18
votes
2answers
1k views

A riddle with a witch and some gnomes

My question concerns a variation and a generalization of the following riddle. The Original Riddle: A wicked witch kidnaps 2 gnomes. She paralyzes them, and places a hat on each of their heads. Each ...
8
votes
1answer
70 views

Blocking lines of length $5$ in a $7 \times 8$ matrix; how can we know the solutions have a specific form?

A friend shared with me the following puzzle he encountered in a Chinese math competition: In a $7 \times 8$ matrix, we place tokens so that any straight line of length $5$ (horizontal, vertical, ...
0
votes
2answers
89 views

Advanced Counting Puzzle

Suppose we have a house in which every room has an even number of doors. Prove that the number of doors from the house to the outside world is also even.
5
votes
1answer
69 views

Drawing previously undrawn cards from a deck

Suppose you have a deck of $y$ cards. First, randomly select $y-x$ distinct cards and sign the face of each, then shuffle all the cards back in to the deck. Proceed as follows: Draw a card. If it is ...
1
vote
4answers
87 views

Two children paradox : where is my reasoning wrong?

I hope here is the good place to be asking this. Apologies otherwise. The statement is as follow : "Ms Michu has two children. We know one of the two is a girl, we call that girl Ludivine. What is ...
3
votes
0answers
102 views

History of a combinatoric problem: exchanging numbers by throwing stones

Another user recently asked a question on the Puzzling stack: Two spies throwing stones into a river. Suitably generalised, it becomes: Two spies (Alice and Bob) need to exchange a message. Each ...
2
votes
0answers
41 views

Number of tries to guess M-1 letters from M-letters-code.

There are N letters in an alphabet. There is a combination lock, the code to it consists from M different letters. You can input M letters combination to try to open the lock. If you guess at least ...
3
votes
1answer
101 views

When to be sure that we have counted all the squares in such problems [duplicate]

My first question is: How would one solve such problems (in general,squares+rectangles). What should be the general technique?How can this problem be reduced to a mathematical problem? My second ...
2
votes
2answers
74 views

Chords of a 20-gon

Twenty points lie on a circle, so as to form a regular polygon. Then they are split into ten pairs, and the points in each pair are connected by a chord. Prove that some pair of these chords have the ...
1
vote
0answers
35 views

Number of solutions to sudoku puzzle

Inspired by this question, consider hints on a Sudoku board. A regular puzzle has a unique solution. It is clear that there are puzzles with 2 or 3 solutions, and therefore, I guess, puzzles with say ...
3
votes
1answer
44 views

Computing probabilities of consecutive letters in a word grid

I'm sure most people are familiar with word grid games like Boggle and the newer digital versions Scramble with Friends and Ruzzle. For anyone not familiar, the idea is to find words by using ...
0
votes
1answer
30 views

Maximal size of set cover

Let $S$ be a set of size $1983$, and let $A_1,..,A_k$ be a familiy of subsets of $S$ such that: The union of every 3 sets of the family is S. For every pair of sets the union of them contains no ...
4
votes
1answer
69 views

Sudoku puzzle with exactly 3 solutions

While published sudoku puzzles typically have a unique solution, one can easily conceive of a sudoku puzzle with two solutions. However, is it possible to construct a sudoku puzzle with exactly 3 ...
11
votes
3answers
167 views

Deducing correct answers from multiple choice exams

I am looking for an algorithmic way to solve the following problem. Problem Say we are given a multiple choice test with 100 questions, 4 answers per question (exactly one of those four being ...
5
votes
1answer
127 views

Rooks in 3D chess board

How many rooks are needed for a 3D chess board of size NxNxN so that every empty cube on the board can be reached by a rook in a single move?
1
vote
1answer
52 views

52-card trick for a larger deck?

