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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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15-game graph contains a Hamiltonian path ? Lovász conjecture for groupoids, loops, quasigroups , etc?

Typically Cayley graphs are defined for groups and generators sets S. But basically one only needs some set S and another set V and partially defined operation SxV->V, then one defines graph with ...
Alexander Chervov's user avatar
5 votes
0 answers
152 views

A puzzle with magic Egyptian tilings

Background I've recently been devising a puzzle that incorporates elements from Egyptian fractions, magic squares, and tilings. The objective of the puzzle is to tessellate a square with sides of ...
Max Muller's user avatar
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19 votes
1 answer
1k views

Does a function from $\mathbb R^2$ to $\mathbb R$ which sums to 0 on the corners of any unit square have to vanish everywhere?

Does a function from $\mathbb{R}^2$ to $\mathbb{R}$ which sums to 0 on the corners of any unit square have to vanish everywhere? I think the answer is yes but I am not sure how to prove it. If we ...
Ivan Meir's user avatar
  • 4,812
2 votes
0 answers
107 views

Proof that a pandiagonal Latin square of order $n$ exists iff $n$ is not a multiple of $2$ or $3$?

A pandiagonal Latin square of order $n$ is an assignment of the numbers $\{0,\ldots,n-1\}$ to the cells of an $n \times n$ grid such that no row, column, or diagonal of any length contains the same ...
Milo B's user avatar
  • 21
13 votes
2 answers
1k views

Optimal search puzzle

Consider the following puzzle: On the integer line from 1 to $t$ (top, let's say 1000 for this example), you have two operators: uniform random on 1 to $t$, and subtract 1. What is the optimal ...
jackisquizzical's user avatar
0 votes
0 answers
148 views

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 ...
Eric's user avatar
  • 2,601
4 votes
0 answers
288 views

References and upper bounds for the SONNAT tiling game?

Introduction In a video released about a month ago, Pembesita describes1 a tiling game called SONNAT: Same Orientation Neighbour Not Allowed, Tiling. In the single-player game2, the player may employ ...
Max Muller's user avatar
  • 4,605
11 votes
0 answers
492 views

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,... ...
Eric's user avatar
  • 2,601
1 vote
0 answers
100 views

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 ...
FD_bfa's user avatar
  • 147
8 votes
1 answer
404 views

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 ...
Yaakov Baruch's user avatar
2 votes
1 answer
231 views

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 ...
Ash Shevlyakov's user avatar
12 votes
0 answers
539 views

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 ...
Max Muller's user avatar
  • 4,605
4 votes
2 answers
270 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? ...
Eric's user avatar
  • 2,601
-3 votes
1 answer
312 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 ...
Martin Weidner's user avatar
15 votes
1 answer
719 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, ...
MKasa's user avatar
  • 151
8 votes
0 answers
277 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 ...
Eric's user avatar
  • 2,601
7 votes
1 answer
277 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 ...
JPMA29's user avatar
  • 79
0 votes
1 answer
259 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 ...
GEV's user avatar
  • 19
2 votes
0 answers
109 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 ...
vidyarthi's user avatar
  • 2,079
0 votes
0 answers
127 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 ...
Steve's user avatar
  • 1
2 votes
1 answer
2k 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 ...
ami232's user avatar
  • 123
12 votes
0 answers
13k 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 ...
David Pepper's user avatar
0 votes
0 answers
87 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 $k=\lfloor\frac{n}{2}\rfloor$ ...
vidyarthi's user avatar
  • 2,079
21 votes
1 answer
540 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 ...
Denis's user avatar
  • 1,291
9 votes
0 answers
295 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, ...
Thomas's user avatar
  • 2,761
27 votes
1 answer
984 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 ...
Eric's user avatar
  • 2,601
1 vote
1 answer
193 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 ...
Dominic van der Zypen's user avatar
1 vote
2 answers
331 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{...
vidyarthi's user avatar
  • 2,079
11 votes
0 answers
557 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,...
Mark S's user avatar
  • 2,133
6 votes
1 answer
268 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 $n+1$, $...
Hans's user avatar
  • 2,229
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 ...
user avatar
2 votes
1 answer
211 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 $B_1, \cdots, B_n$. ...
katana_0's user avatar
  • 353
63 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 ...
Guillaume Aubrun's user avatar
2 votes
1 answer
317 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 ...
Mohammad Al-Turkistany's user avatar
1 vote
0 answers
65 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, $n_1/d_1, n_2/d_2, ..., n_k/d_k$. ...
Mohammad Al-Turkistany's user avatar
3 votes
1 answer
193 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 ...
ilia's user avatar
  • 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 ...
domotorp's user avatar
  • 18.5k
3 votes
0 answers
696 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 ...
Sebastien Palcoux's user avatar
0 votes
0 answers
286 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 ...
Manfred Weis's user avatar
  • 12.8k
26 votes
2 answers
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 ...
Brett Berger's user avatar
21 votes
1 answer
806 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). ...
Christopher King's user avatar
35 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 ...
Christopher King's user avatar
1 vote
2 answers
119 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 ...
vepanimas's user avatar
  • 111
3 votes
0 answers
80 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 ...
pi66's user avatar
  • 1,209
31 votes
4 answers
3k 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-...
user avatar
4 votes
0 answers
210 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 $x\subseteq f(x)$? ...
Yaakov Baruch's user avatar
3 votes
1 answer
194 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 $k$ ...
user avatar
2 votes
0 answers
284 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 ...
Tweakimp's user avatar
6 votes
2 answers
584 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 ...
Vepir's user avatar
  • 611
1 vote
0 answers
216 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 $n-1$ ...
Oblomov's user avatar
  • 2,501