notepadI’ve seen a lot of variants on sudoku but no single comprehensive list of the recurring ones, so here is such a list!
Along with the fundamental rules themselves, I’ve also included corollaries, which are kind of like secret bonus rules you’ll need for non-trivial puzzles. Those are much wordier, and arguably a form of spoiler, so they’re under their own collapsibles.
Sample puzzles are mostly borrowed from the excellent Genuinely Approachable Sudoku series of puzzles, chosen to be solid introductions to the rule. A few more difficult variants have non-GAS puzzles, or puzzles of my own creation. For non-GAS and not-mine puzzles, I give my solve time to give you an idea of the difficulty β my solve time for the GAS puzzles is generally under 6 minutes.
If you want to find more puzzles, you can try the following sources, which I’ve linked to where possible:
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The Cracking the Cryptic catalogue, which includes both GAS puzzles up to a certain point (after which CtC stopped doing GAS solve videos) as well as their more difficult standard fare. Just bear in mind that newer GAS puzzles will never appear in this catalogue, due to no longer appearing in CtC videos.
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The puzzle list on Logic Masters Deutschland, which has gobs and gobs of puzzles, although it doesn’t necesarily have tags for every variant here. Be aware that people are, um, not especially consistent about their tag use here, so you may or may not find what you want. Also some of the results might not actually be sudoku!
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There’s also the GAS website, which has both collected sets of puzzles and a spreadsheet of their past puzzles. Unfortunately, the only way to expand the collapsed rows in that spreadsheet is to duplicate it, and there’s no web interface, so you’re on your own here.
Definitions
A clue is something in the grid that constrains the solution. The most basic clue is a given digit, which is just a digit in a cell.
A region, at least for the purposes of this page, is a set of cells that must contain every digit of the puzzle exactly once: a row, column, box, or occasionally something else.
Two cells can see one another if they’re both in the same region, or other unique group (such as a killer cage). Any region will do. Three or more cells all see one another if they all share at least one region, pairwise. That means they don’t all have to be in the same region, but every possible pair of cells can see one another.
Two cells are adjacent if they share an edge. Two cells touch if they share either an edge or a corner. The latter isn’t a standard term, but it’ll save me repeating myself a lot on this page.
Two cells are adjacent along a line if the line runs immediately from the center of one cell to the center of the other.
A group of cells is orthogonally connected if you can trace a path between any two of them by walking between adjacent cells. That’s a mouthful, though, so let’s just call such a group contiguous.
The break-in is a series of logical steps that lets you start placing the first digits. The term is used for more difficult puzzles, which often have a single intended solve route, and a large part of the puzzle is simply to find it. Easier puzzles often have numerous possible starting points, so talking about “the” break-in makes less sense.
Bifurcation is a process of guessing some digit and then looking numerous steps ahead to see if it leads to a contradiction. It’s generally frowned upon as a means of solving, and especially frowned upon when a puzzle requires it. That said, there’s no clear line as to what constitutes “bad” bifurcation versus merely understanding consequences, and a lot of general sudoku patterns can be phrased in terms of “if you do this, it breaks the puzzle later”.
Just you and the grid
Classic sudoku
The standard rules are to fill the grid with the digits 1 through 9, such that every digit appears exactly once in each row, column, and box (a bolded 3Γ3 region). With very few exceptions, there should be exactly one way to do this.
Most sudoku puzzles are 9Γ9, and that’s what the rest of this page will assume, but various other sizes are possible (though it’s rare for them to be larger). A sudoku is almost always square, and an NΓN sudoku generally expects you to fill it with the digits from 1 to N.
Sample puzzle: Classic Sudoku by clover! (solution video)
Find more puzzles: GAS β¦ CtC
Basic corollaries
There are various ways to phrase the core rule: each digit appears in each group (row/column/box); no digit repeats in a group; each group contains a full set of digits. Since the size of a region is equal to the number of digits you have to fit into them, all of these ideas are equivalent.
Since the values of the numbers are never used, there’s no actual arithmetic involved in a standard sudoku. The numbers could be replaced by letters, colors, star signs, fruit, or whatever else, and the puzzle would still solve exactly the same way. (This is not true for many variant rules.)
Advanced corollaries
There are endless resources regarding how to solve classic sudokus, and I won’t try to recreate them all here within this tiny aside.
The two biggest questions in sudoku are “where in this region can digit X go?” and “what digits can go in this cell?”, and to that end, the two biggest skills are probably scanning and skimming. I think “scanning” is a general term but I definitely made up “skimming” just now.
Scanning is about answering the first question: looking for where else X is in the grid, what that eliminates from some region, and what it leaves. Skimming is about answering the second question: whittling the digits 123456789 down to only the ones a particular cell can’t already see.
Pointing, pairs, and x-wings are also incredibly common, but more advanced classic tactics don’t tend to appear in variant sudoku very often β the hard part is getting your foot in the door with the variant rules, and then the actual sudoku-ing part is often fairly easy.
X sudoku
Each digit must also appear exactly once along each main diagonal, i.e., each diagonal is an extra region. Some puzzles only apply this constraint along one diagonal, in which case it’s not so much an X as a diagonal sudoku.
Sample puzzle: X-Sudoku by clover! (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
If a digit is in the top-left box but not on the diagonal, and that same digit is in the bottom-right box but not on the diagonal, then in the middle box, it must be on the diagonal. The same idea works for various other combinations of boxes.
Advanced corollaries
In a full X sudoku, the four corner cells can all mutually see each other, so they contain a set of four distinct digits. The same is true for the four cells two and three steps in. In the diagram, these are colored in red, green, and blue, respectively.
Hyper sudoku
Also called windoku. Adds four extra 3Γ3 regions, which must also contain a full set of digits each. (In my experience, extra regions are usually shaded; the sample puzzle uses cages, and could equivalently be treated as a killer sudoku with no totals given.)
Sample puzzle: Windoku by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Advanced corollaries
Having all four hyper regions present actually divides the grid into a full set of nine new regions, as shown in the diagram.
