# Questions tagged [puzzle]

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

101
questions

**9**

votes

**0**answers

198 views

### Sum and Product game

Two perfect logicians Steve and Pete, who have never met, before are imprisoned by an eccentric villain. "I have two positive integer numbers x and y" he says to them. "I will tell Steve the sum x+y, ...

**25**

votes

**1**answer

722 views

### The lion and the zebras

The lion plays a deadly game against a group of $N$ zebras that takes place in the steppe (= an infinite plane). The lion starts in the origin with coordinates $(0,0)$, while the $N$ zebras may ...

**1**

vote

**1**answer

146 views

### Bike lock graph

Motivation. I have a bike lock with 4 dials, and I was wondering whether I can reach any combination by always turning a fixed number $k$, say $k=2$, of the dials, by $1$ position, instead of just ...

**1**

vote

**2**answers

236 views

### Constructing a vector consisting of nonnegative entries

Consider constructing a vector $v=(a_1,a_2,\ldots,a_n)$ consisting of nonnegative integers such that $a_1=1$ and, if $a_j$'s are nonzero, then $a_j\equiv a_{n-j+2}+j-1 \pmod m\ \forall 1<j\le\frac{...

**10**

votes

**0**answers

273 views

### Does Chu and Hough's solution to the mixing time of the 15-puzzle carry over to the Rubik's cube?

In his 1988 book Group Representations in Probability and Statistics , Diaconis considers mixing times of the 15-puzzle. He states:
Here is a simplified version: Consider the blank as a $16$th ...

**6**

votes

**1**answer

222 views

### Guessing the number of other $1$'s in a binary sequence

I have posed the following question on math.stackexchange.com but have not received an answer. So I would like to seek experts' opinion here.
Consider the set of all binary sequence of length $n+1$, $...

**0**

votes

**3**answers

420 views

### Given $N$ integers on a circle, how to choose them in pairs to obtain minimum sum?

(Added by YCor 2019 July 7): it has been mentioned in the comments that this is part of a contest "Circular merging, July Challenge 2019 Division 1", where an equivalent question (just more clearly ...

**2**

votes

**0**answers

97 views

### Game on groups (generalization of spinning switches puzzle)

Alice and Bob are playing a game as follows:
Initially
There're two subgroups $A,B$ of Sym(n) known to both Alice and Bob
There're $n$ slots $S_1, \cdots, S_n$ and $n$ boxes $B_1, \cdots, B_n$. ...

**56**

votes

**2**answers

2k views

### Guessing each other's coins

I recently thought about the following game (has it been considered before?).
Alice and Bob collaborate. Alice observes a sequence of independent unbiased random bits $(A_n)$, and then chooses an ...

**2**

votes

**1**answer

75 views

### Lower bound on the number of solutions of N-queens problem

The OEIS lists the number of solutions of N-queens problem
(Number of ways of placing n nonattacking queens on an n X n board). However, no formula is given. It is easy to observe that each number in ...

**1**

vote

**0**answers

57 views

### Computational complexity of fractions multiplication puzzle

I developed a puzzle for which I am seeking an EFFICIENT algorithm (I searched Google but did not find this puzzle in the literature, ):
You have $k$ rationals, $n_1/d_1, n_2/d_2, ..., n_k/d_k$.
...

**3**

votes

**1**answer

138 views

### Matching two sequences between each other

Given the sequence of symbols $A$ (contains ~10,000 symbols) and sequence of blocks $B$ (contains ~3,000 blocks, ~30 symbols inside each block) I need to exclude some blocks from sequence $B$ so that ...

**13**

votes

**2**answers

1k views

### Can we make 101 almost perfect banknotes from 100?

Disclaimer. The practical execution of the algorithm in question might be illegal in certain jurisdictions, and is thus strongly discouraged by the poser of the problem.
This recent post on the ...

**3**

votes

**0**answers

684 views

### Puzzle in 3D grid with black and white boxes, related to shelling

Consider a $n$ by $n$ by $n$ grid represented by the set of $3$-uples $S=\{1,2,\dots, n\}^3$.
A line (resp. slice) of $S$ is a subset of cardinal $n$ (resp. $n^2$) where two components (resp. one ...

**0**

votes

**0**answers

165 views

### Mathematical Aspects of Hectoc-type Puzzles

hectoc is a puzzle, where one is given a sequence of six decimal digits and the task is to intersperse arithmetic operations from the given set $+,-,/,*$ and matching brackets $(,)$ in a way that the ...

