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## Simon Tatham's Puzzles: Strategies

Started by Den, 2013-Oct-23 14:49

#### Den

My favorites puzzles so far (in alphabetical order):
Black Box
Bridges
Filling
Galaxies
Inertia
Light Up
Loopy
Magnets
Map
Mines
Net
Pattern
Range
Rectangles
Signpost
Slant
Tents
Towers
Untangle

Common Mechanics and Concepts

Common mechanics can be found across many games. Concepts like grids, cells, rows, columns, quota, adjacency, parity, paths, etc. are mentioned and expounded in these strategies.
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#### Den

#1
Slant (aka Slalom in Germany and Gokigen Naname in Japan)

I first discovered Slant in Simon Tatham's Puzzle Collection. The rules are simple, however the strategies seem to be more complex than most other puzzles. Here are the rules for Slant:

QuoteYou have a grid of squares. Your aim is to draw a diagonal line through each square, and choose which way each line slants so that the following conditions are met:

• The diagonal lines never form a loop.
• Any point with a circled number has precisely that many lines meeting at it. (Thus, a 4 is the centre of a cross shape, whereas a zero is the centre of a diamond shape – or rather, a partial diamond shape, because a zero can never appear in the middle of the grid because that would immediately cause a loop.)

A half-finished puzzle of Slant looks something like this: Strategy

These are my strategies and hints for hard mode.

Notation and terminology:

The numbered circles are noted as (N), where N is the value. For example, a 1 on the puzzle is noted as (1). Lines may be denoted using slashes and angled brackets. In the future, I will replace them with screenshots of the game.

Most of the strategies work with circled values next to each other horizontally or vertically unless otherwise noted. For two such adjacent circles, they share two INSIDE cells, and the other cells are OUTSIDE either circle. For diagonally adjacent circles, the cell in between is the SHARED cell.

A strategy is CHAINABLE if a series of unbroken (2) come between the two circles in question, all circles are in the same row or column, and no other circles or blanks come in between them. This doesn't affect the strategy; it merely allows it to extend across the grid as if the (2)s didn't exist between the two circles.

Legend on the images: Black lines are existing lines. Green lines are the correct solution. Red lines are illegal moves (either one or all lines are incorrect).

Strategies:

• These always take precedence over the other strategies:

• (0) on the edge. No lines touch the circle.
• (1) in the corner. The line touches the circle.
• (2) on the edge. All lines touch the circle.
• (3) with one line not touching it. All other lines touch the circle.
• (4) All the lines touch the circle.
• Two (1)s diagonally adjacent and neither on the edge. The line in the shared cell does not touch the (1)s. Like so: 1 / 1.
• Other trivial deductions, like < 2 2 < and any (2) in between that would follow the same line pattern.

• These always work. These will help you begin to fill out the grid on hard difficulty where trivial edge hints are usually missing. Most are chainable.

• Two (1) next to each other with at least one not at the edge. For each (1) not at the edge, neither outside lines touch that (1). The result looks like angled brackets surrounding the (1) pair, like so: < 1 1 >. Chainable.
• Two (3) next to each other. For each (3), both outside lines touch that (3). The result looks like reverse angled brackets (or candy wrapping), like so: > 3 3 <. Chainable.
• (1) at the edge next to (3). For the (3), both outside lines touch that (3). The result looks like so: 1 3 <. Chainable.
• This is the general case of the previous step. For (3) next to (1) not at the edge and that (1) has neither lines on the outside touching it, then both outside lines touch the (3). The result looks like so: < 1 3 <. Chainable.
• This is the converse of the previous step. For (3) next to (1) not at the edge and both outside lines touch (3), then neither lines touch the (1). The result looks like so: < 1 3 <. Chainable.

• These are conditional patterns after part of the grid has been filled.

