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So, I've been blasting through the past tests, but not sharing much insight into how I do some of the unique puzzles. In private messages, I've answered some questions about some types of puzzles, or about some specific puzzles, and I thought I'd gather some of these comments here. Its rare to read good articles on how people approach different puzzles, and while I don't intend to start a series of them here, maybe this is as good a space as any to offer my voice to some of the strategies you run into on a USPC or WPC. You might learn a thing or two.
**As general tips are discussed in the context of specific puzzles, I've given links to the tests on which the specific puzzles I describe are included, but you'll still need to go through the steps of printing them out or trying to solve them yourself to get all you can from any of this commentary.
1. Division puzzles: Great Divide - 2001 (other examples - Tile in 2002 or Tile in 2000) Basically, you know the puzzle likely has 180 degree symmetry, so my first step is to mark forced "white" squares given the black squares I see in all mirrored places. This means I put an X in R2C4 (to mirror R5C2), an X in R4C125 and an X in R5C4. Now, the big X is that one in R2C4 which needs to get out, so it forces an X in R2C3 and R3C3 (put in the mirrored black squares as well) and so on. The puzzle actually took 10 seconds. It took me time to count and check the number of line segments needed in the answer. The two Tile examples listed above are a bit harder, but fall to the same notion of splitting a shape along its likely symmetries and just drawing forced "different" squares.
There are other puzzles of this type called dissections - I still list them as a weakness of mine - but you learn there are only so many ways to "cut a pizza". A division of a large rectangle into 3 pieces could be three long rectangles (two parallel cuts along long axis), three fat rectangles (two parallel cuts along short axis) or three imbetween rectangles (with a single cut at ~2/3 on one axis, and a cut in the middle of the thin side that connects only as far as this single cut on the other axis). Those are all the ways really to slice a rectangle. Now, normally there are other oddities in the shape for the rectangle which you must account for and which will force one of these three styles, but experience thinking about how you cut things up into similar pieces gives a quick start on these types. These are often more intuitive than logical puzzles, but a good sense of feel can shave tons of time off your approach.
2. Fences Variation - 2001 version described specifically, also 2002, 2006. We've all practiced on Nikoli but most competition slitherlinks or fences variations will not fall for simply looking for the normal patterns you've learned from practice. On a variation puzzle like this, I might good use of dark and light pencil. Here, there is some good forced logic you can get. Look for things familiar to you that build together to give a starting point. On the 2001 test, the big 2 is the thing that jumped out to me. The big 2 will use a single line from the 3 on the lower-left side and a line from one of two possible threes from the long chain on the right. Mark all the rest of the spaces as not used. This trickles info up through the 1 in the upper-left corner to the 3 to the upper-left of that in a way you should be familiar from Nikoli.
Now, you want to know if that 3 connects to the big 6 or not. Well, if it turns down, then all the other squares around the 6 are joined in a segment (a big 6/7 is super-forced so look for them in these variations just as you look for 3's in the normal puzzle or 5's in hex versions). This fails, so you can direct the 3 so one segment comes out of it along a border of the 6. Now, there are 2 gaps in the big 6 and one of them will be just next to the 3 and the other is unknown. However, the 1's on the top/right edge don't give much freedom. Imagine a gap say on the upper-right segments - how do you fill in the lines? You can't - so fill those in. Work like that. Sometimes draw in a possibility (meant to fail) in light pencil to confirm to yourself that you've discovered something real. These are fun puzzles, and should get you back - as you were when you started Nikoli slitherlinks - to trying to visualize why things must be, and not just looking for 0's next to 3's.
3. Loop-D-Loop (2001) - This puzzle may seem difficult, but it is really easily approached with this simple rule: some squares can only be reached by one other place. Once you've marked all those down, you can repeat it again and again and be done. Its not really a crazy maze, just simple observation. N -> P, B-> N, E -> M, as N is used, C -> O. By position, F-> E and G -> H are forced. This forces H->F. O->G is also now forced. You've finished row 2, so use that in places like column 4 where it is powerful (L->D, P->L). K is only reached by I so I->K, K->J. It trickles out fast from there. Really, if you are pursuing this as a I will start at A and go to B or C or D and see how things go, you are "guessing" where intuition is best. This is just like solving a maze backwards, only here you want to approach it from all possible middles.
4. Alice Maze (2001) - For Alice Maze, a really nice puzzle, I considered a couple choices from the start. In particular, I thought it good to marking all the squares a simple 1 path going down could take me to without changing size. I considered these impossibilities then for any future path if it shrunk to 1 (as most paths seemed to do) and started going to the right. I would specifically avoid all those marked squares if I needed to shrink back to 1, which made 2 ->N -> T -> 0 -> 7 instantly stand out as the necessary path when it arose. I got from there to the nice ever-expanding solution, reread instructions to see if I could go back to S, decided it was ok, and was done in just 3'11".
