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Most American crosswords use rotational (or "radial") symmetry so that squares directly opposite each other (through the center) are identical.
In the Times, I've only seen this *not* be the case when the layout was directly linked to the theme. Is that one of the "rules" you can count on, or are there (modern, W... |
Are there "rules" (or consistent conventions) that limit when the NY Times Crossword can deviate from rotational symmetry? |
I've seen Sudoku puzzles that can be solved two different ways.
1. Is it possible for a Sudoku puzzle to have 3 solutions? If so, what is the maximum number of solutions a single Sudoku puzzle can have?
2. For any given puzzle, is there a way to find out how many solutions it has (other than brute force)? |
What is the maximum number of solutions a Sudoku puzzle can have? |
I've seen Sudoku puzzles that can be solved two different ways. Referring to a traditional 9x9 grid:
1. Is it possible for a Sudoku puzzle to have 3 solutions? If so, what is the maximum number of solutions a single Sudoku puzzle can have?
2. For any given puzzle, is there a way to find out how many solutions it ... |
Most American crosswords use rotational (or "radial") symmetry so that squares directly opposite each other (through the center) are identical.
In the Times, I've only seen this *not* be the case when the layout was directly linked to the theme. Is that one of the "rules" you can count on, or are there (modern, W... |
What is the maximum number of queens that can be placed on a standard 8x8 chessboard such that no one of them attacks any of the others? |
I believe for a number of books I have read and solved myself, it stated in the introduction that guessing should never be necessary for any of the puzzles in the book. Instead, a handful of number of techniques could be used to solve the puzzle completely including hard puzzles.
Are there puzzles out there that do... |
Are there Sudoku puzzles published that do require guessing? |
What is the maximum number of queens that can be placed on a standard 8x8 chessboard such that no one of them is capable of attacking any of the others in a single move? |
How can [Sokoban](http://en.wikipedia.org/wiki/Sokoban) puzzles be procedurally generated with a computer program, guaranteeing at least one solution?
A simple algorithm might be:
{n} times
place a "target" square and a box on the square
end
until {puzzle is done (?)}
move a charac... |
How can I procedurally generate Sokoban puzzles? |
I have seen many variations of the following puzzle:
"Once upon a time, a farmer went to the market and purchased a fox, a goose, and a bag of beans. On his way home, the farmer came to the bank of a river and rented a small boat. The boat was so small that the farmer could carry only himself and a single one of h... |
Sometimes, when solving the Void Rubik's Cube using the [standard beginner method](http://www.youcandothecube.com/), it gets "stuck." What I mean by this is that the top edge of one face and the top edge of an adjacent face's colors are swapped.
Typically, I've just turned random faces until the cube is sufficiently... |
How can I fix "stuck" positions on the Void Rubik's Cube (using the standard beginner's method)? |
I've seen Sudoku puzzles that can be solved two different ways. Referring to a traditional 9x9 grid:
1. Is it possible for a Sudoku puzzle to have more than two solutions? If so, what is the maximum number of solutions a single Sudoku puzzle can have?
2. For any given puzzle, is there a way to find out how many s... |
For those of you familiar with the solution to the Rubik's cube, you will know that a single-corner twist is not a valid position. It can neither be attained by scrambling, nor can it be solved if it occurs.
I understand that there's some fundamental group theory at work here; the corners on a Rubik's cube can only ... |
Why is a single-corner twist not a valid position on a Rubik's cube? |
How many queens can be on a chessboard without attacking each other? |
Given an 8x8 chessboard, your goal is to "cover" each space on the board with the fewest possible number of pieces. A space is "covered" if there is a piece on it, or if a piece on the board can be moved to that space in one move.
A trivially easy solution would be that a board could be covered with 64 pieces. If ... |
How many chess pieces does it take to "cover" all spaces on a chessboard? |
Sometimes, when solving the [Void Rubik's Cube](https://en.wikipedia.org/wiki/Void_Cube) using the [standard beginner method](http://www.youcandothecube.com/), it gets "stuck." What I mean by this is that the top edge of one face and the top edge of an adjacent face's colors are swapped.
