Questions tagged [puzzle]

Recreational mathematics or puzzles with serious mathematical content. Note that math contest problems are generally considered off-topic.

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A fun game related to knot theory

I recently learned the following rather fun game: a group of people is standing up roughly in circle, facing each other. Then participants randomly join hands, in such a way that nobody holds its own ...
Adrien's user avatar
  • 8,234
35 votes
1 answer
3k views

"The Two Sheriffs" puzzle

This puzzle is taken from the book Mathematical puzzles: a connoisseur's collection by P. Winkler. Two sheriffs in neighboring towns are on the track of a killer, in a case involving eight ...
Alexey Ustinov's user avatar
2 votes
0 answers
96 views

Limit shape for oil-shaped stack in the max overhang problem

In the Maximum Overhang paper, the authors mention an oil-shaped configuration (ref. page 19) What is known about the curve that limits this shape?
user66662's user avatar
5 votes
0 answers
464 views

What is God's number for the WrapSlide puzzle?

WrapSlide is a slide-puzzle (reminding of Rubik's Cube) consisting of a 6x6 grid of coloured tiles which are separated into four quadrants of 3x3 tiles. When it is unmixed all the tiles in a quadrant ...
Alewyn Burger's user avatar
13 votes
4 answers
2k views

Undecidable puzzles

There are plenty of popular NP-hard puzzles, for example, generalized Sudoku ($n^2 \times n^2$-board), Flow (I cannot give a source for this), Minesweeper, etc. Recently, I read a bit about aperiodic ...
Per Alexandersson's user avatar
3 votes
2 answers
167 views

All Integers from the Smallest Digit Stream with a Window Filter

Let's represent integers with D digits where each digit has B values (i.e., the base is B and we effectively work only with integers between 1 and B^D). Is it possible to choose a single cyclic/...
bobuhito's user avatar
  • 1,537
5 votes
1 answer
336 views

coin reversal puzzle with one hand and two stacks

Suppose that you have N labeled coins pinched in one stack in your fingertips (your palm is above your fingers and your palm is facing down, so that you can drop as many coins as needed from the ...
bobuhito's user avatar
  • 1,537
10 votes
3 answers
2k views

Could a perfect squared square be split into two perfect squared squares?

This is a geometric puzzle though it might conceivably also define a special class of Pythagorean triples. A perfect squared square PSS is a square (as a plane figure) partitioned into smaller ...
Mirko's user avatar
  • 1,345
0 votes
0 answers
94 views

Minimize the length of two disjoint segments in the string with given property

You are given a string s of size n, consisting of characters A and B only. You have to find minimum sum of size of the two disjoint segments of the string s such that number of A's in them are >= z. ...
Praveen Dhinwa's user avatar
25 votes
2 answers
2k views

An Interesting Optimization Problem

You are given n non-negative integers $a_1, a_2 ,, a_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference ...
Praveen Dhinwa's user avatar
9 votes
1 answer
1k views

How many ways to partition a group of people?

My friend (who is a medical student!) posed me the following question: There are 70 people, and you want to split them up into 10 groups of 7 people each. Two such partitions are "compatible" if no ...
anon's user avatar
  • 303
5 votes
1 answer
575 views

Infinite blue eyed islanders puzzle

Can the well known blue eyed islanders puzzle be extended to an infinite number of islanders? In that puzzle, a set of $k$ islanders, each with either blue eyes or non-blue eyes, each knows the color ...
user44653's user avatar
  • 251
3 votes
1 answer
502 views

Simple reason that a mathematician cannot do better than random when guessing contents of a box?

I have a question about the finite analog of the puzzle proposed here involving mathematicians guessing the contents of boxes. Specifically, suppose there are $k$ unopened boxes each containing a ...
user44653's user avatar
  • 251
15 votes
2 answers
2k views

Can an infinite number of mathematicians guess the number in a box with only one error?

In this question the following observation was made: Consider a sequence of boxes numbered 0, 1, ... each containing one real number. The real number cannot be seen unless the box is opened. Define ...
user44653's user avatar
  • 251
5 votes
1 answer
363 views

Dissecting using a ruler and compass

The problem is to cut a regular hexagon into parts that can be put together (without overlaps or wasting any parts) to make an equilateral triangle using only a ruler and compass (and scissors). What ...
Simd's user avatar
  • 3,195
5 votes
2 answers
448 views

A simple language and systematic computations

The following somewhat popular simple computer language was enjoyed on sci.math, sci.math.research, pl.sci.matematyka, and perhaps before and after at several places (I wish I knew it's exact history)....
Włodzimierz Holsztyński's user avatar
50 votes
8 answers
3k views

Puzzle on deleting k bits from binary vectors of length 3k

Consider all $2^n$ different binary vectors of length $n$ and assume $n$ is an integer multiple of $3$. You are allowed to delete exactly $n/3$ bits from each of the binary vectors, leaving vectors ...
20 votes
1 answer
1k views

