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Questions tagged [puzzle]

This tag is meant for questions about the mathematical principles behind games, riddles, or their possible solutions. If the answer is known to you please do not use this tag to "riddle" other users, but rather to ask about the correctness of a possible solution or ways to extend and improve an existing solution.

268
votes
39answers
46k views

One question to know if the number is 1, 2 or 3

I've recently heard a riddle, which looks quite simple, but I can't solve it. A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl ...
214
votes
4answers
9k views

How many fours are needed to represent numbers up to $N$?

The goal of the four fours puzzle is to represent each natural number using four copies of the digit $4$ and common mathematical symbols. For example, $165=\left(\sqrt{4} + \sqrt{\sqrt{{\sqrt{4^{4!}}}...
148
votes
20answers
10k views

Are there any open mathematical puzzles? [closed]

Are there any (mathematical) puzzles that are still unresolved? I only mean questions that are accessible to and understandable by the complete layman and which have not been solved, despite serious ...
135
votes
3answers
16k views

Slice of pizza with no crust

The following question came up at a conference and a solution took a while to find. Puzzle. Find a way of cutting a pizza into finitely many congruent pieces such that at least one piece of pizza ...
110
votes
13answers
61k views

Taking Seats on a Plane

This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless Imagine there are a 100 people in line to ...
103
votes
18answers
10k views

Fastest way to meet, without communication, on a sphere?

I was puzzled by a question my colleague asked me, and now seeking your help. Suppose you and your friend* end up on a big sphere. There are no visual cues on where on the sphere you both are, and ...
99
votes
12answers
17k views

Logic puzzle: Which octopus is telling the truth?

King Octopus has servants with six, seven, or eight legs. The servants with seven legs always lie, but the servants with either six or eight legs always tell the truth. One day, four servants met. ...
98
votes
5answers
6k views

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif ...
95
votes
11answers
12k views

Riddles that can be solved by meta-assumptions

The website of my university posted a riddle that goes something like this: Riddle There are three men named 1,2 and 3 and each one has two colored dots on his forehead. Possible colors are ...
95
votes
11answers
11k views

Mathematician vs. Computer: A Game

A mathematician and a computer are playing a game: First, the mathematician chooses an integer from the range $2,...,1000$. Then, the computer chooses an integer uniformly at random from the same ...
93
votes
17answers
19k views

How do you find the center of a circle with a pencil and a book?

Given a circle on a paper, and a pencil and a book. Can you find the center of the circle with the pencil and the book?
89
votes
1answer
3k views

The Right Triangle Game

I am looking for a deeper understanding, particularly the optimum strategy and the maximum score as a function of grid size, of the following (single-player) game played with an $n$ by $m$ grid: ($6 ...
88
votes
5answers
5k views

Cutting sticks puzzle

This was asked on sci.math ages ago, and never got a satisfactory answer. Given a number of sticks of integral length $ \ge n$ whose lengths add to $n(n+1)/2$. Can these always be broken (by ...
86
votes
23answers
6k views

Your favourite maths puzzles

Okay, so this question was bound to come up sooner or later- the hope was to ask it well before someone asked it badly... We all love a good puzzle To a certain extent, any piece of mathematics is a ...
84
votes
24answers
18k views

100 blue-eyed islanders puzzle: 3 questions

I read the Blue Eyes puzzle here, and the solution which I find quite interesting. My questions: What is the quantified piece of information that the Guru provides that each person did not already ...
79
votes
9answers
9k views

100 Soldiers riddle

One of my friends found this riddle. There are 100 soldiers. 85 lose a left leg, 80 lose a right leg, 75 lose a left arm, 70 lose a right arm. What is the minimum number of soldiers losing all ...
77
votes
2answers
24k views

Predicting Real Numbers

Here is an astounding riddle that at first seems impossible to solve. I'm certain the axiom of choice is required in any solution, and I have an outline of one possible solution, but would like to ...
75
votes
5answers
11k views

Help me solve my father's riddle and get my book back

My father is a mathteacher and as such he regards asking tricky questions and playing mathematical pranks on me once in a while as part of his parental duty. So today before leaving home he sneaked ...
75
votes
9answers
20k views

Logic problem: Identifying poisoned wines out of a sample, minimizing test subjects with constraints

I just got out from my Math and Logic class with my friend. During the lecture, a well-known math/logic puzzle was presented: The King has $1000$ wines, $1$ of which is poisoned. He needs to ...
74
votes
13answers
8k views

What would have been our number system if humans had more than 10 fingers? Try to solve this puzzle.