Consider the top two hyper regions. They span three rows, which among them contain three full sets of digits. But the hyper regions themselves contain two full sets of digits, so the leftover cells (shown in green) must contain one more full set of digits among them, and that’s the definition of a region.
Repeat this for every pair of adjacent hyper regions, and you get eight regions in total. By similar reasoning, there must be nine leftover cells (shown in purple), and they must contain their own full set of digits, making them a ninth region.
Jigsaw sudoku
Also called irregular sudoku. Instead of boxes, bold lines are used to draw out irregularly shaped regions, which fit together like a jigsaw. Each region still consists of 9 cells.
A prime-sized sudoku, such as 7Γ7, must be a jigsaw sudoku.
Sample puzzle: Irregular by clover! (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
The regions can interact in all kinds of subtle and interesting ways that aren’t possible with a standard grid. You may have to carefully consider how all of the approaches you usually take might apply to each particular puzzle.
Advanced corollaries
Because the regions tend to have a lot of small bits jutting out, it’s very helpful to think of jigsaw sudoku in terms of “innies” and “outies”.
In the diagram, the bottom-left region takes up most of the row. Its outies are the cells that poke out of the row, highlighted in red. The innies are cells from other regions that poke into the row, highlighted in green. Because blue + red is a full set of digits, and blue + green is also a full set of digits, green and red must contain exactly the same digits.
The same idea can be extended across multiple rows and may help you find useful relationships between otherwise distant pairs or groups of cells.
Disjoint groups
Two cells in the same relative positions within their respective boxes cannot contain the same digit.
This is clearly the worst name of anything there has ever been (isn’t sudoku all about disjoint groups in the first place?) so we should really call it antipositional or something.
Doesn’t pair well with jigsaw sudoku.
Sample puzzle: Come Out and Play by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
Equivalently: the top-left corner of every box is a region, the top-middle of every box is another region, etc.
Advanced corollaries
This rule is less constraining than it sounds, since every digit already sees four of the others in its “disjoint region” anyway. The hard part is often just noticing the impact from the other side of the grid.
Gattai sudoku
There are multiple independent puzzles, but their grids overlap somehow, and they share the overlapping cells. They may all need to be solved together, or solving one may provide digits necessary to make progress in another. The geometry may even be ambiguous and need to be specified in the rules.
Samurai is a specific variant consisting of five 9Γ9 sudoku arranged like a shuriken, with four outer puzzles each sharing a corner box with the same central puzzle.
Sample puzzle: Roll Sudoku by clover! (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Advanced corollaries
If both puzzles are the same size, then wherever two regions overlap, the parts of those regions outside the overlap must contain the same digits. In the sample puzzle, the three digits at the far top of every column must be the same as the three digits at the bottom (though not necessarily in the same order).
Chaos construction
As with jigsaw sudoku, but the regions are not given and must be determined.
This rule is only possible with some other strong rule enforcing where the region boundaries must go, and the resulting puzzles tend to be more difficult.
The sample puzzle isn’t GAS, also uses thermos (described below), and in fact is by a constructor known for mind-bending puzzles, but it’s still the simplest example I could find! My time is 10Β½ minutes.
For SudokuPad, you may want to enable the line tool in the settings, which will let you draw borders. Alternatively, you could just use colors to mark regions.
Sample puzzle: Seven by Phistomefel (solution video)
Deconstruction
The usual rules still apply, but the grid is larger than 9Γ9 (typically 11Γ11), and the solver must figure out how to place nine 3Γ3 boxes within it while still obeying other rules. Cells outside any box are generally ignored by other clues.
I swear I’ve also seen this called BYOB, for “bring your own boxes”, but now I can find no evidence of that. Hm.
This rule is only possible with some other strong rule enforcing where the region boundaries must go, and the resulting puzzles tend to be more difficult.
The sample puzzle isn’t GAS, also uses killer cages (described below), and is moderately difficult. However, it’s the purest example of the form, and given the name, it might be the first such puzzle? My time is 25 minutes.
Also, I have to mention my absolute favorite deconstruction, which combines several rules with a grid containing… just about nothing.
It’s also possible to combine this with chaos construction, to create chaos deconstruction, but this is such a ludicrous combination that I’m only aware of it being done once.
Sample puzzle: Deconstruction by Jay Dyer (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
Once you have nine cells in a row or column that you know are part of boxes, the other two must be empty.
Advanced corollaries
11Γ11 is the usual size because being only two rows+columns bigger than 9Γ9, while requiring 3Γ3 boxes to be placed within it, creates nine seed cells β the final boxes must contain the cells highlighted in the diagram, one cell per box.
You can prove this to yourself with a sliding window argument, similar to why a long renban must contain “middle digits”. The top-left box can go no further left than the first column, and it can’t go any further right than the third column, because it has to leave room for two other boxes. (Those boxes can’t simply move down out of the way for the same reason, looking downwards, and so on.) Either way it must contain some cells from the third column, and by the same argument it must contain cells from the third row, so it must contain R3C3.
If the grid were 12Γ12 or larger β i.e., the extra space could fit an extra box β then there’d be enough free space to have only two boxes in a row, and getting started would be much more difficult.
There’s also a whole lot of reasoning you can do mid-puzzle about which cells have to be part of which boxes, but this margin is too small to contain it.
Fog
A rare digital-only rule. Some of the grid is shrouded in fog, obscuring any clues underneath. Placing a correct digit will clear the fog from that cell and all cells it touches.
Also called fog of war, referring to the visibility effect in real-time strategy games, where you can only see parts of the map within a limited radius of your own units.
I’ve never seen a pure-fog puzzle, and I’d bet it’s impossible to construct one, so the sample puzzle also uses kropki dots (described below). Geometry tends to play a large part in fog puzzles; line-based clues and oddball custom rules are common.
Sample puzzle: Fog of Kropki by Bill Murphy (solution video)
Find more puzzles: GAS β¦ CtC
Basic corollaries
If you place a digit and the fog does not clear, you immediately know you’re wrong, which changes the experience somewhat.
