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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.

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 ...
5
votes
1answer
8k views

Riddle (simple arithmetic problem/illusion)

I'm not sure how well known this "riddle" is but here it goes. 3 people go to a restaurant, each buy food worth 10.00. When they're done, they give 30.00 to the waitress. She gives the money to the ...
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: ...
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 ...
8
votes
10answers
6k views

3 Utilities | 3 Houses puzzle?

There's a puzzle where you have 3 houses and 3 utilities. You must draw lines so that each house is connected to all three utilities, but the lines cannot overlap. However, I'm fairly sure that the ...
9
votes
5answers
5k views

Variation on the Monty Hall Problem

Many of us know the Monty Hall Problem But the other day I was asked a variation of this riddle. The answer of the original question is, of course, $ 66\% $ in favor of changing doors, but this is ...
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 ...
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},\...
17
votes
4answers
35k views

How many distinct ways to climb stairs in 1 or 2 steps at a time?

I came across an interesting puzzle: You are climbing a stair case. It takes $n$ steps to reach to the top. Each time you can either climb $1$ or $2$ steps. In how many distinct ways can you ...
19
votes
4answers
22k views

Finding the n-th lexicographic permutation of a string

I have an ordered set of symbols S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }. I want to find the 1,000,000-th permutation in lexicographic order of S. It is a programming puzzle, but I wanted to figure out a ...
10
votes
3answers
2k views

Optimal algorithm for finding the odd sphere with a balance scale

Say we have $N$ spheres indexed as $1,2,3,\dotsc, N$ such that all of them have identical weight apart from one. We have to determine which sphere has the odd weight using just a balance scale. We ...
19
votes
2answers
4k views

After swapping the positions of the hour and the minute hand, when will a clock still give a valid time?

At 12 o'clock, the hour hand and minute hand of the clock can be swapped, and the clock still gives the same time, but at 6 o'clock, it can not be swapped. So in what cases when we swap the hour and ...
5
votes
7answers
5k views

The final state of 1000 light bulbs switched on/off by 1000 people passing by

There are 1000 light bulbs and 1000 tutors. All light bulbs are off. Tutor 1 goes flipping light bulb 1,2,3,4... tutor 2 then flips 2,4,6,8... tutor 3 then 3,6,9...etc until all 1000 tutors have ...
3
votes
4answers
2k views

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ show that $x=-c/b$ when $a=0$

OK, this one has me stumped. Given that the solution for $ax^2+bx+c =0$ $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\qquad(*)$$ How would you show using $(*)$ that $x=-c/b$ when $a=0$ (Please dont use $a=0$ ...
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 ...
12
votes
7answers
31k views

Famous puzzle: Girl/Boy proportion problem (Sum of infinite series)

Puzzle In a country in which people only want boys, every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop. What is ...
4
votes
4answers
9k views

What is the name of the logical puzzle, where one always lies and another always tells the truth?

So i was solving exercises in propositional logic lately and stumbled upon a puzzle, that goes like this: Each inhabitant of a remote village always tells the truth or always lies. A villager will ...
6
votes
2answers
1k views

What is the minimum number of locks on the cabinet that would satisfy these conditions?

Eleven scientists want to have a cabinet built where they will keep some top secret work. They want multiple locks installed, with keys distributed in such a way that if any six scientists are present ...
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 ...
6
votes
1answer
4k views

Fly and Two Trains Riddle

Two trains travel on the same track towards each other, each going at a speed of 50 kph. They start out 100km apart. A fly starts at the front of one train and flies at 75 kph to the front of the ...
9
votes
3answers
3k views

How Strong is an Egg?

You have two identical eggs. Standing in front of a 100 floor building, you wonder what is the maximum number of floors from which the egg can be dropped without breaking it. What is the minimum ...
7
votes
2answers
6k views

why 64 is equal to 65 here? [duplicate]

how is this possible? I know there is some trick, should someone please explain?!
3
votes
2answers
798 views

Name Drawing Puzzle

There is a party with 20 people, and everyone writes their name down on a piece of paper and puts it into a bag. The bag is mixed up, and each person draws one piece of paper. If you draw the name ...
4
votes
3answers
498 views

Maximize :: $A = B \times C$

Given $A = B \times C$ Using any $9$ digits form two numbers $B$ and $C$.All digits must be used exactly once. What is the maximum possible value of $A$?
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 ...
43
votes
2answers
940 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?
7
votes
6answers
12k views

A three-way duel (probability puzzle)