A game on sets of reals

A 2 player game on $\mathcal{P}(\mathbb{R})$: Players take turns playing uncountable sets of reals. Each play must be a subset of the previously played set. Player 1 wins if the intersection of all ...
Monroe Eskew's user avatar
  • 18.1k
13 votes
4 answers
3k views

Mathematical model for Hanoi Towers

The strategy for the Hanoi Tower puzzle is quite simple. It is based on parity only. In an $n$-pieces puzzle, $2^n-1$ moves are sufficient to carry the whole pile from one pole to another one. My ...
Denis Serre's user avatar
  • 51.5k
7 votes
3 answers
3k views

Blue and red balls puzzle

I was sent this puzzle and wondered if it is known or if its origin is known? (I see colored ball puzzles are also in vogue.) Consider a bag with $n$ red balls and $n$ blue balls. At each turn you ...
marcin's user avatar
  • 71
3 votes
1 answer
357 views

two boy scouts problems

As a member of boy scouts I was considering the following problem: suppose you're organising some kind of olympic games... *You divide the boys in $2n$ teams (subsets of equal size) *There are $2n-1$ ...
5th decile's user avatar
  • 1,461
8 votes
5 answers
3k views

Another colored balls puzzle (part II)

The same colleague as in Another colored balls puzzle asked me the following variant which she called "part II". Imagine you have $n$ balls in a bag that are colored from $1$ to $n$. At each turn ...
0 votes
2 answers
2k views

Gödel, Escher, Bach: b is a power of 10. [closed]

I’d like to verify if my formula correctly expresses that a number is a power of $ 10 $, using the $ \sf{TNT} $ language provided by Hofstadter in his famous book Gödel, Escher, Bach: An Eternal ...
almahed's user avatar
7 votes
11 answers
2k views

A function that is defined everywhere but has unknown values [closed]

For pedagogical purposes I am looking for a function $\mathbb{N}\to\mathbb{N}$ that is defined everywhere but has most of its values unknown. Although such a function cannot be simple by definition, ...
Gerard's user avatar
  • 195
8 votes
3 answers
2k views

Bike lock puzzle

I was wondering this when using my bike lock, a combination lock with four dials, each of which has ten digits (0-9) on it in numerical order. Suppose a bicyclist decides that, from now on, after ...
user25491's user avatar
9 votes
2 answers
2k views

Knight tour prime (conjecture)

Hello, I have the following conjecture: Write all numbers from $1$ to $n^2$ over an $n\times n$ board as usually. There not exists $n$ such that we can find a hamiltonian path on primes numbers with ...
Roberto Bosch Cabrera's user avatar
7 votes
1 answer
346 views

Labeling a Square Array

Suppose that the $n^2$ cells of an $n\times n$ array are labeled with the integers $1, \dots, n^2$. Under the traditional left-to-right and top-to-bottom labeling, the labels of horizontally adjacent ...
Martin Erickson's user avatar
22 votes
2 answers
2k views

How to get rich in a Hilberts Hotel?

Suppose you can make infinitely many copies of yourself. Each of them starts his/her life in a Hilberts Hotel, where each room is labeled by an element in the free group with two generators, and ...
Sune Jakobsen's user avatar
22 votes
5 answers
3k views

Irreversible chess

Suppose we play a chess-variant, where any finite number of pieces are allowed, and the board is as large as we wish, but only two kings in total. And there is no 50 move-rule, no castling and no ...
GM2001's user avatar
  • 223
16 votes
2 answers
3k views

God's number for the $n \times n \times n$-cube

This is a question about Rubik's Cube and generalizations of this puzzle, such as Rubik's Revenge, Professor's cube or in general the $n \times n \times n$ cube. Let $g(n)$ be the smallest number $m$, ...
Martin Brandenburg's user avatar
8 votes
2 answers
1k views

Probability of a black path on a random chess board

Take a $2n$ by $2n$ chess board (oriented so the grid lines are vertical and horizontal). Usually there are $2n^2$ squares coloured black and $2n^2$ squares coloured white so that a black square is ...
alext87's user avatar
  • 3,167
28 votes
1 answer
4k views

A Presentation for Rubik's cube group?

