Ï¡U ¡@¡@¢ì¢°¢ê¡U ¡Ï£A¡Ï¢ë¡Ï¡U ¡@¡@¡@¡@¡@¡U ¡Ï¡Ð¡Ï¡@¡Ï¡U ¡Ã¡Ã¡Ã¡Ã¡Ã by counting we have b+d = 1, therefore a and c are both crosses. but by disproof, you need to make an assumption first, you can assume b and d are crosses and get a and c are crosses, you can also first assume b and d and (a/c) are crosses and get (c/a) is a cross, then do the second assumption (assume b and d and (c/a) are crosses), but the second assumption have the same unique factor(b and d are crosses) as the first one to make it wrong, you don't need to continue the assumption after this factor appears and you may deduce (a/c) is also a cross. hope you understand.
sometimes the common point is really difficult to find, and sometimes it include too many situations that increase the difficulty of making assumption(s) to disprove it, just like Jcsl52.
Edit: I miss this:
if ? = line, further deduce 2 lines.
Last edited by Jankonyex  2007.07.10 21:11:37 
Naivoj KwonTom Addict Puzzles: 314 Best Total: 33m 50s  Posted  2007.05.31 05:43:27 A pattern which does not seem do have been posted yet can be found 4 times in #132 (Bang! Bang!, created 2007.05.17), and can also be found in the daily puzzle from Wednesday May23, 2007:
1 Line and 1 Cross
Last edited by Naivoj  2007.05.31 05:46:24 
Jankonyex KwonTom Obsessive Puzzles: 4781 Best Total: 9m 35s  Posted  2007.05.31 16:34:52
Quote: Originally Posted by naivoj A...] 
I've posted this before, showing all relative cases, in this thread. 
blivrail KwonTom Obsessive Puzzles: 559 Best Total: 19m 12s  Posted  2007.05.31 17:19:26 chairman:
I've sat around and thought about your highlander argument and this is what I came up with
Does that agree with what you came up with? This is a useful pattern because usually I can't do any work on a corner if there's no number there!
Last edited by blivrail  2007.05.31 17:20:11 
chairman KwonTom Obsessive Puzzles: 1397 Best Total: 17m 32s  Posted  2007.05.31 17:28:14
Quote: Originally Posted by blivrail ... Does that agree with what you came up with? ... 
This is what I meant indeed. 
Naivoj KwonTom Addict Puzzles: 314 Best Total: 33m 50s  Posted  2007.05.31 22:01:18
Quote: Originally Posted by jankonyex I've posted this before, showing all relative cases, in this thread. 
Can anyone please indicate the date and time when this 1 3 corner pattern was posted in the current thread, as I can not found it?
Last edited by Naivoj  2007.06.01 07:44:11 
gadget1903 KwonTom Addict Puzzles: 325 Best Total: 23m 39s  Posted  2007.06.01 20:01:52 Some more highlanders where the blank squares must be blank (including the middle square)
Jankonyex has some nifty patterns on page 3 that are similar...
But with blanks you get some specialized results (assuming the puzzle is unique)
4 x's
3 lines
I don't think I have seen these patterns anywhere, my apologies if they were already posted.
Last edited by Gadget1903  2007.06.13 03:20:03 
Naivoj KwonTom Addict Puzzles: 314 Best Total: 33m 50s  Posted  2007.06.12 08:34:08
Quote: Originally Posted by gadget1903 Some more highlanders... 4 x's 
A derived highlander pattern.
4 Lines and 3 Crosses
Last edited by Naivoj  2007.06.13 04:19:17 
Naivoj KwonTom Addict Puzzles: 314 Best Total: 33m 50s  Posted  2007.07.05 07:07:05 From the Puzzle from "Logical Thinking" topic
Quote: Originally Posted by gadget1903 sort of related: could the following be another highlander pattern getting 8 x's? there are also possible derivations replacing either of the 2 ?/x pairs with another 1 or by changing the x position. 
This is a variation of chairman corner highlander pattern.
Quote: Originally Posted by chairman The corner and the two adjacent squares must be blank, the question mark can be anything. Two x's can be deduced. 
So when the ? square is a 1 you get 8 x's instead of 2!
You can found this pattern in Speedy user puzzle #147 
Jankonyex KwonTom Obsessive Puzzles: 4781 Best Total: 9m 35s  Posted  2007.07.05 09:31:16
Quote: Originally Posted by naivoj F...7 
I've post this "before" 
Jankonyex KwonTom Obsessive Puzzles: 4781 Best Total: 9m 35s  Posted  2008.01.09 16:01:35 two types of "2":
odd 2
even 2 (or corner{\}2 here)
obviously it's an odd 2 iff x+y = e
it's also an odd 2 iff x+y = e
Edit (2008.02.14 22:22:15): £@¢é£A ¢ì¢Ï¢ê £C¢ë£B
Since it's o iff it's not e, there're only two possibilities: (1): x+y = e iff a+c = e or (2): x+y = o iff a+c = o
For A = 2, A is an odd 2 iff (1). For A = 1 and known position, 2 x's can be deduced. For A = 3 and known position, 2 lines can be deduced.
Edit (2008.02.16 15:21:27):
can be converted into
or other shapes depends on the situation
eg.
*For A = 2, w = y, x = z
Edit: correct word mistake
Last edited by Jankonyex  2008.06.13 07:49:06 
Jankonyex KwonTom Obsessive Puzzles: 4781 Best Total: 9m 35s  Posted  2008.01.09 17:04:01 highlander pattern
2 lines, 2x's:
three possibilities:
Last edited by Jankonyex  2008.01.10 06:09:48 
Naivoj KwonTom Addict Puzzles: 314 Best Total: 33m 50s  Posted  2008.01.10 03:26:35
Quote: Originally Posted by jankonyex highlander pattern 1 line: 
I believe this highlander pattern give a bit more:
2 Lines and 1 Cross
Last edited by Naivoj  2008.01.10 04:20:49 
Jankonyex KwonTom Obsessive Puzzles: 4781 Best Total: 9m 35s  Posted  2008.01.10 05:50:16
Quote: Originally Posted by naivoj I believe this highlander pattern give a bit more 
one more cross, see above.
Last edited by Jankonyex  2008.01.10 06:12:19 
Naivoj KwonTom Addict Puzzles: 314 Best Total: 33m 50s  Posted  2008.01.11 00:30:05
Quote: Originally Posted by jankonyex one more cross, see above. 
Agreed. This final cross was harder to found, good work! 
Jankonyex KwonTom Obsessive Puzzles: 4781 Best Total: 9m 35s  Posted  2008.03.06 10:44:37 example of advanced counting:
