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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God's number for higher dimensional Rubik's cubes

In this MO question, user Martin Brandenburg asks about God's number for $n \times n \times n$-cubes for $n>3$. Here, God's number $g(n)$ was defined as the smallest number $m$ such that every ...
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  • 3,463
4 votes
2 answers
239 views

The mower's challenge

Weeds have taken over the paths (two squares). If mowed, they don't grow back, but unmowed weeds spread at speed $1$ along the road. What's the minimum speed of the mower to get rid of all the weeds? ...
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  • 2,517
-3 votes
1 answer
244 views

What is a good formalization of this classic math puzzle? [closed]

Here is a classic math olympiad problem (but this is NOT my question!): Each of the girls A and B tells the teacher a positive integer but neither of them knows the other's number. The teacher writes ...
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15 votes
1 answer
653 views

English name and references for a combinatorial puzzle from Japan [closed]

I am looking for the name and references of the following puzzle. There are n intersecting circles in a row. At the center of the circle and at the intersection of the two circles, fill the numbers 1, ...
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8 votes
0 answers
269 views

The busy Star Guardian

On an infinite plane, the Prime Star has disintegrated into four constituent stars, the North Star, the South Star, the East Star and the West Star, each traveling at a constant speed of $1$ in their ...
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6 votes
1 answer
240 views

3D Edge matching puzzle generation

I have this weird idea for a puzzle/toy (or torture device, depending on how you look at it) I've been trying to make for years now. I happen to be worse at this kind of math as I thought; and I'd be ...
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  • 69
0 votes
1 answer
198 views

Russell's definite description and vacuous truth: a puzzle? [closed]

According to Russell's definite description theory, "The present King of France is bald" is a false statement. However, since for any property $P$, $P$ is true for the elements of the empty ...
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  • 19
2 votes
0 answers
91 views

Partitioning a set of consecutive nonnegative integers into distinct pairs

Let us have a set of $k$ consecutive natural numbers $2,3,\ldots, k+1$, $k$ even. In addition, we are given a set of $m=\frac{k}{2}$ 'differences' from one among the $k$ numbers $1,2,\ldots, k$. My ...
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0 votes
0 answers
88 views

Spread of a disease on a modular chessboard (torus) - lower bound

I learned about the following result from one of Peter Winkler's books: It is impossible to infect the entire $n\times n$ chessboard (usual chessboard) starting from fewer than $n$ infected cells. The ...
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  • 1
2 votes
1 answer
621 views

Number of 5x5 matrix permutations without repetitions in rows or columns

Context In the boardgame Azul, your goal is to complete as much as possible of a $5\times5$ board by placing 25 tiles of 5 different colours (5 tiles of each colour) so that no colour appears twice in ...
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9 votes
0 answers
2k views

A New York Times tiles-based graph theory question

The New York Times has a daily puzzle named Tiles that works as follows. Start with $m$ squares (in the official version, this is 30, in a 6x5 grid), and a set of $p>4$ possible patterns (typically ...
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0 votes
0 answers
86 views

Maximum number of tuples from $n$ numbers such that no pair is repeated [duplicate]

What is the maximum number of $k$-tuples($3\le k\le n$) of $n$ numbers such that no pair is repeated in any of the tuples? The maximum of number of $k$-tuples occur when $k=\lfloor\frac{n}{2}\rfloor$ ...
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21 votes
1 answer
489 views

Circular track riddle

I'm puzzled by the following riddle, which seems easy at first, but turns out to be more complex than it looks. I would like to go to the bottom of it, and could not find references online. The riddle ...
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  • 1,131
9 votes
0 answers
260 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, ...
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  • 2,265
27 votes
1 answer
901 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 ...
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  • 2,517
1 vote
1 answer
163 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 ...
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1 vote
2 answers
317 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{...
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  • 1,611
11 votes
0 answers
437 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 block,...
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  • 1,585
6 votes
1 answer
235 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 $n+1$, $...
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  • 2,019
0 votes
3 answers
1k 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 ...
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2 votes
0 answers
131 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 $B_1, \cdots, B_n$. ...
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  • 343
59 votes
2 answers
3k 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 ...
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2 votes
1 answer
183 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 ...
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1 vote
0 answers
61 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, $n_1/d_1, n_2/d_2, ..., n_k/d_k$. ...
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3 votes
1 answer
154 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 ...
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  • 153
14 votes
2 answers
2k 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 ...
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3 votes
0 answers
689 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 ...
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0 votes
0 answers
232 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 ...
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25 votes
1 answer
1k 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 ...
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20 votes
1 answer
724 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). ...
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  • 4,150
34 votes
2 answers
4k 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 ...
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  • 4,150
1 vote
2 answers
99 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 ...
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  • 111
3 votes
0 answers
76 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 ...
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  • 1,179
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-...
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4 votes
0 answers
199 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 $x\subseteq f(x)$? ...
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3 votes
1 answer
177 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 $k$ ...
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2 votes
0 answers
242 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 ...
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5 votes
2 answers
461 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 ...
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1 vote
0 answers
204 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 $n-1$ ...
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  • 2,419
23 votes
1 answer
1k 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 ...
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  • 488
15 votes
1 answer
2k 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 ...
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  • 7,422
34 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 ...
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2 votes
0 answers
86 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?
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5 votes
0 answers
425 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 ...
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13 votes
3 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 ...
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3 votes
2 answers
161 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/...
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  • 1,425
5 votes
1 answer
318 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 ...
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  • 1,425
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 ...
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  • 1,315
0 votes
0 answers
91 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. ...
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24 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 ...
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