Try to solve this puzzle: The first expedition to Mars found only the ruins of a civilization. From the artifacts and pictures, the explorers deduced that the creatures who produced this ...
73
votes
14answers
83k views

Dividing 100% by 3 without any left

In mathematics, as far as I know, you can't divide 100% by 3 without having 0,1...% left. Imagine an apple which was cloned two times, so the other 2 are completely equal in 'quality'. The totality ...
70
votes
14answers
6k views

What is $x^y$? How to understand it?

$x+y=z$ I have a pen. He has a pen. Total is two pen. This is plus. $x-y=z$ I had two pens. A pen was lost. So, I have a pen. Total remaining is one. This is minus. $x\cdot y=z$ I have two pens. ...
57
votes
0answers
7k views

“The Bachelorette Problem” (slightly adapted from Tao's Google+ account) [duplicate]

The following puzzle being very much recreational for me, I couldn't resist myself from sharing it with my fellow MSE user friends. Let's have a look at it. You are the most eligible bachelorette ...
55
votes
12answers
7k views

Any smart ideas on finding the area of this shaded region?

Don't let the simplicity of this diagram fool you. I have been wondering about this for quite some time, but I can't think of an easy/smart way of finding it. Any ideas? For reference, the Area is: ...
51
votes
2answers
3k views

A discrete math riddle

Here's a riddle that I've been struggling with for a while: Let $A$ be a list of $n$ integers between 1 and $k$. Let $B$ be a list of $k$ integers between 1 and $n$. Prove that there's a non-empty ...
49
votes
16answers
16k views

Interview riddle

On the Mathematics chat we were recently talking about the following problem @Chris'ssis had to solve during an interview : $$3\times 4=8$$ $$4\times 5=50$$ $$5\times 6=30$$ $$6\times 7=49$$ $$7\...
49
votes
3answers
6k views

Sharing a pepperoni pizza with your worst enemy

You are about to eat a pepperoni pizza, which is sliced into eight pieces. Each pepperoni will unambiguously belong to some slice (no pepperoni is "between" slices). The caveat is that you have to ...
46
votes
3answers
2k views

Expected number of people to not get shot?

Suppose $n$ gangsters are randomly positioned in a square room such that the positions of any three gangsters do not form an isosceles triangle. At midnight, each gangster shoots the person that is ...
45
votes
7answers
6k views

How many scientists can survive?

Yesterday the aliens took 100 scientists from Earth as prisoners. They want to test how smart the humans are. The aliens made 101 headbands, numbered from 1 to 101. On the contest day, they throw ...
45
votes
3answers
2k views

Tall fraction puzzle

I was given this problem 30 years ago by a coworker, posted it 15 years ago to rec.puzzles, and got a solution from Barry Wolk, but have never seen it again. Consider the series: $$1, \frac{1}{2},\...
43
votes
7answers
60k views

How many triangles

I saw this question today, it asks how many triangles are in this picture. I don't know how to solve this (without counting directly), though I guess it has something to do with some recurrence. How ...
43
votes
2answers
941 views

Guessing a subset of $\{1,…,N\}$

I pick a random subset $S$ of $\{1,\ldots,N\}$, and you have to guess what it is. After each guess $G$, I tell you the number of elements in $G \cap S$. How many guesses do you need?
42
votes
2answers
3k views

Proving a jigsaw is possible

This is an offshoot of this question. Suppose we have jigsaw puzzle pieces which are basically squares but where each side can be either straight, concave or convex. An example of three such pieces ...
41
votes
6answers
4k views

Fun Geometric Series Puzzle

I recently was reminded of a puzzle I solved in college and thought I'd give it a shot again. However, being distanced from college math, I am having a harder time remembering how I arrived at the ...
41
votes
6answers
3k views