• The general strategy is to find how many of the inside or outside lines can touch any given circle when influenced by surrounding circles. Usually this involves knowing that only one of the two inside lines touches one of the two neighboring circle. This means the inside lines are parallel, that is, slant the same direction. Another deduction is when it's impossible for both inside lines to touch the same circle.
• Another general strategy is to find and recognize loops and of course trying not to close them, since one of the requirements of the puzzle is that no loops are allowed. The trivial loop is four cells in a square. When lines from three of the four cells form a "U", the last line must point away as to not close the loop.
• Instead of looking for loops, the converse is to look for lines that might be trapped. The idea is that all lines must touch the edge of the screen (result of rule #2, since any loop will cause a line not to touch the edge.) So try to connect the end of a line which has only one exit all the way to the edge. Also beware of connecting this end of the line to a (1) that is not at the edge, which would then close the loop.
• Any number of adjacent (1)s on the edge. All the lines along the edge will slant in the same direction.
• (2) next to (1), where an outside line touches (2). This results in neither outside lines touching (1) and the other outside line not touching (2). The result looks like so: \2 1 > or /2 1 >. Chainable.
• The converse to the previous step. (2) next to (1), where neither outside lines touch the (1) or the (1) is at the edge. This results in exactly one outside line touching the (2), so both outside lines slant the same direction. If the (2) already has an outside line, you can solve the other outside line. Chainable.
• (2) next to (2), where each (2) has an outside line that doesn't touch it. This results in the remaining outside lines touching the (2)s. Chainable.
• The converse to the previous step. (2) next to (2), where each (2) has an outside line that touches it. This results in the remaining outside lines not touching the (2)s. Chainable.
• (2) next to (3), where an outside lines doesn't touch the (2). This results in both outside lines touching (3) and the other outside line touching (2). Chainable.
• The converse to the previous step. (2) next to (3), where both outside lines touch the (3). This results in exactly one outside line touching the (2), so both outside lines slant the same direction. Chainable.

• These are more advanced strategies and situations when all of the above strategies have been exhausted. These may not always tell you how the lines go, but usually tell you how they should NOT go, or maybe only tell you that the lines slant in the same direction, or possibly give hints regarding surrounding circles and cells. Some may be considered trial and error, whereby you imagine how the lines would go and whether the result works or fails.

• (1) diagonally adjacent to (2) which would enclose themselves in a loop if the line in the shared cell would touch both circles. You know this by imagining the remaining lines around the circles by applying the second rule of the game. In this case, the line in the shared cell does not touch the circles. Like so: 1 / 2.
• See if two lines touching a (2) would close off a neighboring loop. Not particularly useful unless there is a line in the third cell not touching the (2). Then you know that the line in the fourth cell touches the (2).
• Similar to the previous step, but more general. See if two adjacent lines touching a (2) would obstruct an adjacent circle, thus causing insufficient lines to touch that circle. For instance, (2) adjacent to (3) would never have both inside lines touching the (2). This is the rationale for the strategies in section 3. Like the previous step, if you have a line in the third cell not touching the (2), then the line in the fourth cell touches the (2).

References and other strategy guides:
http://www.ukpuzzles.org/forum/viewtopic.php?f=10&t=193
http://www.logic-masters.de/Raetselportal/Raetsel/zeigen.php?id=0000XG
http://www.logic-masters.de/Raetselportal/Raetsel/zeigen.php?id=0000XH
http://www.logic-masters.de/Raetselportal/Raetsel/zeigen.php?id=0000XI
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#### Den

#2
Bridges

Even though Bridges (aka Hashi) has circles and lines like many other puzzles, it is actually quite a unique game compared to most puzzles. Here are the rules for Bridges:

Quote
You have a set of islands distributed across the playing area. Each island contains a number. Your aim is to connect the islands together with bridges, in such a way that:

• Bridges run horizontally or vertically.
• The number of bridges terminating at any island is equal to the number written in that island.
• Two bridges may run in parallel between the same two islands, but no more than two may do so.
• No bridge crosses another bridge.
• All the islands are connected together.
There are some configurable alternative modes, which involve changing the parallel-bridge limit to something other than 2, and introducing the additional constraint that no sequence of bridges may form a loop from one island back to the same island. The rules stated above are the default ones.

A half-finished game of Bridges looks something like this: Basic Strategy

Strategy for Max Two Bridges between Islands

The default settings are set to maximum two bridges between a pair of islands. This section is only for two bridges. The next section discusses the advanced strategy for any value as maximum bridges. Nevertheless, learning how to solve two bridges will give you insights to solving three and four bridges.

Notation and terminology:

The numbered circles are noted as (N), where N is the value. For example, a 1 on the puzzle is noted as (1).

Two islands that have the possibility to be linked by a bridge are NEIGHBORS. Two neighboring islands that have been linked by a bridge are BUDDIES.

Opening Moves

I prefer to start at the corners and edges and move inward. It is also common to find inner circles that resemble corners and edges. The special thing about corners and edges is that they have no more than two and three neighbors, respectively.

Specific Moves

• These are hints to solve nodes of specific values under the max two bridges rules.