The first fog puzzles had a limited number of lives and froze the grid if you were wrong three times, but this has not been ported to SudokuPad, probably because it’s not fun. Just consider it a free hint that you’re on the wrong track.
It’s also possible to place a correct digit for the wrong reasons, which might reveal clues you weren’t yet meant to see.
Placement constraints
Antiknight
Two cells a (chess) knight’s move apart cannot contain the same digit.
Sample puzzle: Antiknight by clover! (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
A knight moves one cell in one direction and two cells in an orthogonal direction, in either order.
The diagram shows the range of a knight’s move. The blue cell sees the yellow cells by normal sudoku rules, and it sees the red cells by antiknight rules.
Advanced corollaries
Antiknight has different effects on a neighboring box depending on the position of a digit. Two examples are shown.
There are several kinds of elimination pattern unique to antiknight, where a digit is restricted to two or three cells within a box, and it can then be eliminated from an unusual position in a neighboring box, because some cell sees all the possible candidates by a combination of sudoku and antiknight rules.
One very common example is shown in green: if a digit is restricted to being within a domino, then it can be eliminated from a parallel domino two cells away. No matter where the digit appears in one domino, it’ll be able to see both cells of the other domino.
Coloring can be helpful with more difficult antiknight puzzles, though it can also turn into a sprawling mess.
Antiking
Two cells a (chess) king’s move apart cannot contain the same digit. Equivalently, no digit may touch itself.
Sample puzzle: Pyramid Scheme by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
Because this rule only affects cells that touch, it has no effect on the center cell of a box, nor on the cells in the corners of the grid. Those cells only touch other cells that are already in the same box.
Advanced corollaries
The strongest effect antiking has on a box is on the digits in the middle of a box’s edge, as shown. A digit in the blue cell can be eliminated from five cells in the neighboring box.
It’s a relatively weak rule on its own.
Clone
Shaded regions must contain the same digits in the same arrangement. There may be multiple sets of such regions, in which case they’re usually different shapes and different colors.
Sample puzzle: Duplication Glitch by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
Any particular cell within the clone shares the eliminations of everywhere it appears.
Advanced corollaries
Clones don’t have to contain distinct digits, but unless the clones are all aligned to boxes the same way, they usually will.
In that case, it can be helpful to identify which digits a clone contains, by eliminating those digits that see every cell of the clone.
Palindrome
Gray lines read the same way in both directions, i.e. the digits at both ends are the same, the next digits inward are the same, and so on.
Sample puzzle: Palindrome by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
Because pairs of digits are known to be identical, coloring them can be helpful. One pitfall: you might color two pairs of digits with separate colors when they’re actually all the same digit.
Advanced corollaries
Eliminations on one digit apply to the digit on the other end of the line, and vice versa, which makes them twice as powerful.
This also applies to whole segments of the line at a time, when they happen to fall within a single region.
Quadruples
Digits appearing in large circles (on grid intersections) must appear in the surrounding four cells. If a digit appears in a circle more than once, it appears in the surrounding cells that many times.
Sample puzzle: Quadruples by clover! (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
The same digit cannot appear more than twice in a circle. (The four surrounding cells span two rows, and each row can only have a particular digit once.)
If a circle contains four digits, the surrounding four cells must contain exactly those four digits and nothing else.
If a circle is contained within a single box, its digits can’t appear in the cells in that box that don’t touch the circle.
Advanced corollaries
If a digit appears twice in a circle, the two instances must be placed diagonally.
If two circles are horizontally or vertically aligned, and both share a digit N, you can eliminate N from every cell that doesn’t touch the circles in the two rows/columns shared by the circles.
This is typical sudoku reasoning: you have two N’s to fit into two rows/columns, you already know roughly where they go, and so there can’t be any more N’s elsewhere in those rows/columns.
Pencilmarks
Pencilmarks indicate the only possible candidates for their cells.
A given digit is really like a single pencilmark, if you think about it.
SudokuPad is a little glitchy about these β if you restart the puzzle, it either clears the pencilmarks, or forgets that they are pencilmarks and doesn’t erase them when you enter a digit.
Sample puzzle: The Chocolate Teapots by Bill Murphy (solution video)
Find more puzzles: GAS β¦ CtC
There are no corollaries. They act exactly like your own pencilmarks.
Size
Now the actual values of the digits begin to matter. The corollaries from here on assume a standard 9Γ9 sudoku; they may need to be adjusted, or discarded outright, for puzzles of different sizes or that use a different set of digits.
Greater/less
A chevron between two digits indicates which is larger.
If you could never remember which way around < and > are: the pointy end, which is smaller, points towards the smaller value.
This is equivalent to (and harder to read than) a thermo that only moves orthogonally, so thermos are far more common.
Sample puzzle: Greater Than by clover! (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
1 is not greater than anything. 9 is not less than anything.
Advanced corollaries
A chain of several inequality signs is more powerful. For example, if A > B > C, then all three digits must be different (even if A can’t see C), and A cannot be 1 or 2, because it has to be greater than two different digits.
Since this rule is a weaker form of thermo, thermo strategies are effective.
Fortress
Gray cells contain digits greater than all adjacent white cells. The visual is one of ramparts, which are always higher than their ground-level neighbors.
Sometimes the rule is inverted (shaded digits are less than adjacent white cells), and then it’s called something more like minimum/maximum. In the absence of other rules, this is completely identical, but with the digits swapped around (1 for 9, etc.)
Sample puzzle: Fortress by clover! (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
A white cell with any gray neighbors cannot contain a 9.
A gray cell with any white neighbors cannot contain a 1.
Advanced corollaries
More generally, you can count the number of distinct neighbors to (somewhat) narrow down the range of an unknown cell. If a gray cell has three white neighbors, and they all see each other, then it must contain at least a 4, since it has to be bigger than three different digits.
Thermo
Digits strictly increase along a thermometer, starting from the bulb. Some common subvariants:
Slow thermo β Digits do not decrease along a thermometer, i.e. they can remain equal.
