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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Sequence Limit and Series Ratio [closed]
Does there exist a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that the following properties are satisfied?
$f(n) \leq n$ ;
$\lim_{n\rightarrow \infty}\frac{f(n)}{n} = 0$;
$\lim_{m \...
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0
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100 mathematicians each have a numbers written on their foreheads
100 mathematicians each have a number written on their foreheads, visible to all but themself. One day, a meta-mathematician comes by and remarks, "I see all the numbers are distinct natural ...
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References and upper bounds for the SONNAT tiling game?
Introduction
In a video released about a month ago, Pembesita describes a tiling game called SONNAT: Same Orientation Neighbour Not Allowed, Tiling.
In the single-player game, the player may employ ...
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Making perpetual motion machine from candy-sharing cats
It is well known that cats can be turned into perpetual motion machines under the right circumstances. Candy-sharing cats are such wonderful creatures that come in infinite supplies, labeled 1,2,3,...
...
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How many convex polygons can be made from $n$ identical right angle triangles?
Whilst working on a Tangram problem, I came across the need to find the total number of convex shapes that can be produced from $16$ identical (isosceles) right angle triangles (since the Tangram can ...
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Big triples in a matrix
Consider an $n\times n$ real matrix $A=(a_{ij})$ with non negative entries. Assume that
- the sum of the three largest entries in each row is a constant $R$ (the same for all rows),
- the sum of the ...
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R. Smullyan's "Lady or the tiger", Ch. 5, Island of Questioners, problems 11-12 [closed]
I cannot understand the context and formulation of these problems.
The inhabitants ask only questions answerable by yes or
no. Each inhabitant is one of two types, A and B. Those of
type A ask only ...
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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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2
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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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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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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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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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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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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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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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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
answer
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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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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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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
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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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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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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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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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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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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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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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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 ...
user142755
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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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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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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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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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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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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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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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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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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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1
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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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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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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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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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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-...
user40023
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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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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 $k$ ...
user94040
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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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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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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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1
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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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1
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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 ...
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1
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"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
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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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