5. Path Battleships (2002) - one of my favorite puzzles on this test. Path battleships has a bit of meta logic you need to see, and then you can sort enough of it out to do some trials. Even if you do not see this logic early, if you try some things out you may run into it.
You'll find the hard constraints to satisfy a path are the 7's. If the 7's have sea divisions of 1,1,1 in particular, you cannot make a loop. This is because a loop would have to pass over and back 3 times to fill a 1,1,1 separation of seas and no loop can pass over a line an odd number of times. This kind of thinking is helpful in snake puzzles where many people now how to use a "2", but underuse what they must also know about a "3".
Here, a 1,2 sea division or just 3 division are the only possibilities as you need to get across the gap exactly twice to have a closed loop. Both the top and bottom have such a 7 of relevance. To see that neither use the 3 grouped sea division (a 4/3 battleship, cruiser arrangement is required for this), notice that you must then do a 3+2+2 with a 1/2 gap on the other side, but you cannot fulfill a touching 1 constraint in this way. So I guess its spotting the 7's next to a 1 in each case that tells you the combination of data of 1,2 sea division, plus 2 horizontal and 1 vertical ship. One will be 4+2+V; the other is 3+3+V.
The vertical ships make the most/only sense in columns 4 and 5 as they cannot be in 6 and 7. So, you have a greatly reduced search space to put the 2/3 and 3/4 pairs. The 2 on the left and 4 on the far right jump out when you look at the relative positions of 1's in column 3 and 7. Battleships puzzles often are a lot about feel, and this example had a great way of using "feel" of a loop constraint to influence the battleships. Master the feel of a puzzle and catch onto the most important information and you can conquer seemingly T&E puzzles like this that really aren't.
6. Rotator Mosaics (2003) - Rotator mosaics, as in 2003, or Rotation on the German 2008 Qualifier is the harder form of a puzzle called Spiral Galaxies you will see in 2004 or in hex form in 2005 as spinners. I think atomic fusion in 2006 has some of the same character too as rotator mosaics. I like the "Spiral Galaxies" form most where you are only putting one circle in each object, and Nikoli has published a separate collection of these puzzles now as a pencil puzzle book. Their gimmick is the black circles' regions get shaded in to make a picture, and this often works well.
Anyway, if you can do the regular spiral galaxies - where the basic strategy is to identify cells that can only be linked to one group and/or to mark walls on parts of groups that cannot have adjoining cells (because the rotationally symmetric wall is blocked by something else), you have a starting toolbox for the harder form in 2003. This one was a slower puzzle for me, and still involved some trial with some erasures. My "technique" to start is to try to find really big things that have symmetry, given the average size of each region is 8.1 cells. This means you will have some regions as big as 15-20 most likely. Even if you identify an incorrect big group, just finding many of them will give you a sense of what fits well with what else and you want to match big with big to have that average over 8 cells in a galaxy.
I got a 20 and 11 pretty early and those never had to change. They also pinned a 4 and 2, fully satisfying the lower right, so I liked the feel of them. I had trouble on the left and top, running through a couple versions of what is eventually a 16 region, but eventually saw some other big regions I could do there. One key is places (like the white on an edge in column 1) that cannot match with other things, so you want to something big with the black over white circle that follows to its right. You want to see a white over a black circle - this is only possible on the bottom - you can quick mark in a potential 8 or more like 10 cells that use that shape. If you really bog down in one path, erase and consider some others. Unlike spiral galaxies where the 1 circle per cell lets you almost always use exclusively logical deduction, this rotator mosaic was more about pattern identification in a broader sense. Light and dark pencil is, as always, invaluable.
7. Hex Pie (2003) - On the Hex Pie, I think I've lucked into the answer every time. By that, I guess my intuition works pretty well. I can count triangles in triangle shapes better than lopped off triangles, so I think I've started with AY every time since this quickly gives me 3 identical triangles, and one other shape. I need to test if O or P cut the rest off alright, and one of them does. This is not a standard dissection puzzle, but even without that fact, the O or P divide should look better to you than N/M/L/K. Similarly, thin cuts like U or V or F are going to be really hard to use above as it gives you 1/4/2 triangles in shapes. Its possible one or two shapes will be tricky and have the same count but not look the same, but most of the dissected shapes will be identical to each other for this to really work with just 3 linear cuts.
8. Dutch Week (2003) - another puzzle with seemingly a lot of possible trial and error but no great start point. The Z? Maybe - but two Z's and at least two directions each could go out from so 4 choices just for one corner. What to do?