Typically, I've just turned ... |
Here's a typical, standard payoff Matrix for Prisoner's Dilemma ([adapted from Wikipedia](https://en.wikipedia.org/wiki/Prisoner's_dilemma))
| B coop | B defect
----------------------------
A coop | R, R | S, T
----------------------------
A defect | T, S | P, P
Where `T > R... |
In iterated Prisoner's Dilemma, how would a change in the payoff matrix affect strategy? |
You are in a room with three switches, labeled A, B, and C. Initially, each switch is in an off state. In an adjoining room, there are three incandescent light bulbs. Your task is to correctly label which bulb goes to which switch.
The catch: you may only enter the bulb room once. You have as long as you need ... |
Which switch goes to which bulb? |
What are the criteria for determining the difficulty of Sudoku puzzle?
Is it the subjective estimation based on author's experience, or are there any precise guidelines?
For example, there are some sudokus, where in each step there's always at least one digit that, if you look on some line, must be exactly in tha... |
What are the criteria for determining the difficulty of Sudoku puzzle? |
I have been told that it is possible to guarantee no worse than a tie in normal 3x3 tic-tac-toe, but that the first move is very important. What is that important first move? |
What is the optimal first move in tic-tac-toe? |
If an algo `X` outputs the minimum "number of moves" (not the actual moves, just their count) to solve an input configuration, what's the maximum number `X` can output for any configuration? |
What is the maximum "minimum number of steps" needed to solve a rubiks cube? |
The basic algorithm for procedurally generating mazes goes something like this:
1. Choose a random available direction from the current square.
2. If there is one, go there and repeat from step 1.
3. Otherwise, backtrack to the previous square and repeat from step 1.
This is a fairly good algorithm, but the pro... |
How can I generate mazes that often have multiple "forks" or choices? |
I, being a fan of 2048, was wondering if anyone has any proven strategies for playing [2048](http://gabrielecirulli.github.io/2048)?
This being a hypothetical [tag:tips] question, has some requirements:
- 1 tip or strategy per answer, so they can be isolated as good or bad.
- Upvote the answers you agree with, ... |
I can get to 4096 max. There are videos on youtube of guys making it to 8192. What is the theoretical maximum limit on this?
PS: if you haven't played this, here is the [link to addiction.][1]
[1]: http://gabrielecirulli.github.io/2048 |
In the game "2048" what is theoretically the largest tile possible? |
There is, indeed, a way! No way has yet been found (at least that I've found) to generate them infallibly, though.
In [a paper](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.47.2303&rep=rep1&type=pdf) by Y. Murase, H. Matsubara, and Y. Higara, called "Automatic Making of Sokoban Puzzles", they explain exa... |
According to [Wikipedia](http://en.wikipedia.org/wiki/Verbal_arithmetic), Verbal Arithmetic can be defined as "a type of mathematical game consisting of a mathematical equation among unknown numbers, whose digits are represented by letters. The goal is to identify the value of each letter."
For example, the problem:... |
What is the largest tile possible in 2048? |
Once one has mastered [winning 2048 in general](http://puzzling.stackexchange.com/questions/39/general-strategy-for-2048), it's often an interesting exercise to try to do it as fast as possible.
What strategy is there for speeding up 2048-solving? For example, how can you pass the boring beginning game quickly (ex. ... |
In a crossword some clues have a question mark at the end. Is there any way to tell what these question marks mean based on the puzzle, ie. rhetorical, compoundword? |
There's only one canonical example I can find of a crossword using question marks to mean anything special.
Many American crossword puzzles (for instance, the New York Times) rely on a convention that, if a clue ends in a question mark, it's a play on words.
> Mental Block?