• An island that has only one neighbor must try to connect the max number of bridges to that neighbor or up to the island's value, whichever is lesser. For max two bridges, this only applies to (1)s and (2)s.
• (1) can never be connected to another (1).
• (2) can never have two bridges connected to the same neighboring (2).
• If (2) has exactly two neighbors and one of them is a (1) or (2), then one bridge must connect to the other neighbor. Furthermore, if both neighbors are (2), then one bridge goes to each neighbor.
• (3) with exactly two neighbors has at least one bridge connected to each neighbor.
• (4) with exactly two neighbors has two bridges connected to both neighbors.
• (5) with exactly three neighbors has at least one bridge connected to each neighbor.
• (6) with exactly three neighbors has two bridges connected to all neighbors.
• (7) always has four neighbors, and has at least one bridge connected to each neighbor. Furthermore, three neighbors should be connected with two bridges.
• (8) has two bridges connected to all four neighbors.
• More general tactics that can apply to nodes of any value.

• For each node, see how many bridges still have to be connected and compare that number to its neighbors. If the numbers mismatch, then one or more bridges should connect to other neighbors. For instance, if two (4)s are neighbors, one of them has two bridges unconnected and the other has one bridge unconnected, then the first (4) must have one bridge connected to a third node.
• One of the rules says all islands must be connected together. So after exhausting all moves from above hints, look at each group of connected islands and try to see if neighboring islands must be connected in order to fulfill this rule.

Isolation

After you have played several games using different maximum number of bridges, you may have realized that part of this game's core is the modulus. In modulus math, large and small numbers can turn out identical solutions. In game terms, as large value islands are adjoined to more bridges, the solution to that island may be solved in identical fashion as if it was a smaller value less the number of adjoined bridges.

Example in max two bridges game. Suppose we have a (3) with exactly two neighbors. Based on the above list of specific moves, we must connect one bridge to each neighbor. Now suppose we have a (5) with three neighbors. One of the neighbors already has two bridges connected to this (5). That means no more bridges can be assigned between (5) and this neighbor. That leaves three more bridges that connect (5) to two neighbors. This is, in fact, the exact same situation as the (3) with two neighbors. So, we can safely connect one bridge from (5) to each of the two remaining neighbors.

While this tactic is helpful in max two bridges mode, it is more crucial in higher bridges modes due to the sheer number of bridges involved. Using this technique hugely reduces the time to solve the harder modes. For instance, reducing a (13) with four neighbors to (9) with three neighbors to a (5) with two neighbors, makes it much more manageable to solve.

More tips:
http://www.conceptispuzzles.com/index.aspx?uri=puzzle/hashi/techniques
http://www.conceptispuzzles.com/index.aspx?uri=puzzle/hashi/techniques
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#### Den

#3
Tents

Terminology:
There are four types of cells. A new game begins with only trees and empty cells. Empty cells can be marked and become tents or clear cells. Tents can only be placed next to trees. Clear a cell when you are certain that it is not a tent.

A quota is the value beside each row or column. The number of tents in that row or column must be equal to the quota.

An occupied tree is adjacent to a tent that is certain to be linked to that tree. In contrast, an unoccupied tree has either no tent adjacent to it, or the tent adjacent to it is not certain to be linked to that tree.

A single cell is an empty cell that has no neighboring empty cells along that row or column.

A group is a single or pair of empty cells along the same row or column such that at most one of those cells is a tent. So these count as a group:
A single cell.
Two neighboring empty cells.
Two empty cells along the same row or column that surround a tree. It counts as an additional group if either of these empty cells touches an unoccupied tree.
A contiguous block of 3 or 4 empty cells along the same row or column counts as an additional group. In general, there are ceiling(N/2) groups in a block of N contiguous cells in the same row or column.
Sometimes two groups may overlap. Depending on the situation, one or more groups may be relevant or not.

Tips:
0. Clear out all cells that aren't adjacent horizontally or vertically to a tree.
0a. Per the rules, clear out all cells around a tent, including diagonally.
1. Clear out all rows and columns marked 0 (zero).
2. For a tree that has only one adjacent empty cell, that cell must be a tent.
3. If an unoccupied tree has exactly two adjacent empty cells at different axes (i.e. one cell to the left or right of the tree and another above or below the tree) and a third cell diagonal to the tree that touches both of these empty cells, the third cell cannot be a tent.
4. For a row or column where the number of groups in that row or column is equal to the value of that row, all single cells must be a tent.
4a. With the same criteria above, if a group is exactly two or three cells, clear the empty cell on a neighboring row or column that touches all cells in the group.
5. For a row or column where the number of groups in that row or column is one less than the number of tents left in that row or column, you may be able to eliminate empty cells on neighboring rows or columns based on the following criteria: If two single cells touch the same cell on a neighboring row or column, clear that cell.
6. The relationships between two adjacent rows or columns may give hints to neighboring rows or columns, especially if said rows or columns are isolated by a row or column that has a quota of zero. One such case is the amount of tents and trees in these two rows should match up. If there are extra trees, then they belong to a third row. If one tree is on a row with zero quota, then that tree has a tent on the other side of zero row than the first 2 rows.
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#### Den

