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## Advanced Picross Solving## Zeros and FillsThe first thing I look for is rows and columns that are completely empty or full. The empty rows and column are easy to find being indicated by a "0". The full rows and columns, however, are identified by knowing how wide or tall the puzzle is and looking for a number of that value in each row and column for a solid fill. Both of these situations should be looked for before moving on. They are very simple freebies to solve for. An example of both can be seen in the picture below.## Multiple Number FillsSecond, I'll look for rows and columns that are completely filled with multiple numbers. Each number group must be separated by at least one space, so the minimum space required can be determined easily. Given the examples below we first determine that the puzzle is ten squares wide. The first row gives a "4, 5" so it is known that there needs to be four consecutive numbers then at least one space followed by five consecutive numbers. The minimum spaced required to fulfill these requirements is:The 1 is to account for the required space. With a minimum space of 10 and having 10 available squares to work with there can only one possible arrangement of marked ons. This row can be completely solved for as shown below. I also gave two more examples below that can also be solved for using the same method. ## EdgesNext, I look for known ons that are on the edge of the puzzle. When a row or column only contains one group of numbers and has an edge square marked on only one possibility remains. In the example below you can see that there is a row ten squares wide with the left most square marked on. The only indicated group of numbers is 4. The only possible way to arrange the 4 to use the edge square is to arrange it like the bottom line in the example picture. Then the remaining squares can be marked as known offs.If the row or column contains more than one group of numbers the same method can be applied to the number on either edge or both. In the second given example only the left side can be determined and the remaining squares cannot be filled in because the 1 is in an unknown location. Only the square directly after the 4 can be determined as a known off. The last example shows the method being applied to both sides of the row at once. Another situation that can occur, as shown below, is that a marked on is one unknown square from the edge. When this happens some squares can still be determined on. This can be done by counting in from the edge like before but not marking any squares before the known on. In the first example with the 4 it can be seen that the unknown space is counted over and left unknown. Then one space must be left unknown to the right of the group of 3 before marking unknowns. This is because the group of 4 can be completed by marking the unknown on either side of the 3 group, leaving the remaining unknown as a known off. Again the following two examples in the picture show other possible uses for this method. Note in the second example the known on is two squares away from the edge. Then again the remaining squares cannot be filled in because of the unknown location of the 1. One more quick note on edges. An edge doesn't have to be the edge of the puzzle. A known off can limit the edge of the workspace. Look at the example below, the first situation has a known off directly next to a known on. Given that the row only contains a group of 2 the row can be solved completely. Everything to the left of the known off must be an off because it cannot be connected to the already determined on. Then at this point the rest of the row can be solved as if the known off was on the edge of the puzzle. The second example in the picture shows how the same method can be applied to multi-numbered rows and columns. The group of 2 that must be on the left side of the 1 can only fit in where the existing known on is located. This will mean that everything to the left of that 2 is a known off because the 2 must appear first. However, the location of the 1 is still unknown with the given information. ## LimitsThe use of limits is probably the most important topic to understand when trying to solve difficult Picross puzzles. First I will show a single number example. In the picture below the puzzle is ten squares wide and the only given information to solve for the row is the number 7. The following four rows (grey filled) show the only four possible arrangements of the 7. In all four possible arrangements of the 7, the middle four squares are always on. This allows for the middle four squares to be mark as known ons. You may have noticed already that you would not need the middle two possibilities to determine this. One could just look at the left and right most limits to see the middle four squares overlap and must be on.Here are a few more examples of limits. Each of the following ons were determined by comparing the left and right most limits. Notice nothing can be determined with the 5. It is one square too short to cause for any overlap. Nothing can be determined on this row without guessing. ## Multiple Number LimitsMultiple number limits is probably the most difficult method to explain. It mixes the multiple number fill method with the limits method. The idea is to consider the limits of every number by treating them as one. Looking at the first example in the picture below we are given "5, 2" in a 10 wide row. If the numbers are added (including minimum spaces) then subtracted from the total work area (10 square wide row) the following is determined:The above math shows