> RUBIKSCUBE
It can also be use... |
When playing SlitherLink on [moderate][1] or [large][2] boards I usually find that after using the [basic techniques][3]:
* filling out the values in the corners
* setting lines where there are adjanced 3s
* setting lines where there are 3s diagonally (including cases where they are only separated by 2s)
* cleari... |
The game 2048 can be thought as a two player game, where on player shifts the table around to get to the 2048 tile, while the other player tries to hinder it by putting down 2s and 4s on the table. While the [original game][1] will put down the tiles randomly [there][2] [are][3] variants, where this is not random, and ... |
Growing up on the cryptic crosswords I found in _Games_ magazine and _The New York Times_, I'm used to a very rigid rule for clues: Every clue comprises two parts, one of which directly (if opaquely) clues the word as in a non-cryptic crossword and the other of which does so cryptically. (These may be separated by "is"... |
What rules are there for clue composition in a British cryptic? |
I've tried to explain the solution using as little math as possible, and to give some intuition as to what makes it tick. Nonetheless there will be a little mathematical notation at the end.
### First steps: going beyond the obvious solution in the simplest case (N=2)
The statement of the puzzle as presented here... |
N logicians are wearing hats which can be of N different colors (the colors don't have to be distinct). Each logician can see the colors of all hats except his own. The logicians must simultaneously call out a color; they win if at least one logician calls the color of his own hat. They can agree on a strategy beforeha... |
The MPire coloring game is a game where several "empires" are laid out touching each other. Your task is to color each empire a different color, and not let two of the same color touch. You also have to use as few colors as possible.
I was wondering if anyone had specific strategies for this game? Is it better to st... |
I understand the basic method of generating sudoku puzzles, which is:
1. Put a random number in the current square.
2. If the puzzle is still valid, go to the next square. Otherwise, stay on the same square.
3. Repeat from step 1.
However, sometimes a sudoku puzzle gets "stuck" so that no matter what you put in... |
How to avoid getting "stuck" while generating sudoku puzzles? |
20 teams played a round robin tournament. The winner of each game got 1 point, the loser none, there were no draws. Each team's total score was a square number. At least one team had a unique score. How many upsets were there, defined as times a team with a lower total score beat a team with a higher total score? ... |
I learned how to solve a Rubik’s cube with the normal beginner method. Now I wanted to know how to get from this to speedcubing. Is speedcubing just executing the steps really fast? I don't think so. What are the fastest ways to solve the cube?
|
How to get started in speedcubing? |
The Einstein's puzzle or zebra puzzle is a well-known logic puzzle. However, what are the best ways to solve it? Are there any very easy ways to solve it fast? |
What are strategies for solving Einstein's puzzles? |
The answer quite honestly is: depends on how much you want to memorize.
I'd personally recommend exploring the algorithms and why they work before delving into speedcubing algorithms. The reason for this is simple: over the use of the cube, you'll see faster ways to do what you're trying to do. My solve time using ... |
Can you tell me what is the name of the game I used to play in school?
It's a game for two players. You need grid paper and two pens.
The game is played on a 10x10 arena, shifted by half a grid to one side.
The players take turns, each drawing a one grid length line. One player plays "middles", the other plays... |
What is the name of this line-connecting game? |
How, as a writer of riddles, can I rate their difficulty?
I guess, for me alone, knowing the answer even before the riddle existed, it's pretty much impossible, and I should need some external help. Are there some riddling communities or something like that, who could help rate the difficulty? How would I approach t... |
How to rate difficulty of a riddle? |
The [Einstein's puzzle][1] or [zebra puzzle][2] is a well-known logic puzzle. However, what are the best ways to solve it? Are there any very easy ways to solve it fast?
[1]: http://www.stanford.edu/~laurik/fsmbook/examples/Einstein'sPuzzle.html
[2]: http://en.wikipedia.org/wiki/Zebra_Puzzle |
It is called [Brige-It][1], or [Game of Gale][2], as it was invented by David Gale. It is the rectangular grid equivalent of the [Shannon switching game][3], which is a generalized version of [Hex][4].
[1]: http://en.wikipedia.org/wiki/Bridg-It#Gale
[2]: http://www.papg.com/show?1TPI
[3]: http://en.wikiped... |
To be able to judge the difficulty of a Sudoku, you first need to define "difficulty". The most useful definition seems to rely on the question:
> How long does it take an "average" solver to solve the puzzle?
Making this precise will involve fixing the population of solvers, and setting up some reference. E.g., ... |
I used to play [Einstein's puzzles][1] a lot when I was younger, and very quickly learned to assess quite rapidly whether I thought a puzzle would be a quick solve or more time consuming, but I never worked out the detail on how many clues are required.
