# 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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### 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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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 $n+1$, $... 0 votes 3 answers 1k views ### Given$N$integers on a circle, how to choose them in pairs to obtain minimum sum? (Added by YCor 2019 July 7): it has been mentioned in the comments that this is part of a contest "Circular merging, July Challenge 2019 Division 1", where an equivalent question (just more clearly ... 2 votes 1 answer 200 views ### Game on groups (generalization of spinning switches puzzle) Alice and Bob are playing a game as follows: Initially There're two subgroups$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$. ... 62 votes 2 answers 3k views ### Guessing each other's coins I recently thought about the following game (has it been considered before?). Alice and Bob collaborate. Alice observes a sequence of independent unbiased random bits$(A_n)$, and then chooses an ... 2 votes 1 answer 278 views ### Lower bound on the number of solutions of N-queens problem The OEIS lists the number of solutions of N-queens problem (Number of ways of placing n nonattacking queens on an n X n board). However, no formula is given. It is easy to observe that each number in ... 1 vote 0 answers 63 views ### Computational complexity of fractions multiplication puzzle I developed a puzzle for which I am seeking an EFFICIENT algorithm (I searched Google but did not find this puzzle in the literature, ): You have$k$rationals,$n_1/d_1, n_2/d_2, ..., n_k/d_k$. ... 3 votes 1 answer 165 views ### Matching two sequences between each other Given the sequence of symbols$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$so that ... 14 votes 2 answers 2k views ### Can we make 101 almost perfect banknotes from 100? Disclaimer. The practical execution of the algorithm in question might be illegal in certain jurisdictions, and is thus strongly discouraged by the poser of the problem. This recent post on the ... 3 votes 0 answers 694 views ### Puzzle in 3D grid with black and white boxes, related to shelling Consider a$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 259 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 ... 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 ... 20 votes 1 answer 781 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 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 2 answers 112 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 79 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 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-... 204 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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### 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 $k$ ... 269 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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### 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
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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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### 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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### 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 ...