Dutch Week has a very reasonable logic that you should add to the methods you use. There are many path fitting puzzles that involve checkerboard-like or even/odd parity over a set of squares. Its normally in "fishing" for me, where you must map an outside man to some target in the grid, given the length of his line, and prefiguring which man should go to which fish by parity greatly simplifies the puzzle.
Here, you have 2 Z's, so you know one of them is in the lower left. Also important is that you have 2 E's and they are 5 squares apart, while there are only 3 total E's in the set of days. At least one of these E's must be specified by the odd parity between them. The E's in WOENSDAG and DONDERDAG are 10 apart, which tells you the E in ZATERDAG must be one of the E's in the grid! If the Z in ZATERDAG is the Z on the grid, you have only really the up, right, right path to take, but VRIJDAG cannot be put in the grid as it will be pushed into the already given G. So the Z is in ZONDAG. Again, you are looking to get ZATERDAG in the grid knowing you have GADRE to work backwards from the Z you have to eventually end up on the E square. The G must be up or down (as the corner definitely connects to the Z). G up forces ZONDAG around the bottom corner and to the right and that, like VRIJDAG, fails. G is down, and can start to place pretty sure letters for both groups. I saw that ZATERDAG gives an option that correctly allows the G in the bottom to be used, and took advantage of that. The rest I did not solve logically - a packing rule on an A and E might allow it, but with a completely sure lower left as a starting point, I put in words until they fit. It went pretty fast. The key was identifying two words that must use a particular Z and E in the grid, which is more than many people do when getting started here. Checkerboard-parity made it possible.
9. Piecemaker (2005) - The 13 has to be 4+4+3+2 or 4+3+3+3. In either case, 13 of the 16 squares have their polyominoes left, so the remaining polyomino to add to the set is another 3. As a row with a sum of 5 is definitely 3+2, the 4+4+3+2 must be the 13. The single 2 has to be on the 5 and since it is in column 2, it must also be in column 1. The unknown 3 fills both bottom cells in the rest of that row. Its got a cell coming up so the rest of the row must be the 4 to get 7 total. This forces the 3 to be in column 4 in row 3, the 4 in the first 3 columns. You can work forward from there, but that is the basics of the logical route. Identify all the pieces, then place them as forced. Starting with something you know (how to add to 13), but also seeing the more global constraint that all polyominoes in an n x n square sum to n^2 pieces, takes a seeming trial and error puzzle into a really easy deductive puzzle that takes a matter of minutes.
10. Icon Maze (2005) - I still remember doing this the first time, making a small number of dark lines before my light line guesses converged to an answer. I worked hard this time around to make all steps use just dark lines using my newfound "maximum observation" skills. You have to be careful to define what can get to where (T can go to W, even if W seems to only go to U or Z, but this puzzle is nowhere near as hard as its 25 points.
I can fill in A->D bold, W->Z bold, then C->B->H bold (no other folder can hit B) as a start. This gives D->C bold as nothing else gets to C. With that gone, you can now mark IFEJ but you won't know what direction (just lines here, not arrowed lines as for all earlier ones - which are the best notation here). I cannot connect to J, so J goes to M as well. G's got to get somewhere, so H ->G -> K looks forced now. I is now in real trouble as it cannot go anywhere that isn't F, so I has to go to F and Q must go to I - mark Q->I and follow the arrowheads forward. L now looks lonely to me, so mark K->L->R as you'll see it is forced. N-O-T has to happen in some order. At this point I have so much down I might sketch an idea, but I'll continue the logic path. P can't go to R or M (as things go to those already, so P->S, so now R must go to Q. M must then be what connect to P, leaving N->X (and putting directionality on NOT. T goes to O, so U goes to W. Y must go to V, forcing S to T. connecting XY and VU finishes the grid. Not really trial and error at all if you just, like the Loop-D-Loop above, look for "middles" of the maze that must be one way and no others. Having a lot of dark path segments marked with arrows or just lines when direction isn't known sews together the whole solution rather easily after you are 50-60% done.
11. False Field Fences (2005) - I've looked at this many times and have nothing. Sorry. Ok, not really nothing. I know the 1 in the lower-left was false (the only such corner) and I worked for awhile thinking it was a 2, but you'll see if you just focus on that corner that 2 fails about 4 squares in when you have to fulfill the 322 in the corner which are all "inside" the loop. This just tells you the 1 is actually a 0, which tells you the 3 above it is also not in the loop. Not really a way to get a real nice start. Knowing something about that corner, I felt like trying the upper-left. First, if it were in the loop, the 3 certainly must point to the right. Trickle this out for a while (including the 2 and 1 and next 2). Also use the not loop information, like the 1 below the 3 which must be a 2 and therefore gets a linear segment. The 2 below must then be in the loop as well as the 2 beneath (but not the 3 which you already learned about). You get to an answer. I never feel fast on this puzzle, and my only work in is really choosing a very tempting site after the "not a 1 corner" failed to be a work-in.