#4
Magnet

Like other similar games with a grid and quota values at the sides, a very important strategy is counting the groups of remaining cells against the quota. For example, if the plus and minus quotas is equal to the cells in that row or column, then all cells must be filled in. In this case, the solution is even more obvious if the quota is an odd number, due to parity.

Understanding the alignment of a magnet is useful to solving the puzzle. A piece of magnet can be aligned in two ways along a row or column:

• Inline so that both halves of a magnet lie along the row or column.
• Askew so that only one half of the magnet is in that row or column.

Another concept is the parity, also known as odd and even. Tying this with alignment, understand that inline magnets count as even and askew magnets count as odd. By observing and matching the parity of the magnets and of the quota is helpful to solving the puzzle. For example, if the quota is odd, and all pieces in the row are inline (even) except for one askew (odd), then we know that the askew magnet must be part of this row. Or if the quota is even, then the askew piece is not part of this row.

If two adjacent cells are both ?, then one is plus and the other minus. This is especially helpful if one of the quota is a 1. Then remaining inline magnets must be crossed out. For other quota values, this can let you know which magnet to cross out. For instance, quotas 2+ and 4- in the same row cannot have 6 contiguous +/- cells in that row (which would make 3+/3-). This would mean 3 of each +/-. So the 6 cells must be split by some crossed out cells.

From the previous example, we can extrapolate that rows or columns with plus/minus quota difference of 2 or more can't be contiguous magnets, must be separated by crossed out cells.

Adjacent rows or columns with same sign quotas at half the max cells in that row or column (rounded down), the signs of magnets must be zig-zagged like a checkerboard pattern.
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#### Den

#5
Loopy

Loopy, aka Slitherlink, is not so much about loops, as it is about sides and corners. The loop is incidental to marking the correct sides and corners, rather than vice versa. With harder puzzles, you have to use recursion (aka guess and use trial and error) by testing how the loop goes around. Some recursion is fine, but shouldn't be used as a 50/50 gamble to map out huge sections of the map when you're totally lost. (Maybe I haven't figured out all the patterns and deductions to solve them without doing this, especially on non-square maps.) Ideally the clues on the map should be all you need. Most of the time recursion works because the converse of it works. In other words, there's another deductive pattern or technique that proves this particular recursion tactic worked. Then apply this pattern or technique to general cases, especially on non-square maps.

General Techniques for Any Map Type

Transitive Rule

Outside/Inside
Because there is an enclosed loop, each cell is either inside or outside the loop. While this interface is not conducive to this strategy, it can still apply and speed up the process of solving the puzzle.

Tips for this strategy:
Inside and outside cells must be separated by the lines of the loop. Conversely, two cells separated by a confirmed line are on opposite sides.
Cells on the same side must reach each other. That means cells on the same side of the loop must not be separated by the lines of the loop.

Odd Ends
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#### Den

#6

Terminology

Paths and Sides
Each number or quota around the grid is at the same time an entrance and an exit of a path to and from another number. Each path may be divided by zero or more mirrors, indicated by a slash in a cell. We call a number's side for every cell between the number and the first mirror when traversing the path starting from that number. Conversely, the other side of the path in this direction are every cell from the last mirror to the other number at the end of the path. In other words, each number owns a side that goes no further than the first mirror closest to the number.

Sides will be important for advanced strategies, like parity.

Moreover, everything between the first mirrors from both sides are shared cells. They have nothing to do with parity, and will count the same towards both number's quota. For shared cells, a Vampire counts as 0, and a Zombie or Ghost counts as +1 for both sides.

Parity
Parity in Undead means zero-sum. When counting just the cells for each side (not the shared cells), Zombie is worth 0 points, Vampire worth +1, and Ghost -1. So two Zombies on the same side is worth the same as one Vampire and one Ghost. In other words, the Vampire and Ghost cancel each other. Conversely, these symbols on the other side of a path is worth the inverse amount. That is, a Zombie on the other side is worth 0 points, a Vampire on the opposite side is worth -1, and a Ghost is worth +1. Thus, the same symbol appearing on both sides will cancel each other. A Vampire on one side is +1 and a Vampire on the other side is -1, thus they cancel each other.

