# 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 ...
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? ...
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 ...
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, ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $k=\lfloor\frac{n}{2}\rfloor$ ...
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 ...
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, ...
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 ...
1 vote
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 ...
1 vote
317 views

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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 ... 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 $B_1, \cdots, B_n$. ...
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 ...
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 ...
1 vote
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, $n_1/d_1, n_2/d_2, ..., n_k/d_k$. ...
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 ...
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 ...
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 ...
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 ...
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 ...
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). ...
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 ...
1 vote
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 ...
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 ...