12. Railroads (German 2008 Qualifier) - I've not really met one of these I cannot do by logic and the tough German one was no exception. However, while they most often involve focusing on the stations through which the path does not turn, the German one worked best off exploiting the "all white cell" rule as often as you can. In the bottom-left, for example, I can trickle the whole path in. Most of the thinking is along these lines: this box has 3 potential exits, if I use these two I bring numbers together out of order, or prevent full closure of a loop, so I must use the one of the three I am excluding. Another helpful rule you will see, just as in a slitherlink, is if you have 4 open ends pinned in a 2x2 box, no single end can turn away from that 2x2 box as you need to connect all 4 ends together instead of leaving behind 3 and not being able to close the loop. I ran into this early on, for example, to get R67C12 as such a pinned location, with R67C1 and R67C2 being the forced connections by number order, with an L shape in R4C2, R5C2, R5C3. The connections going down are really important. Use the same "2 out of 3" in R9C4 to show it must use the vertical segment to R8C4. This now prevents the vertical segment to R10C4 (3 is not next to 8) so use the segment to the right, connect other lines, use the forced corners as always which only have 2 choices, you get the whole bottom without really thinking about the numbers. The 8-7 area on the top is not easy, but most of the rest falls once you've got the rails down from attacking the "all white cells used" rule here. Its a fun version of this puzzle type that I really like.
13. Star Battle (German 2008 Qualifier) - Star Battle is an acquired taste. I could point out I made a star battle variant called soduku but I'll neglect that for now. A lot of star battle puzzles involves thinking about how placements will cause the thing to fail (by forcing too many things into a row/column). Even when you cannot firmly place a star (like the either/or choice in R67C3), mark the shaded cells next to it that are touched by both options and therefore invalid. There are sometimes times you can say "I can't fill this row/column unless 2/4/6 stars from this 1/2/3 shapes are in this row/column", almost an innie-outie constraint like on a jigsaw sudoku. Extraneous cells in other rows/columns in those regions can be marked dead. On this puzzle, I think a key square to eliminate was R5C2. Figure out why this is no good from the start. It will build character. Experience with this puzzle type makes R5C2 jump out as easy to eliminate, but the size of the region that contains R5C2 should suggest you want to look at its cells to try to force one or both stars in the shape.
14. Balancing Act (German 2008 Qualifying is discussed here - many USPC examples exist) - Balancing Act is most often a challenge of picking the most critical place where you can force things from, and then running through all possibilities almost blindly. In this one, I love the bottom-most weights. Start high or low, but then work in order from some particular weight. Here, I would have chosen the lower-left weight and started at 8. Instead, I overthought it (by assuming the answer was not the most obvious value of the most obvious starting point) and did 6 and 7 first, then finally got to 8. Anyway, if 8 is there, you can only put 6 2 or 1 3 out. If 7 is there, you can only put 4 2 out. If 6 is there, you can only put 2 1 out. 5 can't be there. 4 can be with 31, nothing else can be there.
The big thing is for any of these pairs to see if there is some partition of the total sum of 36 into the top fulcrum to get the system to balance. I'm working therefore off the sums of the sure divisions I've got (16 from 862), (12 from 813), (13 from 742), (9 from 621). You can do the algebra, or you can notice the right side needs an even weight, or whatever, but most fail. (16 from 862) is ok, and puts a 4 in the far left. Now you just need to figure out how 1357 balance on the right. If this fails (it doesn't), I'd see 12 from 813 cannot work by algebra and therefore know the lower-left weight is not 8. I then go to 7. I'm not trying to be lucky with guesses, much more being sure I try all possibilities and eventually collide face-first with the answer.
I have a love/hate relationship with balancing act. I tend to sense things like I want a big sum on the 2nd fulcrum here from below (so starting at 8 and not at 4 makes sense to me), but at the same time you can often guess from the wrong point and have to go through EVERYTHING to get to the answer. Everything, fortunately, is never 8! or 12! or however many choices, more like 10 things, but catalog what you've tried well at the critical point you start you tests from so that you aren't always guessing and erasing and guessing and erasing and occasionally duplicating past efforts. I've rarely met a balancing act that falls from logic directly, most fall from a combination of deduction and exhaustive coverage of search space.