Fast thermo β Digits increase by at least N along a thermometer. N is usually 2 or 3.
Ambiguous thermo β The bulb is not given. Digits still strictly increase, but the direction must be determined.
Sample puzzle: GG by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
1 cannot appear anywhere on a thermo, except on a bulb.
9 cannot appear anywhere on a thermo, except on the tip.
Advanced corollaries
Longer thermos can have their cells narrowed down to 10 - N candidates, where N is the length of the thermo. For example, a thermo spanning 7 cells can be pencilmarked as 123, 234, 345, … 789.
Between
Digits along a line are strictly between the digits in the circles at either end.
Sample puzzle: Between Your Ears by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
One circle must be smaller than everything on the line, and the other must be bigger. Therefore, a line can never have a 1 or a 9 on it.
Digits in circles cannot appear on the line between them, and vice versa.
Two connected circles can never contain the same digit or two consecutive digits, or there’d be nothing left to put on the line.
Advanced corollaries
Counting the number of distinct digits along a line may be helpful, especially if you already know one circled digit.
A line with six digits on it can only have 12 and 89 at the ends. A line with seven digits can only have 1 and 9!
Skyscraper
Consider each digit in the grid as a building of that height. Clues outside the grid give the number of buildings visible along that row/column, starting from that clue, where taller buildings obscure shorter ones.
Additionally, all of your Elemental HERO monsters gain 1000 ATK.
This is its own genre of pencil puzzle. It’s also one of the puzzle types in [Simon Tatham’s Portable Puzzle Collection](https://www.chiark.greenend.org.uk/~sgtatham/puzzles/), under the name Towers.
Sample puzzle: Skyscrapers by Bill Murphy (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Advanced corollaries
A clue of 9 can only be looking at all nine digits in order.
A clue of 1 must have 9 in the first cell. (Any other clue cannot have 9 in the first cell.)
A clue of 2 cannot have 8 in the section cell β the first cell would be smaller because it can’t be 9, and the 9 would be further along, making a total of 3 visible buildings.
Beyond that, hell, I don’t know. This is just a weird rule and I’ve always kind of bumbled my way through it.
Parity
Even/odd
A cell with a gray square must have an even digit. A cell with a gray circle must contain an odd digit. (Mnemonic: A square has an even number of sides; a circle has an odd number of sides, i.e., one.)
Sample puzzle: I Feel Better by Bill Murphy (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Advanced corollaries
Whether a digit is even or odd is sometimes called its parity.
As with many parity clues, it can be helpful to count (perhaps by coloring) how many even/odd digits are in a region, since there can’t be more than four even digits or five odd digits.
Parity
Digits on the same line have the same parity β i.e., they’re all even or all odd. I’m not aware of a standard color for the line.
Sample puzzle: Ganger by Bill Murphy (solution video)
Find more puzzles: GAS β¦ CtC
Advanced corollaries
Counting and coloring are pretty solid bets here.
Alternating parity
Along a yellow line, digits alternate between odd and even.
I thought this variant was much more common than it seems to be, to the point that I’ve been referring to it as simply “parity lines” β I mean, come on, it’s even the yellow line clue! β and yet I couldn’t find any GAS puzzles using it. The sample puzzle is my own creation and also uses black kropki dots (described below). It’s still around GAS difficulty, though.
Sample puzzle: cheetah by eevee (no solution video, sorry!)
Find more puzzles: GAS β¦ CtC
Advanced corollaries
Again, coloring is very helpful, especially before you manage to pin down which cells are odd and which are even.
Basic arithmetic
Kropki dots
A white dot separates two digits that are consecutive. A black dot separates two digits such that one is twice the other.
A “full kropki” puzzle means that all possible dots are shown, so if two adjacent digits have no dot between them, then neither condition is true: they aren’t consecutive, and one isn’t twice the other. This is also called a negative constraint, since it imposes a condition everywhere the clue doesn’t appear.
Occasionally, a number is written in the dot, indicating that the ratio or difference is something other than 2 or 1.
The word “kropki” is Polish for “dots”, so I suppose calling them “kropki dots” is redundant.
Sample puzzle: Aaaaaaaaah by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
If a chain of adjacent cells are all connected by white dots (and all see one another), then they form a sequence of consecutive digits. The cells in the middle then can’t contain 1 or 9, because you’d need two different digits consecutive with 1 or 9.
The only possible black dot pairs are 1/2, 2/4, 4/8, and 3/6. A black dot can never touch a 5, 7, or 9.
If a cell has both a black and a white dot, and its two dotted neighbors see each other, it cannot contain a 1. This is because 1 is only consecutive to 2, and also would need to double to 2.
Advanced corollaries
When a pair of digits is separated by a white dot, one is odd and one is even. So for example, if a region contains four white dots (with no cells shared, so eight in total), the ninth cell must contain an odd digit.
Two digits separated by a black dot always add up to a multiple of three β that’s adding some number to twice itself, producing thrice itself.
A very common pattern is a chain of three cells joined by black dots, where all three see each other. When this happens, the only possibilities are 1/2/4 and 2/4/8, so the center cell is either 2 or 4, and both 2 and 4 appear somewhere on the chain.
Nonconsecutive
No two adjacent cells may contain consecutive digits, anywhere in the puzzle.
This global constraint, plus antiking and antiknight, were enough to create the original “miracle sudoku“, in which the grid has extremely few clues.
Sample puzzle: Nonconsecutive by clover! (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
This is equivalent to a full white kropki puzzle, with no dots given. Sometimes it might be presented that way.
Advanced corollaries
This rule seems especially prone to roping, although it’s not guaranteed. If you don’t know what that means, don’t worry about it β the knowledge ruined the variant for me!
German whispers
Along a green line, two adjacent digits must have a difference of at least 5.
The name derives from a puzzle made for a world championship held in Germany, which was a riff on another puzzle called “Chinese whispers”, where adjacent digits along a line had a difference of at most 2. The constructor didn’t know this was another name for the game of telephone β hence, digits change only slightly as you move along the line β and simply changed the name of the country.