Is there a mathematical description of the minimum number of cl... |
How many clues are required to ensure Einstein's Puzzle is solveable? |
Once I solve the Rubick's cube I obviously want to start again. I just scramble it until it seems to be enough, however sometimes when I then solve it, I can skip entire steps from the [beginners method][1]. Is this normal or was the cube not scrambled enough? How can one determine if the cube is scrambled enough?
... |
How to determine if a Rubicks Cube is sufficiently scrambled? |
A [River crossing puzzle](https://en.wikipedia.org/wiki/River_crossing_puzzle) puts limits on the number of items that may cross a boundary with limits on what may be left together. The ["Fox, goose and bag of beans"](https://en.wikipedia.org/wiki/Fox,_goose_and_bag_of_beans_puzzle) is one of the earliest examples. Of... |
What is the maximum number of items in river crossing puzzle? |
For another viewpoint, let's take the "classic" version, of `A -> B, B -> C` (that is, fox eats goose, goose eats beans) and add a `D` such that it neither threatens anything nor is threatened by anything (a rock, say), as in the "neutral" element of @Lukasz's answer. Now is it solvable? Well, let's see (`\/` = boat):
... |
[](http://search.dilbert.com/comic/Random%20Number%20Generator)
Each set of moves is designed to correct a parity error in the cube's layout, but each group of parity it corrects is essentially independent and r... |
> This is a follow-up question for the [Sliding Bolt Puzzle](http://puzzling.stackexchange.com/questions/43/sliding-bolt-puzzle). If you have not solved it yet, you might want to head there first, as the extended discussion of its solution in this question will contain **spoilers**.
In the original version of the [S... |
In a sudoku puzzle, what is the absolute minimum number of squares that must be given to specify a layout with a unique solution? Is one in each row, column, and box (9 total) necessary or sufficient?
|
One of the classical puzzles is pouring water into the 5 gallon jug so there is 4 gallons of water remaining. This was popularized by Bruce Willis in "Die Hard with a Vengeance". I was wondering however, if there was a possible way to complete the puzzle using a 3 gallon jug and a 7 gallon jug to still get the 4 gallon... |
Writing riddles is a tricky business, because you must balance difficulty, come up with clues corrupted enough not to be trivial, and simultaneously not too obscure, and get the final result in rhyme form.
What are some techniques of coming up with a decent riddle? |
How do you come up with a good riddle? |
I think the problem with trying to rate these with one number is that *most* of the most popular brainteasers present challenges on up to three different "challenge spectrums":
- **Some are highly dependent on actual skills, knowledge or tools** (like ones that can't easily be solved without a spreadsheet, or that ... |
One of the classical puzzles is pouring water into the 5 gallon jug so there is 4 gallons of water remaining. This was popularized by Bruce Willis in "Die Hard with a Vengeance". I was wondering however, if there was a possible way to complete the puzzle using a 3 gallon jug and a 7 gallon jug to still get the 4 gallon... |
I think the problem with trying to rate these with one number is that *most* of the most popular brainteasers present challenges on up to three different "challenge spectrums":
- **Some are highly dependent on actual skills, knowledge or tools,** like ones that can't easily be solved without a spreadsheet, or that ... |
Here's an old puzzle I found in a book once.
> A box has some number of locks on it, and it will only open if all the locks have keys in them. It is guarded by a Grand Vizier and four slaves, each of which hold keys. The keys they hold are configured such that as long as the Grand Vizier is with one slave, or if any... |
The immediate answer is no: Any valid Sudoku can be solved without guessing, just by exhaustively trying all possibilities.
However, there's two interesting variant interpretations of the question:
### Are there Sudoku puzzles that can't be solved logically?
This one depends on what you accept as "logic". In a... |
Before you proceed, here's the riddle I came up with, through composing this answer. Obviously, the answer = spoiler - follow close behind, so if you want to try to guess it, do it before proceeding below the line.