Ok, that's enough for now (and also where my private messages have run out of puzzles I've written about). I might take requests, or not, but half the fun of puzzles is coming up with your own methods to work through a puzzle so I'm interested to hear how other approaches differ.
**As general tips are discussed in the context of specific puzzles, I've given links to the tests on which the specific puzzles I describe are included, but you'll still need to go through the steps of printing them out or trying to solve them yourself to get all you can from any of this commentary.
1. Division puzzles: Great Divide - 2001 (other examples - Tile in 2002 or Tile in 2000) Basically, you know the puzzle likely has 180 degree symmetry, so my first step is to mark forced "white" squares given the black squares I see in all mirrored places. This means I put an X in R2C4 (to mirror R5C2), an X in R4C125 and an X in R5C4. Now, the big X is that one in R2C4 which needs to get out, so it forces an X in R2C3 and R3C3 (put in the mirrored black squares as well) and so on. The puzzle actually took 10 seconds. It took me time to count and check the number of line segments needed in the answer. The two Tile examples listed above are a bit harder, but fall to the same notion of splitting a shape along its likely symmetries and just drawing forced "different" squares.
There are other puzzles of this type called dissections - I still list them as a weakness of mine - but you learn there are only so many ways to "cut a pizza". A division of a large rectangle into 3 pieces could be three long rectangles (two parallel cuts along long axis), three fat rectangles (two parallel cuts along short axis) or three imbetween rectangles (with a single cut at ~2/3 on one axis, and a cut in the middle of the thin side that connects only as far as this single cut on the other axis). Those are all the ways really to slice a rectangle. Now, normally there are other oddities in the shape for the rectangle which you must account for and which will force one of these three styles, but experience thinking about how you cut things up into similar pieces gives a quick start on these types. These are often more intuitive than logical puzzles, but a good sense of feel can shave tons of time off your approach.
2. Fences Variation - 2001 version described specifically, also 2002, 2006. We've all practiced on Nikoli but most competition slitherlinks or fences variations will not fall for simply looking for the normal patterns you've learned from practice. On a variation puzzle like this, I might good use of dark and light pencil. Here, there is some good forced logic you can get. Look for things familiar to you that build together to give a starting point. On the 2001 test, the big 2 is the thing that jumped out to me. The big 2 will use a single line from the 3 on the lower-left side and a line from one of two possible threes from the long chain on the right. Mark all the rest of the spaces as not used. This trickles info up through the 1 in the upper-left corner to the 3 to the upper-left of that in a way you should be familiar from Nikoli.
Now, you want to know if that 3 connects to the big 6 or not. Well, if it turns down, then all the other squares around the 6 are joined in a segment (a big 6/7 is super-forced so look for them in these variations just as you look for 3's in the normal puzzle or 5's in hex versions). This fails, so you can direct the 3 so one segment comes out of it along a border of the 6. Now, there are 2 gaps in the big 6 and one of them will be just next to the 3 and the other is unknown. However, the 1's on the top/right edge don't give much freedom. Imagine a gap say on the upper-right segments - how do you fill in the lines? You can't - so fill those in. Work like that. Sometimes draw in a possibility (meant to fail) in light pencil to confirm to yourself that you've discovered something real. These are fun puzzles, and should get you back - as you were when you started Nikoli slitherlinks - to trying to visualize why things must be, and not just looking for 0's next to 3's.
3. Loop-D-Loop (2001) - This puzzle may seem difficult, but it is really easily approached with this simple rule: some squares can only be reached by one other place. Once you've marked all those down, you can repeat it again and again and be done. Its not really a crazy maze, just simple observation. N -> P, B-> N, E -> M, as N is used, C -> O. By position, F-> E and G -> H are forced. This forces H->F. O->G is also now forced. You've finished row 2, so use that in places like column 4 where it is powerful (L->D, P->L). K is only reached by I so I->K, K->J. It trickles out fast from there. Really, if you are pursuing this as a I will start at A and go to B or C or D and see how things go, you are "guessing" where intuition is best. This is just like solving a maze backwards, only here you want to approach it from all possible middles.
4. Alice Maze (2001) - For Alice Maze, a really nice puzzle, I considered a couple choices from the start. In particular, I thought it good to marking all the squares a simple 1 path going down could take me to without changing size. I considered these impossibilities then for any future path if it shrunk to 1 (as most paths seemed to do) and started going to the right. I would specifically avoid all those marked squares if I needed to shrink back to 1, which made 2 ->N -> T -> 0 -> 7 instantly stand out as the necessary path when it arose. I got from there to the nice ever-expanding solution, reread instructions to see if I could go back to S, decided it was ok, and was done in just 3'11".
