instruction stringlengths 15 30k |
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There is a puzzle
> Suppose a standard 8x8 chessboard has two unknown (any) cells removed. The only thing known is that the cells had *different* colours. Is it always possible to place 31 dominoes of size 2x1 so as to cover all of these squares?
The answer is
>! yes
and there is very clever and fast prove:
... |
> I was at a dinner party the other night and I spoke to Mr Smith. He mentioned having two kids, both with unisex names: Sam and Alex. I remember he mentioned at one point having a son, but I don't recall which child he was talking about, and I don't know if he has a daughter or not. What are the odds that he has two s... |
> I was at a dinner party the other night and I spoke to Mr Smith. He mentioned having two kids, both with unisex names: Sam and Alex. I remember that he told a story about "my son", but I don't recall which child he was talking about, and I don't know if he has a daughter or not. What are the odds that he has two sons... |
What is correct answer for this IQ task and **why**?
![enter image description here][1]
[1]: https://i.stack.imgur.com/9fv2s.jpg |
How to solve this puzzle from IQ test? |
The answer to whether this could be brute forced is yes. However, would it be feasible to brute force the answer? Probably not. Let's look at why:
A chessboard has 64 squares - 32 white, and 32 black. Removing one of each color gives us 32^2 possible ways to arrange it. However, the symmetry of the board reduces tha... |
There is a puzzle
> Suppose a standard 8x8 chessboard has two unknown (any) cells removed. The only thing known is that the cells had *different* colours. Is it always possible to place 31 dominoes of size 2x1 so as to cover all of these squares?
The answer is
>! yes
and there is very clever and fast prove:
... |
![enter image description here][1]
I have been shown this "Chinese puzzle from textbook for 2nd grade" like a joke. But it is intriguing, so I would like to know the answer, but I am quite bad in associative puzzles and can't find the pattern.
The translation of the 3 lines of text should be the following:
> ... |
How to solve this crazy Chinese puzzle for children? |
![enter image description here][1]
I have been shown this "Chinese puzzle from textbook for 2nd grade" like a joke. But it is intriguing, so I would like to know the answer, but I am quite bad in associative puzzles and can't find the pattern.
The translation of the 3 lines of text should be the following:
> ... |
There is well known task:
> How to divide a pie between two people, so each will think that he got equal or more that the other one?
It has the following solution:
>! First person divide the pie on 2 equal (in his opinion) parts. Second one choses the biggest one.
I know that it is possible to generalise in... |
![enter image description here][1]
I have been shown this "Chinese puzzle from textbook for 2nd grade" like a joke. But it is intriguing, so I would like to know the answer, but I am quite bad in associative puzzles and can't find the pattern.
The translation of the 3 lines of text should be the following:
> ... |
H.
Various categories of symbol appear on alternate rows and columns:
Odd row, odd column - rectangles shaded in two colours, with diagonal dividing line.
Odd row, even column - single headed arrows
Even row, odd column - double headed arrows
Even row, even column - cross
Each category of symbol appears to ... |
>A lighthouse keeper decides to take a bath an hour before the next lighthouse keeper comes to relieve him of his shift.
>This lighthouse keeper is having a bad day. First he accidentally locks himself in the bathroom, so he's not going to be able to get out until the next lighthouse keeper arrives and is able to u... |
Here is the solution I found. Please tell, if you see any mistakes.
Let's call \$M'\$ maximum number of jokers, which can be among group of chosen \$N'\$ man according to our knowledge.
Our start group will include all men, \$N' = N\$. Our knowledge is that \$K+M=N, K>M\$, therefore \$M'= (N-1)/2\$.
The goal ... |
The answer to whether this could be brute forced is yes. However, would it be feasible to brute force the answer? Probably not. Let's look at why:
A chessboard has 64 squares - 32 white, and 32 black. Removing one of each color gives us 32^2 possible ways to arrange it. However, the symmetry of the board reduces tha... |
How I understand this puzzle:
In the corners (cells: 11,13,31,33) we have some objects. In the middle edges (12,21,23,32) we have operations on neighbouring two objects. In the centre (22) we have nothing, since there is no neighbouring objects.