**24**

votes

**1**answer

944 views

### Has there been any new development on the Freudenthal Problem?

Background
I have seen a few variants of this Sum-and-Product puzzle in the past. The premise of these puzzles is as follows
Sam hears the sum of two numbers, Polly the product. The numbers are ...

**19**

votes

**1**answer

624 views

### Who wins the Rubik's cube game?

This game has two players, Spoiler and Solver. We start with a solved 3x3x3 rubik's cube (to make the problem easier).
Solver and Spoiler take turns making 90 degree twists (starting with Solver). ...

**34**

votes

**2**answers

3k views

### Who wins two player sudoku?

Let's say players take turns placing numbers 1-9 on a sudoku board. They must not create an invalid position (meaning that you can not have the same number in within a row, column, or box region). The ...

**1**

vote

**2**answers

92 views

### Fill the board with zeroes, inverting the intersections of rows and columns

Given $n \times n$ board randomly filled with $x \in \{0, 1\}$. When you invert value in cell $x_{i,j}$, all corresponding values in $row_i$ and $col_j$ are inverted too. The goal is to fill the board ...

**3**

votes

**0**answers

73 views

### Find four sets of coins with similar weights

There are $n\geq 4$ coins. You are allowed to ask for the weight of any set of coins. What is the worst-case (asymptotic) minimum number of questions after which you can divide the coins into four ...

**31**

votes

**4**answers

2k views

### A puzzle with some jumping frogs

(The following puzzle is ispired by this nice video of Gordon Hamilton on Numberphile)
In a pond there are $n$ leaves placed in a circle, for convenience they are numbered clockwise by $0,1,\ldots,n-...

**4**

votes

**0**answers

197 views

### How many inclusion preserving maps of subsets?

Let $S$ be a set with $n$ elements and $\Sigma_k= \{ R\subseteq S \mid |R|=k \}$. For $k\le n/2$ how many bijections $f$ are there between $\Sigma_k$ and $\Sigma_{n-k}$, such that $x\subseteq f(x)$?
...

**3**

votes

**1**answer

170 views

### Generalized Shared Birthday

Suppose a year has $d$ days. How many people should be in a room so that there are at least $2k$ people in the room with birthdays shared with each other (all could be same day or there could be $k$ ...

**2**

votes

**0**answers

235 views

### How many different solutions does this cube puzzle have?

I designed a 4x4x4 soma cube in AutoCad and then built it with wood cubes.
Now I want to know how many different solutions there are for it.
Similar to the Bedlam Cube, there are twelve pentacube and ...

**5**

votes

**2**answers

429 views

### Separating Heavier from the Lighter Balls

This Question was originally posted Here, where I'm more interested in the methods for manual solutions yielding $n$ or less moves on average.
I wanted to post it here as well, to see what the people ...

**1**

vote

**0**answers

200 views

### Concrete solution to the (oriented) Oberwolfach problem with one table

I asked the following on MSE, but it received little attention...
The oriented Oberwolfach problem (with only one table) and its solution are the following.
In a meeting of $n$ people during $n-1$ ...

**21**

votes

**1**answer

985 views

### Covering of a surface of a cube $n\times n \times n$ by pieces of paper $1\times 6$

When I was too young one of my problems was in the list of problems of All-Russian Olympiad. The problem is the following:
Problem. We have a surface of a cube $n\times n \times n$ such that each ...

**15**

votes

**1**answer

1k views

### 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 ...

**33**

votes

**1**answer

2k 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 ...

**2**

votes

**0**answers

85 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?

**5**

votes

**0**answers

403 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 ...

**12**

votes

**3**answers

1k 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 ...

**3**

votes

**2**answers

159 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/...

**5**

votes

**1**answer

297 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 ...

**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 ...

**0**

votes

**0**answers

90 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.
...

**24**

votes

**2**answers

1k 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 ...

**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 ...

**6**

votes

**1**answer

468 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 ...

**3**

votes

**1**answer

448 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 ...

**10**

votes

**1**answer

1k 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 ...

**4**

votes

**0**answers

239 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 ...

**5**

votes

**2**answers

432 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)....

**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 ...

**18**

votes

**1**answer

975 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 ...

**13**

votes

**4**answers

2k 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 ...

**7**

votes

**3**answers

2k 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 ...

**3**

votes

**1**answer

343 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 $2n-1$ ...

**8**

votes

**5**answers

2k 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 ...