5. Path Battleships (2002) - one of my favorite puzzles on this test. Path battleships has a bit of meta logic you need to see, and then you can sort enough of it out to do some trials. Even if you do not see this logic early, if you try some things out you may run into it.
You'll find the hard constraints to satisfy a path are the 7's. If the 7's have sea divisions of 1,1,1 in particular, you cannot make a loop. This is because a loop would have to pass over and back 3 times to fill a 1,1,1 separation of seas and no loop can pass over a line an odd number of times. This kind of thinking is helpful in snake puzzles where many people now how to use a "2", but underuse what they must also know about a "3".
Here, a 1,2 sea division or just 3 division are the only possibilities as you need to get across the gap exactly twice to have a closed loop. Both the top and bottom have such a 7 of relevance. To see that neither use the 3 grouped sea division (a 4/3 battleship, cruiser arrangement is required for this), notice that you must then do a 3+2+2 with a 1/2 gap on the other side, but you cannot fulfill a touching 1 constraint in this way. So I guess its spotting the 7's next to a 1 in each case that tells you the combination of data of 1,2 sea division, plus 2 horizontal and 1 vertical ship. One will be 4+2+V; the other is 3+3+V.
The vertical ships make the most/only sense in columns 4 and 5 as they cannot be in 6 and 7. So, you have a greatly reduced search space to put the 2/3 and 3/4 pairs. The 2 on the left and 4 on the far right jump out when you look at the relative positions of 1's in column 3 and 7. Battleships puzzles often are a lot about feel, and this example had a great way of using "feel" of a loop constraint to influence the battleships. Master the feel of a puzzle and catch onto the most important information and you can conquer seemingly T&E puzzles like this that really aren't.
6. Rotator Mosaics (2003) - Rotator mosaics, as in 2003, or Rotation on the German 2008 Qualifier is the harder form of a puzzle called Spiral Galaxies you will see in 2004 or in hex form in 2005 as spinners. I think atomic fusion in 2006 has some of the same character too as rotator mosaics. I like the "Spiral Galaxies" form most where you are only putting one circle in each object, and Nikoli has published a separate collection of these puzzles now as a pencil puzzle book. Their gimmick is the black circles' regions get shaded in to make a picture, and this often works well.
Anyway, if you can do the regular spiral galaxies - where the basic strategy is to identify cells that can only be linked to one group and/or to mark walls on parts of groups that cannot have adjoining cells (because the rotationally symmetric wall is blocked by something else), you have a starting toolbox for the harder form in 2003. This one was a slower puzzle for me, and still involved some trial with some erasures. My "technique" to start is to try to find really big things that have symmetry, given the average size of each region is 8.1 cells. This means you will have some regions as big as 15-20 most likely. Even if you identify an incorrect big group, just finding many of them will give you a sense of what fits well with what else and you want to match big with big to have that average over 8 cells in a galaxy.
I got a 20 and 11 pretty early and those never had to change. They also pinned a 4 and 2, fully satisfying the lower right, so I liked the feel of them. I had trouble on the left and top, running through a couple versions of what is eventually a 16 region, but eventually saw some other big regions I could do there. One key is places (like the white on an edge in column 1) that cannot match with other things, so you want to something big with the black over white circle that follows to its right. You want to see a white over a black circle - this is only possible on the bottom - you can quick mark in a potential 8 or more like 10 cells that use that shape. If you really bog down in one path, erase and consider some others. Unlike spiral galaxies where the 1 circle per cell lets you almost always use exclusively logical deduction, this rotator mosaic was more about pattern identification in a broader sense. Light and dark pencil is, as always, invaluable.
7. Hex Pie (2003) - On the Hex Pie, I think I've lucked into the answer every time. By that, I guess my intuition works pretty well. I can count triangles in triangle shapes better than lopped off triangles, so I think I've started with AY every time since this quickly gives me 3 identical triangles, and one other shape. I need to test if O or P cut the rest off alright, and one of them does. This is not a standard dissection puzzle, but even without that fact, the O or P divide should look better to you than N/M/L/K. Similarly, thin cuts like U or V or F are going to be really hard to use above as it gives you 1/4/2 triangles in shapes. Its possible one or two shapes will be tricky and have the same count but not look the same, but most of the dissected shapes will be identical to each other for this to really work with just 3 linear cuts.
8. Dutch Week (2003) - another puzzle with seemingly a lot of possible trial and error but no great start point. The Z? Maybe - but two Z's and at least two directions each could go out from so 4 choices just for one corner. What to do?