Let's look on (12) operation (one arrow), it leads to simple vertica... |
F.
How I understand this puzzle:
In the corners (cells: 11,13,31,33) we have some objects. In the middle edges (12,21,23,32) we have operators on neighbouring two objects. In the centre (22) we have meta operator on neighbouring operators.
Let's look on (12) operator (one arrow), it leads to simple vertical st... |
The idea is to ask question about question. In that case `lie+lie = truth+truth = truth` (or `lie+truth = truth+lie = lie`).
____
An example of such a question is:
**What would you tell me if I ask you whether this road ends to death?**
If answer is "yes" you better do not take this road. If answer is "No" yo... |
The answer to whether this could be brute forced is yes. However, would it be feasible to brute force the answer? Probably not. Let's look at why:
A chessboard has 64 squares - 32 white, and 32 black. Removing one of each color gives us 32^2, or 1024 possible ways to arrange it. However, there are 4 lines of symmetr... |
What is the maximal number of steps computer would need to solve an arbitrary 2x2x2 rubiks cube configuration (if he does it as fast as it can)?
Are there known algorithm, which will perform similarly fast? |
What is the maximal number of steps one needs to solve 2x2x2 rubiks cube? |
There is well known problem:
> The picture is attached to a string and you want to hang it on N nails so that if one takes out any of the nails, the picture falls. How do you do it?
*For those who have not heard this puzzle:
No brainteaser tricks are here, this is basically topology problem: "How to put N poin... |
How to generalise hang picture puzzle? |
My first question is:how would one solve such problems(in general). What should be the general technique?
![enter image description here][1]
[1]: https://i.stack.imgur.com/uD7XC.jpg
My second question is;When to be sure that we have counted all the squares in such problems?
PS: Please add the approp... |
This is another example of an information puzzle. You are trying to find the most numbers a pair of spies can send to each other given that there are 26 stones in the river.
Furthermore, the stones are all identical, and the only way the spies can communicate is by throwing a certain *number* of stones into the pond... |
Another strategy would be as follows:
1. How many 1x1 squares can we have, ignoring the smaller squares? 4 rows of 4 --> 16
2. How many 2x2 squares can we have, ignoring the smaller squares? 3 rows of 3 --> 9
3. How many 3x3? 2 rows of 2 --> 4
4. How many 4x4? 1 --> 1
This gives us (16+9+4+1)=30 squares.
No... |
My first question: how would one solve such problems (in general)? What should be the general technique?
![enter image description here][1]
[1]: https://i.stack.imgur.com/uD7XC.jpg
My second question: When to be sure that we have counted all the squares in such problems?
|
Well we can easily rephrase the question as follows (the subtractor being `k + (k - 1) / 2` obviously for mod k)
(n - 1) ≡ 0 mod 2
(n - 3) ≡ 0 mod 3
(n - 10) ≡ 0 mod 5
(n - 21) ≡ 0 mod 7
(n - 45) ≡ 0 mod 10
using congruent math we can prettify this:
n ≡ 1 mod 2
n ≡ 0 mod 3
... |
This is another example of an information puzzle. You are trying to find the most numbers a pair of spies can send to each other given that there are 26 stones in the river.
Furthermore, the stones are all identical, and the only way the spies can communicate is by throwing a certain *number* of stones into the pond... |
This is another example of an information puzzle. You are trying to find the most numbers a pair of spies can send to each other given that there are 26 stones in the river.
Furthermore, the stones are all identical, and the only way the spies can communicate is by throwing a certain *number* of stones into the pond... |
There is a puzzle about two spies:
> Two spies must pass each other two secret numbers (one number per spy), unnoticed by their enemies. They have agreed on a method for doing this using only 26 indistinguishable stones in advance.
> They meet at a river, where there is a pile of 26 stones. Starting with the fir... |
There is a puzzle about two spies:
> Two spies must pass each other two secret numbers (one number per spy), unnoticed by their enemies. They have agreed on a method for doing this using only 26 indistinguishable stones in advance.
> They meet at a river, where there is a pile of 26 stones. Starting with the fir... |
F.