Sample puzzle: German Whispers by clover! (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
5 can never appear on a German whispers line, because no other digit is 5 away from it.
A 4 can only be next to a 9, and a 6 can only be next to a 1. Therefore, 4 and 6 can’t appear in the “middle” of a German whispers line, if their two neighbors can see each other.
Advanced corollaries
Digits along a German whispers line alternate between low (1234) and high (6789), because a step of 5 or more will always cross over the middle digit, 5. The first step in a German whispers puzzle is often to color the biggest lines in two alternating colors to encode this idea.
Note that there can’t be more than four cells of a color in any region, since the colors represent sets of only four digits.
Dutch whispers
Along an orange line, two adjacent digits must have a difference of at least 4.
The name is presumably a further riff on “German whispers”. It might trace back to the puzzle “Dutch Whispers” by Aad van de Wetering (a Dutch constructor), solved on Cracking the Cryptic on Oct 6, 2021, although the SudokuPad version of the puzzle has disappeared.
Sample puzzle: Dutch Whispers by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC
Basic corollaries
These are more difficult to work with than German whispers, as every digit can appear on a Dutch whispers line, and no digit only has one possible neighbor.
Advanced corollaries
Digits do not strictly alternate between low and high like German whispers lines, because it’s possible for the sequence to have a “hiccup” by passing through 1β5β9. That said, if 5 can be pinned down or ruled out, segments of a Dutch whispers line may still alternate.
Renban
Each magenta line contains a set of consecutive digits, in some order.
“Renban”, or ι£ηͺ, is Japanese for “consecutive number”.
Sample puzzle: Renban by clover! (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
Digits cannot repeat on renban lines.
A renban line cannot be longer than 9 cells.
Advanced corollaries
Imagine the set of available digits as a strip, and the digits on some renban line as a window sliding along it.
Then a renban line of 5 or more cells must have a 5 on it, somewhere, because every possible window includes the 5. The far extremes are 12345 and 56789, which both include a 5.
Similarly, a renban line of 6 or more cells must include all of 456, a renban line of 7 or more cells must include all of 34567, and so on.
Because digits don’t repeat, any tactics for handling overlapping regions could be helpful.
Nabner
A sage line contains no repeats and no consecutive digits.
It’s usually called a gold line, but, I mean, look at that. That’s not gold, right? I even used a tool to find names for colors (restricted to just Wikipedia color names) and it said sage and I agree that is deadass sage. It’s not just me!
“Nabner” is “renban” backwards. Appropriately, it’s also called anti-renban.
Sample puzzle: Li’l Nabner by Philip Newman (no solution video, sorry!)
Find more puzzles: GAS β¦ CtC
Basic corollaries
The longest a nabner line can be is 5 cells: the odd digits.
Advanced corollaries
A 4-length nabner line contains a digit from 123, a digit from 345, a digit from 567, and a digit from 789.
Entropic
Along a peach line, any set of three adjacent digits contains one digit from 123, one digit from 456, and one digit from 789.
“Entropic” refers to the notion of entropy, a sort of measure of disorder.
Sample puzzle: Entropic Lines by clover! (solution video)
Find more puzzles: GAS β¦ CtC
Basic corollaries
A digit may not repeat within two steps along the line.
Advanced corollaries
The digits along a line will alternate between the three groups in a consistent cycle. This is because any three digits must be of the form ABC, and a fourth digit would be part of a triplet already containing B and C, so it could only be A again.
Coloring can be very helpful.
Modular
Along a teal line, any set of three adjacent digits are all different, modulo 3.
That means they all have different remainders after dividing by 3; for a 9Γ9 sudoku, the groups are 147, 258, and 369, which are also the columns on a numeric keypad.
The sample puzzle is 6Γ6, so the groups are just 14, 25, and 36 instead. (The puzzle is actually from a pack of four, where the same grid can be solved four different ways using different rules for the lines: nabner, zipper, whispers, and modular.)
Sample puzzle: B1G3: Crossing the Streams by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC
Basic corollaries
In the absence of other clues, this functions exactly the same as entropic lines, just with the digits rearranged.
Advanced corollaries
Just like with entropic lines, the digits along a line will alternate between the three groups in a consistent cycle, so coloring can be very helpful.
Sums
A number of clue types require that some set of digits add up to some total (which may or may not be given). The digits don’t always have to be unique, but either the particular rule or the shape of the grid often force them to be.
This is such a broad topic that I’m going to give the very concept of sudoku sums its own corollaries.
Basic corollaries
Some totals can only be made in a few ways, depending on how many digits you need to use. It is indispensible to know the most constrained arrangements.
| One way | Two ways | Three ways | |
|---|---|---|---|
| Two digits | 3 β 12 4 β 13 16 β 79 17 β 89 |
5 β 14 or 23 6 β 15 or 24 14 β 59 or 68 15 β 69 or 78 |
7 β 16 or 25 or 34 8 β 17 or 26 or 35 12 β 39 or 48 or 57 13 β 49 or 58 or 67 |
| Three digits | 6 β 123 7 β 124 23 β 689 24 β 789 |
8 β 125 or 134 22 β 589 or 679 |
9 β 126 or 135 or 234 21 β 489 or 579 or 678 |
| Four digits | 10 β 1234 11 β 1235 29 β 5789 30 β 6789 |
12 β 1236 or 1245 28 β 4789 or 5689 |
(not worth remembering) |
| Five digits | 15 β 12345 16 β 12346 34 β 46789 35 β 56789 |
17 β 12347 or 12356 33 β 36789 or 45789 |
(not worth remembering) |
Fear not! Most of these are just the smallest/largest digits, plus a little wiggling.
The further away from these extremes a total is, the more possible ways it can be reached. For example, for four digits, 10 and 30 are the least and greatest totals you can make (and both only one way). Halfway between them is 20, which can be made a staggering twelve ways, the most of any number.
