# Tagged Questions

Recreational mathematics or puzzles with serious mathematical content. Note that math contest problems are generally considered off-topic.

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### Separating Heavier from the Lighter Balls

This Question was originally posted Here, and received no significant progress so far. I wanted to post it here as well, to see what the people of mathoverflow think about it. I think we are ...
163 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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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### How many ways to partition a group of people?

My friend (who is a medical student!) posed me the following question: There are 70 people, and you want to split them up into 10 groups of 7 people each. Two such partitions are "compatible" if no ...
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### Infinite blue eyed islanders puzzle

Can the well known blue eyed islanders puzzle be extended to an infinite number of islanders? In that puzzle, a set of $k$ islanders, each with either blue eyes or non-blue eyes, each knows the color ...
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### Simple reason that a mathematician cannot do better than random when guessing contents of a box?

I have a question about the finite analog of the puzzle proposed here involving mathematicians guessing the contents of boxes. Specifically, suppose there are $k$ unopened boxes each containing a ...
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### Can an infinite number of mathematicians guess the number in a box with only one error?

In this question the following observation was made: Consider a sequence of boxes numbered 0, 1, ... each containing one real number. The real number cannot be seen unless the box is opened. Define ...
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### Dissecting using a ruler and compass

The problem is to cut a regular hexagon into parts that can be put together (without overlaps or wasting any parts) to make an equilateral triangle using only a ruler and compass (and scissors). What ...
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### A simple language and systematic computations

The following somewhat popular simple computer language was enjoyed on sci.math, sci.math.research, pl.sci.matematyka, and perhaps before and after at several places (I wish I knew it's exact history)....
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### Puzzle on deleting k bits from binary vectors of length 3k

Consider all $2^n$ different binary vectors of length $n$ and assume $n$ is an integer multiple of $3$. You are allowed to delete exactly $n/3$ bits from each of the binary vectors, leaving vectors ...
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### a game on sets of reals

A 2 player game on $\mathcal{P}(\mathbb{R})$: Players take turns playing uncountable sets of reals. Each play must be a subset of the previously played set. Player 1 wins if the intersection of all ...
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### Blue and red balls puzzle

I was sent this puzzle and wondered if it is known or if its origin is known? (I see colored ball puzzles are also in vogue.) Consider a bag with $n$ red balls and $n$ blue balls. At each turn you ...
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### two boy scouts problems

As a member of boy scouts I was considering the following problem: suppose you're organising some kind of olympic games... *You divide the boys in $2n$ teams (subsets of equal size) *There are $2n-1$ ...
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### Another colored balls puzzle (part II)

The same colleague as in Another colored balls puzzle asked me the following variant which she called "part II". Imagine you have $n$ balls in a bag that are colored from $1$ to $n$. At each turn ...
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### Gödel, Escher, Bach: b is a power of 10. [closed]

I’d like to verify if my formula correctly expresses that a number is a power of $10$, using the $\sf{TNT}$ language provided by Hofstadter in his famous book Gödel, Escher, Bach: An Eternal ...
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### A function that is defined everywhere but has unknown values [closed]

For pedagogical purposes I am looking for a function $\mathbb{N}\to\mathbb{N}$ that is defined everywhere but has most of its values unknown. Although such a function cannot be simple by definition, ...
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### Bike lock puzzle

I was wondering this when using my bike lock, a combination lock with four dials, each of which has ten digits (0-9) on it in numerical order. Suppose a bicyclist decides that, from now on, after ...
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### Knight tour prime (conjecture)

Hello, I have the following conjecture: Write all numbers from $1$ to $n^2$ over an $n\times n$ board as usually. There not exists $n$ such that we can find a hamiltonian path on primes numbers with ...
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### Labeling a Square Array

Suppose that the $n^2$ cells of an $n\times n$ array are labeled with the integers $1, \dots, n^2$. Under the traditional left-to-right and top-to-bottom labeling, the labels of horizontally adjacent ...
1k views

### How to get rich in a Hilberts Hotel?

Suppose you can make infinitely many copies of yourself. Each of them starts his/her life in a Hilberts Hotel, where each room is labeled by an element in the free group with two generators, and ...
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### Irreversible chess