_______
I googled and found the test this task is taken from: http://www.iqtest.dk/testEnglish.swf
If I answer 1-10 questions correctly and 38 H, I get IQ 92.
If I answer 1-10 questions correctly and 38 F, I get IQ 93.
So I conclude that correct answer is F.
_______
How I understand this puzzle:
... |
The idea is to ask question about question.
In that case `Lie about Lie = Truth about Truth = Truth ` (if you ask person about question to himself)
or `Lie about Truth = Truth about Lie = Lie ` (if you ask person about question to his antipode).
Therefore it won't be matter who you are asking.
____
An exam... |
Imagine very big number (2N) of people, which do not know each other, play the following game:
> 1. Players are divided on two equal groups: A and B. In each group players are numbered from 1 to N.
>
2. Each A-player chooses position for a white king on chessboard.
>
3. Each B-player chooses position for a bl... |
Imagine very big number (2N) of people, which do not know each other, play the following game:
> 1. Players are divided on two equal groups: A and B. In each group players are numbered from 1 to N.
>
2. Each A-player chooses position for a white king on chessboard.
>
3. Each B-player chooses position for a bl... |
Imagine very big number (2N) of people, which do not know each other, play the following game:
> 1. Players are divided on two equal groups: A and B. In each group players are numbered from 1 to N.
>
2. Each A-player chooses position for a white king on chessboard.
>
3. Each B-player chooses position for a bl... |
I never hold this awesome thing in my hands, but there is 2 very nice videos on youtube, that fully describes the approach:
https://www.youtube.com/watch?v=UBqfzmCcEeM
https://www.youtube.com/watch?v=KWVVF4hC7p4
They are made in assumption that you know how to solve 3x3x3 Rubik's.
I will do a short summary... |
N logicians are wearing hats which can be of N different colors (each hat has one color; there can be multiple hats with the same color). 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. The... |
I was just thinking about this.
Let's say that we have a card deck with 54 cards and the $n$th card from the top is flipped. We repeat a short process of cutting the deck directly in the middle and then interleaving the cards from each pile (in other words, a perfect shuffle).
For example, if the cards were numb... |
Yes, there is an algorithm.
Oh, you wanted to know what the algorithm is? Here is one, which presents an algorithm but not all the reasoning behind it. For that, and a proof, and the variant where the odd ball must be identified but not the direction of its deviance, read [Thomas Pornin's excellent presentation on t... |
As the question is [currently written](http://puzzling.stackexchange.com/revisions/619/1), there is a trivial answer. While this answer is boring in terms of algebra, it demonstrates the need for careful wording when writing mathematical reasoning. I expect that this is not what is meant, but it's what is written.
>... |
The question of [twelve balls and a scale](http://puzzling.stackexchange.com/questions/183/twelve-balls-and-a-scale) is probably the best-known example of the "find the ball of a different weight" problem. But does it generalize? Is there a general way to find a weighing algorithm for $N$ balls on a scale where $N \ne ... |
This is another example of an information puzzle. You are trying to find the most numbers a pair of spies can send to each other given that there are 26 stones in the river.
Furthermore, the stones are all identical, and the only way the spies can communicate is by throwing a certain *number* of stones into the pond... |
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... |
[This question](http://puzzling.stackexchange.com/questions/371/how-do-you-solve-the-circular-table-problem) was asked about a group of people whom a king wishes to play a cruel game with. Going around in a circle, the king kills every other person.
The question concerned how one could compute the answer without ex... |
Generically, what do you do if there are $c$ colors, and $n$ people? Imagine further the aliens say the following:
> You can plan your strategy now, but we are only going to tell you what the colors are going to be (including their hex representation) and how many there are after you are no longer allowed to communi... |
This [answer][1] reminded me about one nice puzzle. I present it as a challenge for you.
> N thieves have stolen an magical artefact together. They bought one lock with N keys and attached the artefact to a loop on a wall in the garage, so that each of them can use an artefact at any time. Immediately they understo... |
This [answer][1] reminded me about one nice puzzle. I present it as a challenge for you.