Long ago someone demonstrated the following card trick with a standard 52-card deck: (1) A volunteer selects 5 cards from a shuffled deck, which the performer does not see. (2) The assistant puts ...
1
vote
1answer
34 views

Show that there exists a satisfactory assignment for the unstandard language of arithmetic $\{\textbf{0}, ', <_1\}$

Consider: $A1: \textbf{0} \not = x'$ $A2: x'=y' \rightarrow x = y$ $A3: \neg x < \textbf{0}$ $A4: x < y' \leftrightarrow (x < y \vee x = y)$ $A5: \textbf{0} < y ...
1
vote
0answers
70 views

Number of paths in a grid

A common puzzle problem is to count the number of paths that start from the bottom-left-hand corner of a grid and end at the top-right hand corner, with the restriction that you can only move upwards ...
3
votes
0answers
72 views

Given a number of items, how many sets of three are there where no two sets are two thirds similar?

Sorry if the title isn't proper math-talk. Hopefully I can explain it better here. So let's say we have a set. 1, 2, 3, 4, 5, 6, 7, 8, 9. I want to know how many groups of three can be made where no ...
2
votes
2answers
103 views

Is there a solution to this Seating Plan problem?

So a colleague asked me for some Help on an interesting Problem, which we both couldn't find the optimal answer for. The event which needed it is already in the past, so this is just me trying to ...
3
votes
3answers
119 views

solve the puzzle how many liars?

Each boy in a group of $20$ boys either always tells thuth or always tells a lie. These boys are sitting around a table. Each boy says that his neighbours are liars. Prove that at least $7$ out of ...
2
votes
2answers
119 views

Winning a restricted game of Nim?

Given the following piles, find the Grundy number of the initial position and make the first move in a winning strategy given that no more than two sticks may be removed from a pile at any time. Pile ...
4
votes
1answer
112 views

How many answers to this combinatorial puzzle?

Take a square. How many ways are there to draw or not draw a line from the center to each of its sides? 16, of course. Here are all the different squares: Now, how many ways are there to put ...
22
votes
5answers
1k views

Number of vectors so that no two subset sums are equal

Consider all $10$-tuple vectors each element of which is either $1$ or $0$. It is very easy to select a set $v_1,\dots,v_{10}= S$ of $10$ such vectors so that no two distinct subsets of vectors $S_1 ...
3
votes
2answers
116 views

Number-Theoretic Coin Puzzle

There are three piles of coins. You are allowed to move coins from one pile to another, but only if the number of coins in the destination pile is doubled. For example, if the first pile has 6 coins ...
0
votes
1answer
388 views

How many triangle can be drawn with those points? [duplicate]

There are 7 points on the circumference of a circle.How many acute triangle can be drawn with those points. please help me to solve this problem.
2
votes
1answer
70 views

Maximal number of kings on a chessboard, but this time two can be adjacent.

How many kings can be placed on an $8 \times 8$ chessboard such that every king can capture (is adjacent to) at most one other king? I can do 26, but can not prove that this is optimal.
0
votes
1answer
57 views

Maximum score for the game

Here is a game: There is a list of distinct numbers. At any round, a player arbitrarily chooses two numbers $a, b$ from the list and generates a new number $c$ by subtracting the smaller number from ...
0
votes
2answers
318 views

How do I find the maximum number of knights on a chess board?

I came across this problem and after thinking a lot I could not get any idea how to calculate it. Please suggest to me the right way to calculate it. Given a position where a knight is placed on ...
0
votes
0answers
63 views

Optimized search for lock combinations

I came across an interesting puzzle the other day expressed as follows. You have a combination which has a dial on its face with the values of {1-30}. The combination that will open the lock is an ...
8
votes
4answers
299 views

Kings on a chessboard

In how many different ways can six kings be placed on a $6\times 6$ chessboard so that no one attacks the others? If the problem was asked for a $3 \times 3$ board and $3$ kings, then the answer ...
3
votes
3answers
123 views

TicTacToe with considerations of symmetry

It is not difficult to determine the number of possible games of tic toe, but what about when rotationally symmetric positions are considered equal? Please do not simply give me the number, I would ...
2
votes
1answer
130 views

probability of a word in a string

What is the probability of a word n characters long appearing in a string of m characters, in an alphabet of x characters? A word here is simply a string of characters contained in another string of ...
12
votes
1answer
160 views

Coloring 5 Largest Numbers in Each Row and Column Yields at Least 25 Double-Colored Numbers

This is a question from a very old American Mathematical Monthly, if I recall correctly. It has a very nice solution and illustrates an often useful technique. If it is unsolved after a while, I will ...
15
votes
2answers
374 views

A “What's my vector?” game

Alice chooses a binary vector $V$ of length $n$ which is unknown to Bob. In each round Bob can choose any number of indices $i$ and then Alice tells Bob how many of the $V_i$ are set to $1$. The ...
1
vote
1answer
192 views

How many unique patterns exist for a 5x5 grid with paths of spaces intersecting at 1 space and leading to each edge of the grid?