> *Eight hours of death under your head*.<br>
> *Colorful box that's full of dreams*<br>
> *Listen... |
Writing riddles is a tricky business, because you must balance difficulty, come up with clues corrupted enough not to be trivial, and simultaneously not too obscure, and get the final result in rhyme form.
What are some techniques of coming up with a decent classic, rhymed riddle? |
How, as a writer of riddles, can I rate their difficulty?
I guess, for me alone, knowing the answer even before the riddle existed, it's pretty much impossible, and I should need some external help. Are there some riddling communities or something like that, who could help rate the difficulty? How would I approach t... |
Here's an old puzzle I found in a book once.
> A box has some number of locks on it. It is guarded by a Grand Vizier and four slaves, each of which hold keys to the locks.
> The Grand Vizier and slaves each hold a different set of keys to the locks, and each key they hold opens a different lock.
> The keys the... |
Here's an intuitive explanation: let's set the squares to the things we know:
? 8 ?
? 5 ?
? ? ?
We can figure out the bottom one now, since `15-8-5=2`:
? 8 ?
? 5 ?
? 2 ?
The other two numbers on the bottom row now must add up to 13, which can only be 9/4 or 7/6. Now let's split this up into ... |
As Kendall mentions, it is impossible to move pieces out of a 3x3 grid in this manner. This is called the [Lo Shu square](http://en.wikipedia.org/wiki/Lo_Shu_Square), and is the only unique solution to the 3x3 magic grid puzzle.
The reason for this is actually rather simple. If you look at the Lo Shu square, you'll ... |
Before you proceed, here's the riddle I came up with, through composing this answer. Obviously, the answer = spoiler - follow close behind, so if you want to try to guess it, do it before proceeding below the line.
> *Eight hours of death under your head*.<br>
> *Colorful box that's full of dreams*<br>
> *Listens ... |
A group of N pirates has come by a chest containing 200 gold coins. Their rules require that the coins be distributed by the following approach. The pirates are ranked from fiercest to meekest and all pirates know the ranking. The fiercest must propose a division, which is put to majority vote with the fiercest winn... |
When solving Sudoku on paper, I sometimes get to the point where I need to resort to guessing or bifurcation. At that point, I might guess one number, and continue solving, changing that guess if I arrive at a contradiction.
I've experimented with a couple of approaches, e.g., circling the guess and writing the cons... |
What is a good notation for trial-and-error when solving Sudoku on paper? |
> #Names in Boxes
> - The names of 100 prisoners are placed in 100 wooden boxes, one name to a box, and the boxes are lined up on a table in a room.
> - One by one, the prisoners are led into the room; each may look in at most 50 boxes, but must leave the room exactly as he found it
> and is permitted... |
I'm asking about classic riddles as they appear in books and kids' literature, based on metaphors - not maths problems or tasks, or deceitful questions but just an old good-natured rhyme describing the subject in a way that makes it tricky to guess. Let me give some examples from J.R.R. Tolkien for what I mean:
> Th... |
How do you come up with a good classic rhyme riddle? |
Before you proceed, here's the riddle I came up with, through composing this answer. Obviously, the answer = spoiler - follow close behind, so if you want to try to guess it, do it before proceeding below the line.
> *Eight hours of death. Beneath your head -*.<br>
> *a colored box that's full of dreams.*<br>
> *I... |
A group of N pirates has come by a chest containing 200 gold coins. Their rules require that the coins be distributed by the following approach. The pirates are ranked from fiercest to meekest and all pirates know the ranking. The fiercest must propose a division, which is put to majority vote with the fiercest winn... |
Before you proceed, here's the riddle I came up with, through composing this answer. Obviously, the answer = spoiler follows close behind, so if you want to try to guess it, do it before proceeding below the line.
> *Eight hours of death. Beneath your head -*.<br>
> *a colored box that's full of dreams.*<br>
> *It... |
In [this question](http://puzzling.stackexchange.com/questions/112/the-puzzle-of-the-locked-box), I presented a puzzle involving seven locks, with a Grand Vizier and four slaves.
But I would like to use this style of puzzle in some of my own secret-unlocking riddles, in order to require that a person solve a specifi... |
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