> N thieves have stolen an magical artefact together. They bought one lock with N keys and attached the artefact to a loop on a wall in the garage, so that each of them can use an artefact at any time. Immediately they understo... |
This [answer][1] reminded me about one nice puzzle. I present it as a challenge for you.
> N thieves have stolen an magical artefact together. They bought one lock with N keys and attached the artefact to a loop on a wall in the garage, so that each of them can use an artefact at any time. Immediately they understo... |
This [answer][1] reminded me about one nice puzzle. I present it as a challenge for you.
> N thieves have stolen an magical artefact together. They bought one lock with N keys and attached the artefact to a loop on a wall in the garage, so that each of them can use an artefact at any time. Immediately they understo... |
This [answer][1] reminded me about one nice puzzle. I present it as a challenge for you.
> N thieves have stolen an magical artefact together. They bought one lock with N keys and attached the artefact to a loop on a wall in the garage, so that each of them can use an artefact at any time. Immediately they understo... |
There are N letters in an alphabet. There is a combination lock, the code to it consists from M different letters.
You can input M letters combination to try to open the lock.
If you guess M-1 letters from the code correctly the lock will be opened. The order does not matter.
**How many** tries you need to gua... |
There are $N$ letters in an alphabet. There is a combination lock, the code to it consists from $M$ different letters.
You can input $M$ letters combination to try to open the lock.
If you guess $M-1$ letters from the code correctly the lock will be opened. The order does not matter.
**How many** tries you nee... |
There are $N$ letters in an alphabet. There is a combination lock, the code to it consists from $M$ different letters.
You can input $M$ letters combination and try to open the lock. (But you can't check the lock unless you have inputted all $M$ letters).
If you guess $M-1$ letters from the code correctly the loc... |
There are $N$ letters in an alphabet. There is a combination lock, the code to it consists from $M$ different letters.
You can input $M$ letters combination and try to open the lock. (But you can't check the lock unless you have inputted all $M$ letters).
If you guess at least $M-1$ letters from the code correctl... |
This [answer][1] reminded me about one nice puzzle. I present it as a challenge for you.
> N thieves have stolen an magical artefact together. They bought one lock with N keys and attached the artefact to a loop on a wall in the garage, so that each of them can use an artefact at any time. Immediately they understo... |
Oskar van Deventer has created „***the haunted vending machine puzzle***“. You have to insert coins into the slots 1,2,3 or 4. A coin rests on a green switch or changes its position if it hits one of his legs. A starting position is shown on the left side the goal is to have 7 coins in the machine(one on each switch).... |
Oskar van Deventer has created the “haunted vending machine puzzle”. You have to insert coins into the slots 1,2, 3 or 4. A coin that hits a green switch on its head rests there; a coin that hits a switch on its leg changes its position. A starting position is shown on the left side. The goal is to have 7 coins in the... |
Oskar van Deventer has created the “haunted vending machine puzzle”. You have to insert coins into the slots 1,2, 3 or 4. A coin that hits a green switch on its head rests there; a coin that hits a switch on its leg changes its position. A starting position is shown on the left side. The goal is to have 7 coins in the... |
This to add to [Joe Z.'s answer][1], who achieved 1861. Hope this will help the other people to improve the answer.
Joe Z.'s idea with some tuning allows to achieve 1891. The idea, which makes it possible to proceed further is to choose between two algorithms basing on the first turn result (and is a special case of... |
This to add to [Joe Z.'s answer][1], who achieved 1861. Hope this will help the other people to improve the answer.
Joe Z.'s idea with some tuning allows to achieve 1891. The idea, which makes it possible to proceed further is to choose between two algorithms basing on the first turn result. *[Actually, this is a sp... |
Two perfect logicians, Summer and Proctor, are told that integers $x$ and $y$ have been chosen such that $1 < x < y$ and $x+y < 100$. Summer is given the value $x+y$ and Proctor is given the value $x \cdot y$. They then have the following conversation.
- _Proctor:_ "I cannot determine the two numbers."
- _Summ... |
> I'm thinking of a number, $s$, which equals either $0$ or $1$ with equal probability. If $s=1$, then $p$ is either $0$ or $1$ with an unknown probability distribution - otherwise $p=0$.
