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3
votes
2answers
118 views

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 ...
5
votes
1answer
128 views

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 ...
9
votes
3answers
726 views

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 ...
0
votes
0answers
72 views

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. ...
22
votes
2answers
788 views

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 ...
9
votes
1answer
392 views

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 ...
3
votes
0answers
184 views

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 ...
3
votes
1answer
366 views

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 ...
10
votes
1answer
701 views

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 ...
3
votes
0answers
139 views

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 ...
5
votes
2answers
412 views

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 ...
45
votes
8answers
2k views

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 ...
15
votes
1answer
544 views

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 ...
7
votes
3answers
767 views

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 ...
3
votes
1answer
295 views

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 $2n-1$ ...
8
votes
5answers
1k views

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 ...
0
votes
2answers
602 views

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 ...
6
votes
11answers
2k views

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, ...
5
votes
2answers
1k views

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 ...
8
votes
2answers
970 views

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 ...
0
votes
1answer
548 views

Thomas Clausen's puzzle [closed]

Please refer to this link for the puzzle: http://3.bp.blogspot.com/_jNRwOIxTW3U/S2mWv0uUKmI/AAAAAAAABK0/cFxX971MdV4/s1600-h/Thomas+Clausen+problem.png Thanks!
7
votes
1answer
324 views

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 ...
21
votes
2answers
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 ...
14
votes
5answers
2k views

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 ...
12
votes
1answer
778 views

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 ...
7
votes
2answers
1k views

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 ...
22
votes
1answer
2k views

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 ...
2
votes
5answers
824 views

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 ...
1
vote
0answers
554 views

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 ...
23
votes
3answers
2k views

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 ...
4
votes
7answers
2k 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, ...
40
votes
5answers
6k views

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 ...
6
votes
1answer
425 views

Guessing a subset of {1,…,N}

I pick a random subset $S$ of $\lbrace1,\ldots,N\rbrace$, 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 ...
24
votes
1answer
1k views

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 ...
38
votes
3answers
4k views

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 ...
5
votes
1answer
1k views

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 ...
3
votes
1answer
593 views

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. ...
7
votes
2answers
3k views

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 ...
6
votes
2answers
524 views

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. ...
18
votes
5answers
1k views

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 ...
5
votes
1answer
436 views

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 $\sum_{i=1}^8 f(a_i) = 0$ ...
15
votes
3answers
1k views

What is this subgroup of $\mathfrak S_{12}$ ?

On some occasion I was gifted a calendar. It displays a math quizz every day of the year. Not really exciting in general, but at least one of them let me raise a group-theoretic question. The quizz: ...
20
votes
6answers
5k views

A balls-and-colours problem

A box contains n balls coloured 1 to n. Each time you pick two balls from the bin - the first ball and the second ball, both uniformly at random and you paint the second ball with the colour of the ...
3
votes
1answer
452 views

Is always possible to slice a pizza in a fair way

Given a pizza, represented by the unit disk $D_1(0,0)=\{(x,y)\in\mathbb{R}^2\mid \|(x,y)\|\leqslant 1\}$, and given $N$ slices of $r$-pepperoni, represented by disks ...
7
votes
1answer
2k views

12 balls weighing puzzle

In an article describing the twelve balls weighing problem, the author mentions a solution that involves the finite projective plane of order 3, discovered by Rick Wilson. Does anyone know what this ...
0
votes
1answer
858 views

How to tackle this puzzle? [closed]

Disclaimer: This is not a homework problem. I stumbled on this puzzle on internet and I also have the answer. However I am not able to figure out whats the method to be used to arrive at the answer. ...
7
votes
2answers
8k views

100 Prisoners, 100 Boxes: Proof of Optimality

There's a chestnut about 100 prisoners, labeled 1 through 100, and 100 boxes, each with a number 1 through 100 in it. Each prisoner, completely independently of the others, tries to find the box which ...
8
votes
1answer
551 views

Sticks and thread

In this recent question Math puzzles for dinner we had a nice time as we were asked to provide new maths puzzles for dinners. I suggested the following: Given three equal sticks, and some ...
90
votes
67answers
28k views

Math puzzles for dinner [closed]

You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a ...
4
votes
1answer
1k views

Ulam spiral coordinate system [closed]

Inspired by a Project Euler problem, I recently started playing around with Ulam spirals. My first thought was that an Ulam spiral could be a (rather useless) coordinate system, and how I might be ...