Let $G$ be Rubik's cube group. It is generated by the rotations by 90 degrees $L,R,D,U,F,B$ (left, right, down, up, front, behind), but what relations beyond $L^4=R^4=...=B^4=1$ do they satisfy? Thus ...
Martin Brandenburg's user avatar
6 votes
6 answers
2k views

Circumference of Convex Shapes

Here is a puzzle I found in Mitteilungen der DMV (roughly, "Letters of the German Society of Mathematicians"), issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch ...
Matthias Goergens's user avatar
1 vote
0 answers
916 views

Honest and Deceitful Professors Problem [closed]

I found this in An Introduction to Bioinformatics Algorithms. I've paraphrased for clarity. There are 100 professors. Some are honest, while others are dishonest. There are more honest professors ...
randomPerson's user avatar
22 votes
3 answers
2k views

An elementary problem in Euclidean geometry [closed]

This problem was first put to me by Luke Pebody (who did not know the answer at the time) and after some work I am yet to find a proof or counterexample. I would be grateful of any insights. Call a ...
Chris Taylor's user avatar
8 votes
8 answers
5k views

probability and math puzzle books/references [closed]

Hi All, I'd like to solve some math puzzles, especially in the context of probability theory, but I'm open to other areas too. The kind of problems that does not require much knowledge of mathematics, ...
70 votes
7 answers
13k views

Identifying poisoned wines

The standard version of this puzzle is as follows: you have $1000$ bottles of wine, one of which is poisoned. You also have a supply of rats (say). You want to determine which bottle is the poisoned ...
Qiaochu Yuan's user avatar
12 votes
1 answer
905 views

Guessing a subset of {1,...,N}

I pick a random subset $S\subseteq\lbrace1,\ldots,N\rbrace$, 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 to ...
Dave R's user avatar
  • 856
24 votes
1 answer
2k views

A puzzle about finding three points $(x,y)$, $(x,z)$ and $(y,z)$ in a subset of a square.

I was asked (by myself) to give a proof of the following seemingly simple geometric statement, but after thinking a little I now suspect it could be less elementary than I thought (or am I being silly?...
Pietro Majer's user avatar
  • 56.5k
54 votes
3 answers
7k views

cube + cube + cube = cube

The following identity is a bit isolated in the arithmetic of natural integers $$3^3+4^3+5^3=6^3.$$ Let $K_6$ be a cube whose side has length $6$. We view it as the union of $216$ elementary unit ...
Denis Serre's user avatar
  • 51.5k
13 votes
1 answer
3k views

Hard Cube Puzzle

You are and your friend are given a list of $N$ distinct integers and are told this: Six distinct integers from the list are selected at random and placed one at each side of a cube. The cube is ...
user12265's user avatar
  • 133
3 votes
1 answer
801 views

The Mystic Rose

Consider $n$ points equally spaced around the unit circle, joined by all possible combinations of lines to make a complete graph. Let $g(n)$ be the number of triangles formed in the resulting diagram. ...
Chris Taylor's user avatar
13 votes
1 answer
7k views

Generalization of a horse-racing puzzle

A well-known puzzle goes: "Suppose that you have 25 horses and a racetrack on which you can race up to 5 horses. If the outcome of each race only tells you the relative speeds of the horses in the ...
Chris Taylor's user avatar
6 votes
2 answers
616 views

Point in Polygon algorithm from the viewpoint of a robot

I've come across the following puzzle: You're on an island, on which there is a fence (which is a simple closed contour). You need to determine whether you're inside or outside the fence. ...
ohadsc's user avatar
  • 163
29 votes
6 answers
7k views

Integers in a triangle, and differences

I read about the following puzzle thirty-five years ago or so, and I still do not know the answer. One gives an integer $n\ge1$ and asks to place the integers $1,2,\ldots,N=\frac{n(n+1)}{2}$ in a ...
Denis Serre's user avatar
  • 51.5k
6 votes
1 answer
504 views

function that sums to zero over cube vertices

Does anyone have an answer to the three-dimensional analogue of the 2009 Putnam Competition A1 problem, viz., if $f\colon \mathbb{R}^3 \rightarrow \mathbb{R}$ satisfies $\sum_{i=1}^8 f(a_i) = 0$ ...
Greg Marks's user avatar
  • 1,198
26 votes
3 answers
3k views

What is this subgroup of $\mathfrak S_{12}$?

On some occasion I was gifted a calendar. It displays a math quizz every day of the year. Not really exciting in general, but at least one of them let me raise a group-theoretic question. The quizz: ...
Denis Serre's user avatar
  • 51.5k
21 votes
6 answers
13k views

A balls-and-colours problem

A box contains n balls coloured 1 to n. Each time you pick two balls from the bin - the first ball and the second ball, both uniformly at random and you paint the second ball with the colour of the ...
Hedonist's user avatar
  • 1,269
3 votes
1 answer
582 views

Is always possible to slice a pizza in a fair way

Given a pizza, represented by the unit disk $D_1(0,0)=\{(x,y)\in\mathbb{R}^2\mid \|(x,y)\|\leqslant 1\}$, and given $N$ slices of $r$-pepperoni, represented by disks $D_r(a_i,b_i)=\{(x,y)\in\mathbb{R}^...
Sirolf's user avatar
  • 493
8 votes
1 answer
3k views

12 balls weighing puzzle

In an article describing the twelve balls weighing problem, the author mentions a solution that involves the finite projective plane of order 3, discovered by Rick Wilson. Does anyone know what this ...
Charles Chen's user avatar