>
> If your number is $n$, where $n\neq 4$, is the $\displaystyle\lim_{x\to n}\frac{1}{x-2-s-p}\geq0$?
- If $x=1$, then the ... |
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 ... |
There is a puzzle about two spies:
> Two spies must pass each other two secret numbers (one number per spy), unnoticed by their enemies. They have agreed on a method for doing this using only 26 indistinguishable stones in advance.
> They meet at a river, where there is a pile of 26 stones. Starting with the fir... |
Oskar van Deventer has created the “haunted vending machine puzzle”. You have to insert coins into the slots 1,2, 3 or 4. A coin that hits a green switch on its head rests there; a coin that hits a switch on its leg changes its position. A starting position is shown on the left side. The goal is to have 7 coins in the... |
Oskar van Deventer has created the “haunted vending machine puzzle”.
> ![enter image description here][1]
> You have to insert coins into the slots 1, 2, 3 or 4. A coin that hits a green switch on its head rests there; a coin that hits a switch on its leg changes switch position. A starting position is shown on ... |
Similarly to http://puzzling.stackexchange.com/questions/464/how-many-digits-can-be-removed-from-a-multiplication-puzzle-and-still-give-only I am curious about division puzzles.
I know this puzzle:
<img src="https://i.stack.imgur.com/NDuXn.jpg" width="200">.
It has only 2 out of 42 digits $\approx$ 4.8% digi... |
How many digits can be removed from a division puzzle? |
[This question](http://puzzling.stackexchange.com/questions/1715/oskar-van-deventers-haunted-vending-machine-puzzle) asks about a strange vending machine in which coins can fall out as well as being deposited in, and all the coins it asks for must be deposited before the machine will accept them.
That got me thinkin... |
[This question](http://puzzling.stackexchange.com/questions/1715/oskar-van-deventers-haunted-vending-machine-puzzle) asks about a strange vending machine in which coins can fall out as well as being deposited in, and all the coins it asks for must be deposited before the machine will accept them.
That got me thinkin... |
I heard Martin Gardner in one of his books presented a crossword, which looks like 3x3 square. Could you help me to find it?
Also, is there a smaller crossword puzzle? |
I heard Martin Gardner in one of his books presented a crossword, which looks like 3x3 square. Could you help me to find it?
Also, have been there a smaller crossword puzzle created? |
It's not hard to construct a 1x1 crossword:
> +---+
> |1 |
> | |
> +---+
>
> Across:
>
> 1. A Roman one.
>
> Down:
>
> 1. The shortest pronoun.
There's also a cheating one going around, with clues "*What letter am I thinking of?*" and "*What is the answer to 1 Across?*", and clai... |
If "PSAKLJIUASP" is "PJIUAZAKLKP" then, what is the word for "LIOQWERRPOIL"?
I noticed that first 3 characters are reverse of last 3 characters in both words in question. A is mapped to I and I is mapped to A. But same logic don't work for other cases. Any clues please? |
I don't see a nice answer, but there are some things you should notice. First, the wording is ambiguous: We are told "PSAKLJIUASP" is "PJIUAZAKLKP", but it is not clear whether "LIOQWERRPOIL" should be the source or destination. I would notice the groups of three letters AKL and JIU which are maintained but shifted... |
First of all if either of the top two switches has a coin on it, drop another to clear it. Now, if either of them are leaning in (and by leaning I always mean the _top_ of the switch leaning that way), drop a coin on their inside so that lean out. From now on we will only ever hit a top-row switch with pairs of coins (... |
![enter image description here][1]
If one were to work through the way you describe (each player taking into account only the previous players plan), then a 12 unit loop would quickly be observed. Case \$M=2\$ is the same as case \$M=14\$ which is the same as case \$M=26\$ etc. This doesn't really help anything.... |
I heard Martin Gardner in one of his books presented a crossword, which looks like 3x3 square. **Could you help me to find it?**
Also, have been there a smaller crossword puzzle created? |
There are arithmetic puzzles with some numbers (typically four) and rules for combining them (perhaps just the four basic arithmetic operators or including things like combining digits 2 and 3 to make numbers like 23).
http://puzzling.stackexchange.com/questions/623/write-twenty-four is an example which was recently... |
Is there a more specific name for this type of arithmetic puzzle? |
> You have N numbers written in one row. One needs to put any amount of these four symbols: "(",")","+","*" between them in such a way that the resulting expression is dividable by M.