Suppose we play a chess-variant, where any finite number of pieces are allowed, and the board is as large as we wish, but only two kings in total. And there is no 50 move-rule, no castling and no ...
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### God's number for the $n \times n \times n$-cube

This is a question about Rubik's Cube and generalizations of this puzzle, such as Rubik's Revenge, Professor's cube or in general the $n \times n \times n$ cube. Let $g(n)$ be the smallest number $m$,...
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### Probability of a black path on a random chess board

Take a $2n$ by $2n$ chess board (oriented so the grid lines are vertical and horizontal). Usually there are $2n^2$ squares coloured black and $2n^2$ squares coloured white so that a black square is ...
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### A Presentation for Rubik's cube group?

Let $G$ be Rubik's cube group. It is generated by the rotations by 90 degrees $L,R,D,U,F,B$ (left, right, down, up, front, behind), but what relations beyond $L^4=R^4=...=B^4=1$ do they satisfy? Thus ...
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### Circumference of Convex Shapes

Here is a puzzle I found in Mitteilungen der DMV (roughly, "Letters of the German Society of Mathematicians"), issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch ...
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### Honest and Deceitful Professors Problem [closed]

I found this in An Introduction to Bioinformatics Algorithms. I've paraphrased for clarity. There are 100 professors. Some are honest, while others are dishonest. There are more honest professors ...
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### An elementary problem in Euclidean geometry [closed]

This problem was first put to me by Luke Pebody (who did not know the answer at the time) and after some work I am yet to find a proof or counterexample. I would be grateful of any insights. Call a ...
3k views

### probability and math puzzle books/references

Hi All, I'd like to solve some math puzzles, especially in the context of probability theory, but I'm open to other areas too. The kind of problems that does not require much knowledge of mathematics, ...
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### Identifying poisoned wines

The standard version of this puzzle is as follows: you have $1000$ bottles of wine, one of which is poisoned. You also have a supply of rats (say). You want to determine which bottle is the poisoned ...
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### Guessing a subset of {1,…,N}

I pick a random subset $S\subseteq\lbrace1,\ldots,N\rbrace$, and you have to guess what it is. After each guess $G$, I tell you the number of elements in $G \cap S$. How many guesses do you need to ...
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### A puzzle about finding three points $(x,y)$, $(x,z)$ and $(y,z)$ in a subset of a square.

I was asked (by myself) to give a proof of the following seemingly simple geometric statement, but after thinking a little I now suspect it could be less elementary than I thought (or am I being silly?...
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### cube + cube + cube = cube

The following identity is a bit isolated in the arithmetics of natural integers $$3^3+4^3+5^3=6^3.$$ Let $K_6$ be a cube whose side has length $6$. We view it as the union of $216$ elementary unit ...
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### Hard Cube Puzzle

You are and your friend are given a list of N distinct integers and are told this: Six distinct integers from the list are selected at random and placed one at each side of a cube. The cube is placed ...
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### The Mystic Rose

Consider $n$ points equally spaced around the unit circle, joined by all possible combinations of lines to make a complete graph. Let $g(n)$ be the number of triangles formed in the resulting diagram. ...
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### Generalization of a horse-racing puzzle

A well-known puzzle goes: "Suppose that you have 25 horses and a racetrack on which you can race up to 5 horses. If the outcome of each race only tells you the relative speeds of the horses in the ...
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### Point in Polygon algorithm from the viewpoint of a robot

I've come across the following puzzle: You're on an island, on which there is a fence (which is a simple closed contour). You need to determine whether you're inside or outside the fence. ...
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### Integers in a triangle, and differences

I read about the following puzzle thirty-five years ago or so, and I still do not know the answer. One gives an integer $n\ge1$ and asks to place the integers $1,2,\ldots,N=\frac{n(n+1)}{2}$ in a ...
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### function that sums to zero over cube vertices

Does anyone have an answer to the three-dimensional analogue of the 2009 Putnam Competition A1 problem, viz., if $f\colon \mathbb{R}^3 \rightarrow \mathbb{R}$ satisfies $\sum_{i=1}^8 f(a_i) = 0$ ...