This puzzle is taken from Mathematical Puzzles: A Connoisseur's Collection [Peter Winkler]. I don't understand the solution. Alice, Bob, and Carol arrange a three-way duel. Alice is a poor shot, ...
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 ...
11
votes
5answers
6k views

Better than random

I have been trying to solve this question, but in vain. Please help. You are given two boxes with a number inside each box. The two numbers are different but you have no idea what they are. You ...
8
votes
2answers
2k views

Frog Jump Problem

Lotus leaves are arranged around a circle. A Frog starts jumping from one leaf in the manner described below. In the first jump it skips one leaf,next jump it skips two,three the next jump and ...
-1
votes
1answer
793 views

Integers and integer functions

Let $\Bbb{Z}^+$be the set of all non-negative integers where $n$ and $k$ are given natural numbers. We consider the following non-decreasing function, $$f:\Bbb{Z}^+ \to \Bbb{Z}^+$$ such that \begin{...
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 ...
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!}}}...
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 ...
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 ...
24
votes
3answers
2k views

Covering ten dots on a table with ten equal-sized coins: explanation of proof

Note: This question has been posted on StackOverflow. I have moved it here because: I am curious about the answer The OP has not shown any interest in moving it himself In the Communications of the ...
22
votes
3answers
758 views

Why everytime the final number comes the same?

I have come across an interesting puzzle. Write $20$ numbers. Erase any two number say $x$ and $y$ and and replace with $\text{Number}_{new} = xy/(x + y)$ OR $\text{Number}_{new}= x + y + xy$ ...
14
votes
5answers
19k views

How many bananas can a camel deliver without eating them all?

This is a fun puzzle I was assigned on the first day of highschool (over a decade ago). I just dug it up randomly from under my bed and thought I'd share it with the SE community. At the time, I ...
17
votes
4answers
3k views

9 pirates have to divide 1000 coins…

A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins. Arriving on a deserted island, they now have to split up the ...
13
votes
6answers
3k views

A strange puzzle having two possible solutions

A friend of mine asked me the following question: Suppose you have a basket in which there is a coin. The coin is marked with the number one. At noon less one minute, someone takes the coin ...
11
votes
4answers
1k views

Can the product $AB$ be computed using only $+, -,$ and reciprocal operators?

Can the product of $A, B$ be computed using only $+, -,$ and reciprocal operators using a calculator? You can use calculator's memory function (multiply and divide are broken though). Additional: I ...
10
votes
3answers
4k views

Why does the infinite prisoners and hats puzzle require the axiom of choice?

Infinite prisoners puzzle. The link to Wikipedia describes the puzzle, and the solution. The axiom of choice is used to pick a sequence from each equivalence class, which the prisoners memorize ...
2
votes
5answers
3k views

balance scale problem for 13 (not 12) items

The 12-item balance scale puzzle is very familiar. The object is to find the lone non-standard item (if one exists) out of a group of 12 seemingly identical items, using a balance scale and a maximum ...
7
votes
2answers
4k views

Lights Out Variant: Flipping the whole row and column.

So I found this puzzle similar to Lights Out, if any of you have ever played that. Basically the puzzle works in a grid of lights like so: 1 0 0 00 0 0 00 1 0 0 0 0 1 0 When you selected a light (...
6
votes
1answer
2k views

Four turtles/bugs puzzle

I was reading about the the four turtles/bugs math puzzle Four bugs are at the four corners of a square of side length D. They start walking at constant speed in an anticlockwise direction at all ...
5
votes
3answers
2k views

Chameleons Riddle

There are 10 red, 11 blue, 12 green chameleons. Sometimes, two chameleons meet. If they are the same color, nothing happens. If they are different colors, they will both change to the third color. ...
9
votes
4answers
70k views

Missing dollar problem [duplicate]

This sounds silly but I saw this and I couldn't figure it out so I thought you could help. The below is what I saw. You see a top you want to buy for $\$97$, but you don't have any money so you ...
6
votes
2answers
2k views

reversing digits and squaring

If we reverse the digits of $12$ we will get $21$. $12^{2}=144$. If we reverse its digits we will get $441$ which is $21^{2}$. Here is the puzzle. How many such two digit numbers are there? Digits ...
2
votes
1answer
8k views

How Old Are Children?

Two old friends, Jack and Bill, meet after a long time. Jack: Hey, how are you man? Bill: Not bad, got married and I have three kids now. Jack: That’s awesome. How old are they? Bill: The product of ...
3
votes
1answer
559 views

Find possible number of triangles with integer sides for a given inradius

inradius = $5$ possible triangle sides $(25, 20, 15)$ $(37, 35, 12) (39, 28, 17)$ ... Find formula to find other possible sides of triangles.