These are the ones I know more or less offhand. A puzzle that expects you to find patterns beyond this is difficult enough that you’re probably not reading this page for guidance. At that point you might want to use the killer calculator/tracker built into something like SudokuPad, consult tables of combinations, or just work out all the combinations yourself as notes on the grid.
Advanced corollaries
You should be familiar with the “triangular numbers”, which are the sums of the numbers from 1 to N. The first few are 1, 3, 6, 10, and 15, which are, not coincidentally, the smallest total you can make out of one, two, three, four, and five distinct digits.
The ninth triangular number, the sum of the numbers from 1 to 9, is 45. nine triangular numbers (the sums of the numbers from 1 to N), which tell you the minimum that N distinct digits can possibly add up to, or at least know that the sum of the numbers from 1 to 9 is 45. That means every row, column, box, and any other 9-cell region must also sum to 45. (This is sometimes called “the secret”, after a sort of running joke on Cracking the Cryptic.)
It’s very often helpful to find groups of summed digits that share (or mostly share) the same region(s), add their totals, and do some arithmetic with the total of the regions (which will be 45 times the number of regions).
XV pairs
An X separates two digits that add to 10. A V separates two digits that add to 5.
A “full XV” puzzle has a negative constraint similar to kropki: if two adjacent digits do not have an X or V between them, they do not sum to either 5 or 10.
Occasionally, an XV may be used, to separate two digits that add to 15.
Generally incompatible with greater/less, for obvious reasons.
Sample puzzle: XV Pairs by clover! (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
5 cannot be part of an X cell.
These are really just special cases of killer cages (see below), so strategies for those may be helpful.
Advanced corollaries
There are only two ways to make 5 with two digits: 1+4 and 2+3. If a puzzle has a number of V pairs that can partially see one another, it may be helpful to color them.
An X always spans one low (1234) and one high (6789) digit.
Killer
The grid contains “cages” of any number of cells, indicated by dotted lines. Digits cannot repeat within a cage, and the digits within a cage must add up to the total given in that cage’s upper-left corner.
Occasionally, cages may have looser constraints like “> 5”, “even”, or nothing at all. A common subvariant is to not give any of the cage totals, but specify that they must all be one of a handful of specific numbers, or must all be different.
The name was coined by The Times in their article introducing it to the paper in 2005, and while they don’t explicitly explain why, it seems likely to be derived from the included quip from inventor Tetsuya Nishio, that logic is “a knife that kills”.
Sample puzzle: Sequence Brick by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
As noted above, there are several combinations of size and total that can only be made in one way.
Even if there are several ways to make a total, you might still glean some information. For example, 8 can be made from three digits in two ways, but both ways require a 1, so such a cage must contain a 1 somewhere.
Killer cages are exceptionally prone to creating pairs/triples/etc., since you’ll often work out which digits appear in a cage before you know how they’re arranged.
Advanced corollaries
Because digits don’t repeat, any tactics for handling overlapping regions could be helpful.
Little killer
Clues outside the grid (with diagonal arrows) give the sum of the digits along the indicated diagonal.
This variant can be a little fiendish; unlike regular killer and most other sum-type rules, the digits need not be unique (and usually aren’t).
Sample puzzle: The Most Gorgeous Day In History by Bill Murphy (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
If the diagonal only passes through a single box, the digits are unique, and killer combinations are useful.
It may help to color or draw a line through the diagonal to keep track of which digits are being added together.
Advanced corollaries
Little killer totals often have very few degrees of freedom, which can highly restrict the digits along their diagonal.
For example, if a little killer clue requires that two digits add up to 16, you have two degrees of freedom, because the most two digits can possibly make is 18 = 9 + 9. If you start from that total, you have to subtract 2 in some combination from the two digits in order to make the total, meaning the smallest either digit can possibly be is 9 - 2 = 7.
Arrow
Digits along an arrow must add up to the total given in the attached circle. Subvariants include:
Average arrow β The circled digit is the average of the digits on the arrow, instead of the sum.
Product arrow β The circled digit is the product of the digits on the arrow, instead of the sum.
Pill arrow β Instead of circles, some or all arrows connect to a “pill” that spans two or three cells, for spelling out a larger total. Not sure if there’s a real name for this.
Double arrow β Lines have circles at both ends, resembling between lines. The sum of the digits on the line equals the sum of the two digits in the attached circles. So there aren’t any actual arrows.
Split pea β Like double arrow, but the sum of the digits on the line is a two-digit number, and the digits of that sum appear in the circles in some order. Circles and lines are usually drawn in green.
Sample puzzle: Blades of Glory by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
Unless it only passes through one cell, an arrow can never have a 9 on it. Similarly, unless its arrow(s) only pass through one cell each, a circle can never contain a 1.
Advanced corollaries
Digits may repeat on an arrow, which makes them somewhat trickier. On the other hand, unless the puzzle uses pills, the total can never exceed 9. Also, it’s very common for arrows to remain contained within a single region, in which case the digits must be distinct.
In harder puzzles, an arrow that spans three cells in the same region is valuable, because the arrow must have at least 1 + 2 + 3, so the circle can only contain 6, 7, 8, or 9.
Zipper
Along a lavender line, pairs of digits equidistant from the center (generally marked with a dot) all have the same sum. If the line has an odd length, that sum appears in the middle cell.
Sample puzzle: Z is for Zipper by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC
Basic corollaries
A 9 cannot appear on a zipper line, except on the center dot.
If two summed digits see each other, killer combinations are helpful.
Region sum
Blue lines are split into segments by box borders, and each segment of the same line must have the same sum. If combined with jigsaw sudoku, then the jigsaw regions’ edges split the lines, although it’s not clear what it means for a line to clip a corner.
(The sample puzzle also makes simple use of white kropki dots.)
Sample puzzle: Slip Away by Bill Murphy (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
The digits on each segment are distinct, so killer combinations are useful.
If a line segment only passes through one cell, that cell contains the line’s sum. If this happens more than once on the same line, all such cells contain the same digit.