Quota Difference
The shortcut to solving the quotas is to get the difference of the numbers at both ends of the path. In other words, subtract the two numbers. Then using parity, find out how many extra Vampires and Ghosts are at either side. Because of parity concept explained above, Vampires are worth +1 for that side and Ghosts -1; and vice versa if they appear on the other side. Generally, the higher number has more Vampires and fewer Ghosts (not always true, but that's the basic idea.) Use Vampires if you need to raise a side to match the difference and Ghosts to lower a side.

Example 1: Suppose the two sides of a path are 3 and 1. The difference is 2. That could mean: the 3 side has two more Vampires, or the 1 side has two more Ghosts, or the 3 side has a Vampire and the 1 side has a Ghost. This is the simple way to look at it if only two cells are involved.

Example 2: Suppose the two sides of a path are 3 and 1. The difference is 2. But the 3 side already has a Vampire, which is worth +1. That means we need either another Vampire at the 3 side or a Ghost at the 1 side. Both solutions give +1 to the 3 side.

Example 3: Suppose the two sides of a path are 3 and 1. The difference is 2. But the 3 side already has a Ghost, which is worth -1. We need to raise +3 in order to match the difference of 2. That means we need either three more Vampires at the 3 side or three more Ghosts at the 1 side, or some combination in between (i.e. 1 Vampire 2 Ghosts, 2 Vampires 1 Ghost).

2 for 1
A Ghost and a Vampire on the same side or both in shared cells will do two things: cancel each other to make zero parity, and cause two cells to provide only one toward the quota. This is useful if you have just enough cells to fill a quota or you need to eliminate extra cells.

Example 1: Suppose two sides of a path are both 2, and exactly two cells and one mirror in the path. Then you know both cells are Zombies, filling 2 to both quotas. A Vampire or Zombie would give 1 less toward their quotas.

Example 2: Suppose two sides of a path are both 2, and exactly three cells and one mirror in the path. The number of cells in the path is greater than the quota of one or both sides. Thus you know two of those cells must be Vampires and/or Ghosts, and the third cell is Zombie. The Vampire/Ghost will take up two cells to fill 1 quota.

Zombie Count

Another form of parity is odd/even parity (in general mathematics.) This may be useful to find whether there are odd or even number of Zombies on both sides of a path.

• Let E = number of cells along the path that are not shared.
• Let D = the quota difference for this path.
• E - D = Zp, the odd/even parity of Zombies for this path.
• Zp = odd means odd number of Zombies.
• Zp = even means even number of Zombies.

0 (Zero) Quota
A path that involves a 0 (zero) endpoint or quota is the most trivial to solve and will help fill out the cells. There are two simple rules to remember:

• All cells on the 0's side (before the first mirror) are Ghosts.
• All cells beyond the first mirror are Vampires.

No Mirrors

A path with no mirrors is only possible if a path runs straight across the same row or column. The quotas or numbers on the endpoints will be identical. The number of Ghosts on this path is equal to number of cells minus either quota.

Effective Single Mirror
One or more consecutive mirrors with no empty cells in between effectively act as a single mirror, unless the entrance and exit of a possible effective single mirror crosses the same cell. A special case is a path between two endpoints with only a true effective single mirror and no other mirrors (or if all shared cells are Vampires). Then the following are true.

• Let E = the number of non mirror cells along this path.
• Let P and Q be the quotas of the two endpoints of this path.
• 0 <= P <= E and 0 <= Q <= E.
• P and Q may or may not be equal.
• Let T = P + Q, the total quota of both endpoints.
• Let M = 2*E, the maximum total quota along this path.
• E is also the minimum total quota along this path.
• Then E <= T <= M must be true.
• M - T = G + V, the number of non Zombies along this path.
• T - E = Z, the number of Zombies along this path.
• When T = M, then all cells along this path are Zombies. In other words, P = Q = Z = E.
• When T = E, then all cells are non Zombies.

Multi-Mirror
Multi-mirror paths can be reduced to effective single mirror paths, if all shared cells in the path are known. Then subtract values as follows, and solve like an effective single mirror path.

• Let E = cells in the path that are not shared.
• For each Ghost and Zombie in shared cells, subtract 1 (one) from both P and Q.
• Solve like an effective single mirror path, except using these new values.
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#### Den

#7
Turn on Color Code as Letters

2-cell Elimination

3-cell Elimination

Unique Solution

Kempe chains?
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