> What are $[N,M]$ pairs it is always possible to do with (independently of the given sequence of numbers)?
For example, with $N=2... |
When can one construct expression dividable by M from any N numbers? |
> You have N numbers written in one row. One needs to put any amount of these four symbols: "$($", "$)$", "$+$", "$\cdot$" between them in such a way that the resulting expression is dividable by M.
> What are $[N,M]$ pairs it is always possible to do with (independently of the given sequence of numbers)?
For exa... |
> You have N numbers written in one row. One needs to put any amount of the four symbols $($, $)$, $+$, and $\times$ between them in such a way that the resulting expression is divisible by $M$.
> What are $[N,M]$ pairs it is always possible to do with (independently of the given sequence of numbers)?
For example... |
While you can use software and premade corpuses (whatever the latin declination would be?), I think that's a bit of a cop out. The first creators had no such tools at their disposal!
The first thing to do is to forget about trying to get the black cells in a pattern. I find it easiest to just start with a good long ... |
Is it common to see *non-words* (like the homophone [verticals in this answer][1]) used in English crosswords? Ignoring abbr., initials, odd names, various symbols, "Xth and Yth letter", "middle of X" and other similar simple fillers.
Was that particular puzzle just extra cryptic (for fun/art) or is this common?
... |
How common are non-words in crosswords? |
I see http://puzzling.stackexchange.com/questions/582/identify-unique-verbal-arithmetic-puzzles deals with trying to deal with uniqueness when creating Verbal Arithmetic puzzles, but I have a question that deals with something earlier in the creation process, namely **how does one go about picking words for the puzzle ... |
How does one go about picking words when creating Verbal Arithmetic puzzles? |
I've been struggling for a while with the [Between Worlds](https://play.google.com/store/apps/details?id=com.BitShiftInteractive.Games.BetweenWorlds) Android game, and have not yet solved it. I've only noticed a couple vague properties that hold in localized cases.
![enter image description here][1]
To operate th... |
How do I solve the "Between Worlds" Android puzzle game? |
First of all if either of the top two switches has a coin on it, drop another to clear it. Now, if either of them are leaning in (and by leaning I always mean the _top_ of the switch leaning that way), drop a coin on their outside so that lean out. From now on we will only ever hit a top-row switch with pairs of coins ... |
There is not winning strategy for 9cell 3x3 board.
There is a winning strategy for 16cell 4x4 board, or even for 10cell "3x3 + 1_cell_near_a_corner" board.
I wonder is there a board with 9 or less cells where a player has a winning strategy? So the question:
What is the smallest tic-tac-toe board to have a winni... |
What is the smallest tic-tac-toe board to have a winning strategy? |
There is not winning strategy for 9cell 3x3 board.
There is a winning strategy for 16cell 4x4 board, or even for 10cell "3x3 + 1_cell_near_a_corner" board.
I wonder is there a board with 9 or less cells where a player has a winning strategy? So the question:
What is the smallest tic-tac-toe board to have a winni... |
There is not winning strategy for 9cell 3x3 board.
There is a winning strategy for [16cell 4x4 board][1], or even for 10cell "3x3 + 1_cell_near_a_corner" board (see M.Gardner "Mathematical games").
I wonder is there a board with 9 or less cells where a player has a winning strategy? So the question:
What is the ... |
There is not winning strategy for 9cell 3x3 board.
There is a winning strategy for [16cell 4x4 board][1], or even for 10cell "3x3 + 1_cell_near_a_corner" board (see M.Gardner "Mathematical games"):
$$
\begin{array}{|c|c|}
\hline
& & & - \\
\hline
& & & - \\
\hline
\space\space & \space\space & ... |
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