I'm try to design a game in which the board is made up of a 3x3 grid of square tiles. Each tile is a 5x5 grid of spaces. Each tile has 4 exit spaces each located on 1 of the middle 3 spaces along ...
3
votes
3answers
155 views

Combinatorics: Lock puzzle , minimum combinations

A safe has three locks of which every lock has 8 possibilities 1, 2 ...8. Safe gets opened if any 2 of 3 locks gets opened. So, a possible way to open safe is try 2 locks, for each possible pair of ...
4
votes
1answer
109 views

What is the minimum number of locks on the cabinet that would satisfy these conditions?

Eleven scientists want to have a cabinet built where they will keep some top secret work. They want multiple locks installed, with keys distributed in such a way that if any six scientists are present ...
1
vote
2answers
307 views

Cutting a hexagon to make an equilateral triangle

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. The cuts should all be straight. What is the ...
2
votes
0answers
60 views

Minimum Overlap

You have a set of ten numbers, and you are trying to cover all 4-element subsets of this set. To do this, you choose 5 elements from the set every time and you cover all 4-element subsets of your ...
3
votes
1answer
74 views

Minimal diameter of set of fractions

Let $p_n$ be a pairwise partition of $\{1,2,...,2n\}, n\in \bf N$ where $(a,b)\in p \implies a<b$, and $P_n$ the set of all such pairwise partition. $d(n) := \min_{p_n\in ...
1
vote
1answer
224 views

flower pot puzzle

Sara has 6 flower pots, each having a unique flower. Pots are arranged in an arbitrary sequence in a row. Sara rearranges the sequence each day but not two pots should be arranged adjacent to each ...
1
vote
2answers
147 views

Probabilistic puzzle

There are $n+1$ boxes and every box contains $n$ balls. For every $k\in\left\{ 0,1,\ldots,n\right\} $ there is exactly $1$ box containing $k$ white balls and $n-k$ black balls. A box is picked out and ...
1
vote
0answers
472 views

what is maximum number of points of intersection between the diagonals of a convex octgon?

What is the maximum number of points of intersection between the diagonals of a convex octagon (8-vertex planar polygon)? Note that a polygon is said to be convex if the line segment joining any two ...
3
votes
2answers
138 views

Efficiently identifying spam honeypots

I realise that the title is computing specific, but I think the underlying problem is general - I just don't know how to phrase it more generally (which may be part of my problem). So I am asking ...
1
vote
1answer
547 views

The smallest amount

Using a pool of problems, 20 tests will be formed. -Every test should have the same number of problems. -Any problem should be included in at most 10 tests. -For every 5 tests, there should be at ...
10
votes
2answers
167 views

Given a desired coloring scheme for a stick, how can I brush it with the fewest steps?

If I want to color a stick (regarded as a line segment in one-dimensional space) to a desired coloring scheme using brush, how can I make it with the fewest steps? Notice that, new color will just ...
0
votes
1answer
71 views

Combinatorics puzzle

I am having a problem dealing with following topic: lets say we have set of numbers n = [ 1 2 2 3 ] and we want be able to get all two-elementary combinations out ...
5
votes
2answers
192 views

A puzzle on game theory

Bob and Alice are playing a game. They will start with an integer $n$. Alice goes first, in each turn, a player can choose an integer between 1 and 13 and that number is to be subtracted from $n$. ...
2
votes
0answers
143 views

Make one cube out of 8 little cubes

As part of a puzzle, you have to stack 8 little $1\times 1\times 1$-cubes so that they form one big $2\times 2\times 2$-cube. Now I want to check all possible solution to the puzzle and therefor I'm ...