A variant of the Monty Hall problem

Everybody knows the famous Monty Hall problem; way too much ink has been spilled over it already. Let's take it as a given and consider the following variant of the problem that I thought up this ...
39
votes
6answers
4k views

Drinking habits riddle: the village is $90$%$+80$%$+70$%$+60$%$ = 300$ % saturated with liquids

In a small village $90\%$ of the people drink Tea, $80\%$ Coffee, $70\%$ Whiskey and $60\%$ Gin. Nobody drinks all four beverages. What percentage of people of this village drinks alcohol? I got this ...
37
votes
9answers
76k views

Maximum board position in 2048 game

A game called 2048 is making rounds on social media. I am trying to determine the maximum score attainable for this game. Let's assume WLOG that only 2s are returned (if 4s are possible the max score ...
36
votes
5answers
2k views

Can this ant find its way back to the nest?

So the puzzle is like this: An ant is out from its nest searching for food. It travels in a straight line from its nest. After this ant gets 40 ft away from the nest, suddenly a rain starts to pour ...
36
votes
5answers
8k views

Chance of getting a date

A girl you like tells you that there's a $1$% chance she'll go out on a date with you. What is the expected number of times you have to ask her out before you get a date? (Not a homework problem - I ...
35
votes
4answers
948 views

Fibonacci numbers from $998999$

Is there a nice explanation of $$\frac{1}{998999}=0.000\,\underbrace{001}_{F_1}\,\underbrace{001}_{F_2}\,\underbrace{002}_{F_3}\,\underbrace{003}_{F_4}\,\underbrace{005}_{F_5}\,\underbrace{008}_{\...
33
votes
5answers
691k views

How many squares actually ARE in this picture? Is this a trick question with no right answer?

This is one of those popular pictures on sites like Facebook. I always see a huge variation of answers such as $8, 9, 16, 17, 24, 28, 30, 40, 41, 52,$ etc., yet I've never seen a definitive answer on ...
32
votes
9answers
4k views

Probability brain teaser with infinite loop

I found this problem and I've been stuck on how to solve it. A miner is trapped in a mine containing 3 doors. The first door leads to a tunnel that will take him to safety after 3 hours of travel. ...
32
votes
6answers
11k views

Is there no solution to the blue-eyed islander puzzle?

Text below copied from here The Blue-Eyed Islander problem is one of my favorites. You can read about it here on Terry Tao's website, along with some discussion. I'll copy the problem here as ...
32
votes
3answers
2k views

Guaranteed Checkmate with Rooks in High-Dimensional Chess

Given an infinite (in all directions), $n$-dimensional chess board $\mathbb Z^n$, and a black king. What is the minimum number of white rooks necessary that can guarantee a checkmate in a finite ...
32
votes
1answer
800 views

A fun Pascal-like triangle

Inspired by Pascal, I put on some shackles and a thorny belt. Inspiration came pouring in, and I thought of the following triangle: $$ \begin{array}{rcccccccccc} & & & & &...
31
votes
10answers
4k views

how to find the balls?

You have 15 balls, 2 of them are radioactive. You have to run 7 tests (no more) on the balls which will sort out the 2 radioactive ones guarenteed every time. You can test them in groups, or even one ...
31
votes
1answer
1k views

Mosquito in the Room (Puzzle)

Pondering about a very real mosquito in my room I came up with this puzzle: You are in a room of dimensions $10$ m $\times$ $10$ m $\times$ $3$ m. There is also a (point-shaped) mosquito in the ...
30
votes
9answers
6k views

Which simple puzzles have fooled professional mathematicians?

Although I'm not a professional mathematician by training, I felt I should have easily been able to answer straight away the following puzzle: Three men go to a shop to buy a TV and the only one ...
30
votes
6answers
11k views

100 prisoners and a lightbulb

100 prisoners are imprisoned in solitary cells. Each cell is windowless and soundproof. There's a central living room with one light bulb; the bulb is initially off. No prisoner can see the light bulb ...
29
votes
2answers
1k views

Bruteforcing a keypad lock

A particular lock at my university has a keypad with the digits 1-5 on it. When pressed in the correct permutation of those five digits, the lock will open. Obviously, since there are only 120 ...