X-Sums
Clues outside the grid give the sum of the first X digits along that row or column (starting from the clue), where X is the first such digit.
(I’d never seen circles used for X-Sums before, but all the GAS puzzles seem to use them!)
Sample puzzle: Where 4 Art Thou? by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
The summed digits are all in the same row or column, so killer combinations are useful, although you don’t know how many digits you’re adding.
Most of the digits can be freely rearranged, but not the first one!
Advanced corollaries
The presence of the count of digits within the sum does some strange things to the possibilities. An 11 clue can never be the sum of four digits, because the only way to do that is 1235, which doesn’t include 4!
Sandwich
Clues outside the grid give the sum of the digits between the 1 and 9 in that row or column. (The 1 and 9 are not included.)
Sample puzzle: The Sum of Monte Cristo by Philip Newman (solution video)
Find more puzzles: GAS β¦ CtC β¦ LMD
Basic corollaries
A zero clue means the 1 and 9 are adjacent.
A sandwich clue can never be 1.
Sandwich messes with your killer muscle memory a little, since you don’t know how many digits are in the sum, and 1 and 9 are never part of it.
Advanced corollaries
For large clues, adding the 1 and 9 to the clue and subtracting from 45 will give the sum of the digits outside the 1 and 9. For example, a 31 clue can only have digits adding to 4 on the outside, and since the 1 is not available, the only way to make that is to have a lone 4, which limits the outermost cells to 149.
Working out how many digits could make the sum (a minimum, maximum, or both) is often important.
Counting circles
A digit in a circle indicates how many circles contain that digit.
This is a somewhat trickier rule, and the sample puzzle isn’t GAS, but it’s only moderately difficult.
Sample puzzle: Clockwork by shamShaman (solution video)
Find more puzzles: GAS β¦ CtC
Basic corollaries
Hmm, that’s odd. This rule is about counting. Why’s it listed with all the sum clues?
Advanced corollaries
If N is circled, then it accounts for N circles. Therefore, the sum of the unique circled digits equals the total number of circles. These digits don’t repeat, so they form a killer combination. The catch is that you don’t know how many digits there are.
If any single region contains N circles, there must be at least N circled digits.
If any single region contains no circles, then 9 is never circled.
10-lines
Each line can be divided into one or more segments, each of which sums to 10.
The target sum may also be something else, in which case the more general name is N-lines.
The ambiguity in this rule makes it inherently more difficult, and there are no GAS puzzles that make use of it. I’m including it anyway because I’ve seen it enough that it’s become lodged in my brain as “one of the types of line”, and it’s not a variant of any other line rule. The sample puzzle is the original 10-lines puzzle, which for reference took me about 45 minutes, whereas the GAS puzzles on this page take me 4~6 minutes.
Sample puzzle: 10 Lines by zetamath (solution video)
Find more puzzles: GAS β¦ CtC
Advanced corollaries
A single cell can never add to 10 on its own. Therefore, a line of length 4 can only be a single segment or two 2-segments, not a 1 and a 3. Similarly, a line of length 5 must be split into a 2- and 3-segment.
Pencil puzzle hybrids
There are many other logic puzzles that can be played on a grid, and so there are sudoku puzzles that borrow their rules to make a hybrid puzzle. These puzzles tend to be more difficult, so very few of the samples here are GAS.
The sky is the limit here, so these are just the most common examples; in principle, you could cross a sudoku with any kind of pencil puzzle, or even your own bespoke coloring rules.
Most of these involve shading the grid in two different colors. The rules may phrase this as “shaded” and “unshaded”, as would be necessary for a puzzle done on paper with only a single color of pencil. But since we’re usually doing these digitally, I prefer to phrase it in terms of two separate colors (and then blank cells can mean “unknown”).
Coloring does nothing on its own, so there must be some extra rule that ties the coloring to the digits. A few common examples:
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Kurodoko β certain cells (often containing circles) count the number of cells of the same color that they can “see”, horizontally or vertically, including themselves, up until a cell of the other color. For example, the center of a plus sign shape can see 5 cells of its color.
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Rule creation β contiguous groups of cells of the same color may act as e.g. killer cages.
-
Invisibility β the digits in cells of a particular color may be ignored by some other rule, e.g. not count towards killer totals or act as though they aren’t on a renban line.
-
Rule modification β the digits in cells of a particular color may interact differently with some other rule, e.g. count as double or negative for summing purposes.
Loop
Draw a contiguous one-cell-wide loop in the grid that doesn’t cross itself.
If it’s just a path and not a loop, it’s called a snake, but they often play much the same way.
Most puzzles either specify that the loop must pass through every cell or that the loop may never touch itself (diagonal touching may or may not be allowed), to keep the loop from being too freeform.
The sample puzzle is GAS, and uses the parity of circled digits to inform whether the loop is bent or straight at that cell.
Sample puzzle: Loopdoku by clover! (solution video)
Yinβyang
Shade the grid with two colors, such that each color forms a single contiguous area, and no 2Γ2 block is a single color.
This seems to be a pretty popular variant. The final grid is usually a striking pattern, too.
The sample puzzle uses the parity of circled digits to distinguish the two colors. It’s my own creation, an attempt at a GAS-level yin-yang puzzle. Unfortunately, yin-yang is the kind of puzzle where you need to infer a couple of corollaries to get very far on non-trivial puzzles, and while I did my best to nudge you towards them within this single puzzle, it’s gonna be hit or miss. Do feel free to read them below if the puzzle seems completely impenetrable.
Sample puzzle: approachable balance by eevee (no solution video, sorry!)