Dutch Week has a very reasonable logic that you should add to the methods you use. There are many path fitting puzzles that involve checkerboard-like or even/odd parity over a set of squares. Its normally in "fishing" for me, where you must map an outside man to some target in the grid, given the length of his line, and prefiguring which man should go to which fish by parity greatly simplifies the puzzle.
Here, you have 2 Z's, so you know one of them is in the lower left. Also important is that you have 2 E's and they are 5 squares apart, while there are only 3 total E's in the set of days. At least one of these E's must be specified by the odd parity between them. The E's in WOENSDAG and DONDERDAG are 10 apart, which tells you the E in ZATERDAG must be one of the E's in the grid! If the Z in ZATERDAG is the Z on the grid, you have only really the up, right, right path to take, but VRIJDAG cannot be put in the grid as it will be pushed into the already given G. So the Z is in ZONDAG. Again, you are looking to get ZATERDAG in the grid knowing you have GADRE to work backwards from the Z you have to eventually end up on the E square. The G must be up or down (as the corner definitely connects to the Z). G up forces ZONDAG around the bottom corner and to the right and that, like VRIJDAG, fails. G is down, and can start to place pretty sure letters for both groups. I saw that ZATERDAG gives an option that correctly allows the G in the bottom to be used, and took advantage of that. The rest I did not solve logically - a packing rule on an A and E might allow it, but with a completely sure lower left as a starting point, I put in words until they fit. It went pretty fast. The key was identifying two words that must use a particular Z and E in the grid, which is more than many people do when getting started here. Checkerboard-parity made it possible.
9. Piecemaker (2005) - The 13 has to be 4+4+3+2 or 4+3+3+3. In either case, 13 of the 16 squares have their polyominoes left, so the remaining polyomino to add to the set is another 3. As a row with a sum of 5 is definitely 3+2, the 4+4+3+2 must be the 13. The single 2 has to be on the 5 and since it is in column 2, it must also be in column 1. The unknown 3 fills both bottom cells in the rest of that row. Its got a cell coming up so the rest of the row must be the 4 to get 7 total. This forces the 3 to be in column 4 in row 3, the 4 in the first 3 columns. You can work forward from there, but that is the basics of the logical route. Identify all the pieces, then place them as forced. Starting with something you know (how to add to 13), but also seeing the more global constraint that all polyominoes in an n x n square sum to n^2 pieces, takes a seeming trial and error puzzle into a really easy deductive puzzle that takes a matter of minutes.
10. Icon Maze (2005) - I still remember doing this the first time, making a small number of dark lines before my light line guesses converged to an answer. I worked hard this time around to make all steps use just dark lines using my newfound "maximum observation" skills. You have to be careful to define what can get to where (T can go to W, even if W seems to only go to U or Z, but this puzzle is nowhere near as hard as its 25 points.
I can fill in A->D bold, W->Z bold, then C->B->H bold (no other folder can hit B) as a start. This gives D->C bold as nothing else gets to C. With that gone, you can now mark IFEJ but you won't know what direction (just lines here, not arrowed lines as for all earlier ones - which are the best notation here). I cannot connect to J, so J goes to M as well. G's got to get somewhere, so H ->G -> K looks forced now. I is now in real trouble as it cannot go anywhere that isn't F, so I has to go to F and Q must go to I - mark Q->I and follow the arrowheads forward. L now looks lonely to me, so mark K->L->R as you'll see it is forced. N-O-T has to happen in some order. At this point I have so much down I might sketch an idea, but I'll continue the logic path. P can't go to R or M (as things go to those already, so P->S, so now R must go to Q. M must then be what connect to P, leaving N->X (and putting directionality on NOT. T goes to O, so U goes to W. Y must go to V, forcing S to T. connecting XY and VU finishes the grid. Not really trial and error at all if you just, like the Loop-D-Loop above, look for "middles" of the maze that must be one way and no others. Having a lot of dark path segments marked with arrows or just lines when direction isn't known sews together the whole solution rather easily after you are 50-60% done.
11. False Field Fences (2005) - I've looked at this many times and have nothing. Sorry. Ok, not really nothing. I know the 1 in the lower-left was false (the only such corner) and I worked for awhile thinking it was a 2, but you'll see if you just focus on that corner that 2 fails about 4 squares in when you have to fulfill the 322 in the corner which are all "inside" the loop. This just tells you the 1 is actually a 0, which tells you the 3 above it is also not in the loop. Not really a way to get a real nice start. Knowing something about that corner, I felt like trying the upper-left. First, if it were in the loop, the 3 certainly must point to the right. Trickle this out for a while (including the 2 and 1 and next 2). Also use the not loop information, like the 1 below the 3 which must be a 2 and therefore gets a linear segment. The 2 below must then be in the loop as well as the 2 beneath (but not the 3 which you already learned about). You get to an answer. I never feel fast on this puzzle, and my only work in is really choosing a very tempting site after the "not a 1 corner" failed to be a work-in.