that we are short of a multiple number fill by only two squares. This means that the whole group of numbers can be shifted to the right or left by 2 squares. This also means that the gap between any number group can be up to three(2+1). However, this information can still be useful because the 5 is greater than the calculated 2. Known ons can be determined by filling in the row as if the row was the calculated difference shorter (2 squares) but not marking the first calculated difference squares counted. Counting it out only a part of the number group 5 can be determined, but the 2 group is less than or equal to the calculated 2 so it cannot determine any known ons. Do not forget to count an empty square between each number group. The counting starting from the left would be as follows: Note that when finished counting there are two remaining squares. This is because the total of the numbers including spaces is two short of the row length. The number of squares left over should be equal to the difference calculated. The second example pictured below uses the same method with the following math: Counting it out a part of every number group can be determined because every number is greater than the calculated difference of 1. Again do not forget to count an empty square between each number group. The counting starting from the left would be as follows: ## ImpossibilitiesAt this point I try to look for squares that cannot be on or off because they will make the requirements for that row or column impossible. It is a bit of a guess and check process, but it uses simple logic to figure the guess. Then the guess is checked to be possible or not. Given the "2, 3" situation below, no previously discussed method will solve any additional squares on this row. However, this row is completely solvable with the information given. The space separating the ons can be determined as a known off because it is impossible for it to be on. If it was on it would have to be part of a 4 or greater. Seeing as how the row calls for a 2 and 3 it cannot be on. Once it is determined off the edges method can be used to finish up.The "1, 3" situation below is another example of where the previous methods will not work but considering impossibilities will. It can be determined that because to ons are directly next to each other they must belong to a number group of 2 or more. There is only one number group equal to or larger than 2 (the 3) so it must be part of that number group. Then because of the order of the "1, 3" the 1 must be to the left of the 3, and that there must be a space separating them. This is a minimum of two spaces need to the left of the 3 group and there is only two spaces. This leaves only one possible way to position the 1 so that 3 is the the right of it. Once the 1 is in place the edges method can be used to finish off the 3 and the rest of the row. This last example is not completely solvable but a bit of more easy information can be determined. Once again no other method will be able to determine any addition information. Using impossibilities it becomes obvious that the three consecutive ons in the row must be the 3 in the "1, 3, 1". This means both squares on each side can be marked as known offs. The 1s in this case cannot be determined with the given information. ## Puzzle BreakdownThis is not a necessary method given that using the other methods will arrive at the same solution, however it is worth mentioning to save some time while puzzle solving. The idea is to simply break down the puzzle to smaller sections to analyze knowing that the rest is finished. For example the following picture shows a "3, 2, 2" with the 3 and the right most 2 solved for leaving a three square wide unknown area. We can imagine the puzzle as only a 2 in a three wide unknown row. Then use the limits method to determine the middle square as an on. This same situation can be solved using the multiple number limit method; however it is faster to just analyze the small section. The second example is the same concept with different numbers.## Multiple SolutionsThis last topic is a difficult one, but it must be addressed when dealing with randomized puzzles. Every now and then with randomized puzzles (and badly made non-random puzzles) the player will be faced with a multiple solution situation. There is no real method to solving this that I know of other then taking a guess as to which square is an on and trying to finish the puzzle. Sometimes you'll get lucky and guess one of the possible solutions. Inferior Soft's version of Picross will accept any solution to multiple solution puzzles (when the puzzle is randomly generated). To show how a situation like this can occur look at the picture below. It is clear that no method previously discussed will allow for any known ons. This is because this puzzle has six solutions, so nothing can be determined absolute.A small portion of a puzzle can deliver a similar situation as the one above. They usually involve 1s in a 2x2 square arrangement. Where the upper left and lower right could be on or the upper right and the lower left could be on. The following picture is just to show how having six solutions to the previous puzzle is possible. ## Thanks!Thank you for reading! I hope you enjoyed the reading and learned something about Picross. Just remember most puzzles can be solved without guessing. Always try to use logic to solve a Picross puzzle. We all know how just randomly clicking in a game of Minesweeper never seems to work out, the same can be said about Picross. Don't forget to download Inferior Soft's version of Picross available here. Enjoy! | ||||