Find more puzzles: GAS β¦ CtC
Because all cells of a single color must be connected, you have to avoid cutting a chunk off from the rest of its color. I usually hear this called “escape analysis”, because you’re working out how a color can escape being penned in. In the diagram, the red areas are almost completely surrounded, and there’s only one route out of the yellow. All of the cells marked with circles can therefore be colored red. Of course, this only applies if you know there are other red cells to connect to somewhere (which is why I marked one off in the distance). In yin-yang, because of the 2Γ2 rule, there will be a cell of both colors in every 2Γ2 block in the whole grid, so you can usually take for granted that a color needs to escape. Also, the 2Γ2 rule means that the coloring will look like a series of hallways, with no “rooms”, if that helps. No 2Γ2 block may be colored with a checkerboard pattern. If you start with such an arrangement, pick one of the cells, and try to draw any contiguous path to the other cell of the same color, you will necessarily cut the cells of the other color off from each other. This is just a consequence of needing to be contiguous β one color needs to reach its partner, the other color needs to get out from between the two of them, and there’s no way to have those two paths at the same time, because contiguous paths cannot cross. Also, the border may only have at most one contiguous span of each color β equivalently, it may only change color twice. This is often key to the break-in. The reasoning is the same: if the border had two spans of each color, any way of connecting one color would separate the spans of the other color. This is hard to convey in a convincing way with a single picture, so I encourage you to play with it yourself. Just open a puzzle, ignore the clues, and mess around with the coloring.Basic corollaries
Advanced corollaries
Cave
Shade the grid with two colors, cave and wall. Cave cells form a single contiguous area. Wall cells may form multiple contiguous areas, but each area of wall cells must connect to the outside border of the puzzle.
Also known as Corral and Bag. Note that 2Γ2 blocks aren’t forbidden.
The original pencil puzzle is about drawing a loop, which is functionally identical (you’re just drawing the boundary between wall and cave). The clues in that puzzle are kurodoko clues, i.e. they give the number of cave cells visible from the clue in all four directions, up to a wall or edge of the grid, including the clue itself. Cave sudoku often use the same type of clue.
The sample puzzle is hard β my time is 45 minutes β but it’s the purest (and possibly first?) example of the form. It’s also so old that it predates SudokuPad’s ability to list the rules, so to summarize:
Shade the grid by cave rules. Yellow cells are cave, and the digits in them are kurodoko clues. Digits do not repeat within a contiguous chunk of wall.
Sample puzzle: Cave Sudoku by Phistomefel (solution video)
Find more puzzles: GAS β¦ CtC
Like yin-yang, escape analysis applies to cave cells, since they all need to connect to each other. It usually doesn’t apply to wall cells… except that a chunk of wall in the middle of the grid does still need to reach the edge, so if it’s mostly surrounded by cave, it needs to escape somehow. Cave rules are similar to yin-yang rules, if the puzzle were surrounded by a outer ring of wall cells. That means the 2Γ2 checkerboard rule still applies.Basic corollaries
Advanced corollaries
Nurikabe
Shade the grid with two colors, island and ocean. Ocean cells form a single contiguous area, and no 2Γ2 block may be all ocean. Island cells form any number of contiguous areas.
Without further rules, nothing is stopping you from simply shading the entire grid as island, so there’s usually a significant cap on the number of island cells somehow.
Nurikabe is a pencil puzzle where the clues are the sizes of the islands, and it’s named after an invisible wall youkai that blocks travellers. I don’t entirely understand the inspiration there.
The sample puzzle is not GAS and is moderately difficult β my time is 26 minutes. Like the original nurikabe puzzle, it uses circled cells to give the size of islands, with some extra rules to help fill in the rest of the digits.
Sample puzzle: The Caribbean by Niverio (solution video)
Find more puzzles: GAS β¦ CtC
Like with yin-yang, escape analysis is extremely important, although most of the time it only applies to the ocean. In many puzzles (including the sample puzzle), islands are defined as distinct in some way, so separate islands must be kept separate. This can sometimes force the placement of ocean cells. The 2Γ2 checkerboard rule doesn’t apply, since there may be any number of islands, and cutting them off from one another isn’t a problem. It may still apply if you know that both island cells are meant to be part of the same island, in which case completing it will necessarily cut off an ocean cell.Basic corollaries
Advanced corollaries
Meta
These are general patterns or riffs on other types of clues, which makes them hard to demonstrate in isolation or even talk about very specifically, so I’ll only mention them in brief. Even most of these names aren’t really set in stone.
Indexing β The value of a digit is used as a coordinate in the grid somehow. A basic example is to color the first, fifth, and ninth columns red, and state that the digits in those columns tell you which column the 1, 5, and 9 are in in that row.
Liar β The digits in the final grid do not match the given clues. A liar clue is considered fulfilled if any part of its usual rules is violated; for example, a liar killer cage could be correctly filled with digits that add to its total, as long as at least one of those digits repeats.
For extra difficulty, only some of the clues may be liars, commonly one clue of each type in a puzzle that uses several variants together. In this case the puzzle is also sometimes called wrogn.
Multitask β The same clue uses several rules at once, e.g., a single line might be both a renban and a region sum line.
Ambiguous β The puzzle uses several variant rules with clues that are depicted in similar ways, and it doesn’t tell you which rule each clue uses. Most common with lines, since there are so many types of line rules, but it’s also possible to have e.g. clues outside the grid that might be X-sums or sandwich sums or skyscraper counts.
Modifier cells β Usually takes the form that there is one “special” cell to locate in each row, column, and box; each special cell contains a different digit; and each cell’s value (for the purposes of other clues) is altered in some way. This adds some ambiguity to the given clues, but also constrains it to only the special cells.
Common types of modifiers are doublers (the cell’s value is twice its digit), halvers, negaters, adding or subtracting 1, and copycat (the cell’s value is the digit on the opposite side of the grid).
SchrΓΆdinger cells β Some cells contain multiple digits simultaneously (often following the same rules as modifier cells). Both digits contribute in some way to other rules; for example, the cell’s value might be the sum of its digits.
Knapp daneben β Clues or givens are actually one higher or lower than their true values.
Finks the rat β Honorable mention because I love Finks. A series of puzzles by Marty Sears where you have to draw a path (as a snake) from Finks the rat’s starting location to a goal cupcake, so that she can get the cupcake. The grid contains walls, the path has to obey a different constraint in each puzzle, other common clues are recontextualized as physical objects within the “maze”, and also there are teleporters. It’s great.