12. Railroads (German 2008 Qualifier) - I've not really met one of these I cannot do by logic and the tough German one was no exception. However, while they most often involve focusing on the stations through which the path does not turn, the German one worked best off exploiting the "all white cell" rule as often as you can. In the bottom-left, for example, I can trickle the whole path in. Most of the thinking is along these lines: this box has 3 potential exits, if I use these two I bring numbers together out of order, or prevent full closure of a loop, so I must use the one of the three I am excluding. Another helpful rule you will see, just as in a slitherlink, is if you have 4 open ends pinned in a 2x2 box, no single end can turn away from that 2x2 box as you need to connect all 4 ends together instead of leaving behind 3 and not being able to close the loop. I ran into this early on, for example, to get R67C12 as such a pinned location, with R67C1 and R67C2 being the forced connections by number order, with an L shape in R4C2, R5C2, R5C3. The connections going down are really important. Use the same "2 out of 3" in R9C4 to show it must use the vertical segment to R8C4. This now prevents the vertical segment to R10C4 (3 is not next to 8) so use the segment to the right, connect other lines, use the forced corners as always which only have 2 choices, you get the whole bottom without really thinking about the numbers. The 8-7 area on the top is not easy, but most of the rest falls once you've got the rails down from attacking the "all white cells used" rule here. Its a fun version of this puzzle type that I really like.
13. Star Battle (German 2008 Qualifier) - Star Battle is an acquired taste. I could point out I made a star battle variant called soduku but I'll neglect that for now. A lot of star battle puzzles involves thinking about how placements will cause the thing to fail (by forcing too many things into a row/column). Even when you cannot firmly place a star (like the either/or choice in R67C3), mark the shaded cells next to it that are touched by both options and therefore invalid. There are sometimes times you can say "I can't fill this row/column unless 2/4/6 stars from this 1/2/3 shapes are in this row/column", almost an innie-outie constraint like on a jigsaw sudoku. Extraneous cells in other rows/columns in those regions can be marked dead. On this puzzle, I think a key square to eliminate was R5C2. Figure out why this is no good from the start. It will build character. Experience with this puzzle type makes R5C2 jump out as easy to eliminate, but the size of the region that contains R5C2 should suggest you want to look at its cells to try to force one or both stars in the shape.
14. Balancing Act (German 2008 Qualifying is discussed here - many USPC examples exist) - Balancing Act is most often a challenge of picking the most critical place where you can force things from, and then running through all possibilities almost blindly. In this one, I love the bottom-most weights. Start high or low, but then work in order from some particular weight. Here, I would have chosen the lower-left weight and started at 8. Instead, I overthought it (by assuming the answer was not the most obvious value of the most obvious starting point) and did 6 and 7 first, then finally got to 8. Anyway, if 8 is there, you can only put 6 2 or 1 3 out. If 7 is there, you can only put 4 2 out. If 6 is there, you can only put 2 1 out. 5 can't be there. 4 can be with 31, nothing else can be there.
The big thing is for any of these pairs to see if there is some partition of the total sum of 36 into the top fulcrum to get the system to balance. I'm working therefore off the sums of the sure divisions I've got (16 from 862), (12 from 813), (13 from 742), (9 from 621). You can do the algebra, or you can notice the right side needs an even weight, or whatever, but most fail. (16 from 862) is ok, and puts a 4 in the far left. Now you just need to figure out how 1357 balance on the right. If this fails (it doesn't), I'd see 12 from 813 cannot work by algebra and therefore know the lower-left weight is not 8. I then go to 7. I'm not trying to be lucky with guesses, much more being sure I try all possibilities and eventually collide face-first with the answer.
I have a love/hate relationship with balancing act. I tend to sense things like I want a big sum on the 2nd fulcrum here from below (so starting at 8 and not at 4 makes sense to me), but at the same time you can often guess from the wrong point and have to go through EVERYTHING to get to the answer. Everything, fortunately, is never 8! or 12! or however many choices, more like 10 things, but catalog what you've tried well at the critical point you start you tests from so that you aren't always guessing and erasing and guessing and erasing and occasionally duplicating past efforts. I've rarely met a balancing act that falls from logic directly, most fall from a combination of deduction and exhaustive coverage of search space.
Ok, that's enough for now (and also where my private messages have run out of puzzles I've written about). I might take requests, or not, but half the fun of puzzles is coming up with your own methods to work through a puzzle so I'm interested to hear how other approaches differ.
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