1
vote
0answers
55 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
48 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
74 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 ...
1
vote
3answers
94 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
82 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
104 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
967 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
100 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
135 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
62 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
46 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
196 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
56 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
269 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
111 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 ...
1
vote
1answer
92 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 ...
11
votes
1answer
147 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
339 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
130 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
114 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
99 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
258 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
57 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
73 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
202 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
140 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
217 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
113 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
541 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
160 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
67 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
177 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
134 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 ...
2
votes
3answers
119 views

$6$ people, $3$ rooms, $1$ opening door

$6$ people spread in $3$ distinguishable rooms, every room needs one person who opens the door. There are ${6 \choose 3}\cdot 3 \cdot 2$ options to choose the three door opener persons and let them ...
2
votes
3answers
116 views

Multilingual hedge fund - Puzzle

I'm having difficulty finding the solution for the following problem: A hedge fund has 70 employees. For any two employees $X$ and $Y$ there is a language that $X$ speaks but $Y$ does not, and there ...
1
vote
3answers
325 views

black and white balls in the box

A box contains $731$ black balls and $2000$ white balls. The following process is to be repeated as long as possible. (1) arbitrarily select two balls from the box. If they are of the same color, ...
5
votes
2answers
238 views

Lights Out Variant: Flipping the whole row and column.

So I found this puzzle similar to Lights Out, if any of you have ever played that. Basically the puzzle works in a grid of lights like so: 1 0 0 00 0 0 00 1 0 0 0 0 1 0 When you selected a ...
1
vote
2answers
202 views
2
votes
1answer
99 views

How to approach the following combinatorics questions

I've come across the following Combinatorics problems: A group of people go to an amusement park with $10$ rides, each of them goes on exactly $5$ rides and any two different people go on at most ...
1
vote
2answers
764 views

Counting squares of maximum size in a rectangle

Given a rectangle of sides $m$ and $n$. $( m,n \in [1,1000] )$ We can cut the rectangle into smaller identical pieces such that each piece is a square having maximum possible side length with ...
6
votes
1answer
231 views

Colored balls puzzle

Imagine you have $n$ balls in a bag that are colored from $1$ to $n$. At each turn you take two balls at random out that have different colors and color one the color of the other. You then put them ...
2
votes
1answer
95 views

Riddle - cover a $62 \times 66$ board using only $341$ straight rows of $12$ squares each

Is it possible to cover a $62 \times 66$ board using only $341$ straight rows of $12$ squares each?
0
votes
2answers
90 views

A Nim variant: Number of stones

Alice and Bob play the following game. There is one pile of $N$ stones. They take turns to pick stones from the pile, Alice will play first. In each turn, a player can only pick $k$ $(a \le k \le b)$ ...
1
vote
2answers
47 views

Counting ordered triples of sets, with empty intersection.

I was recently asked this question which I couldn't solve. Give the number of ordered triples $(A_1, A_2, A_3)$ of sets which have the property that $A_1 \cup A_2 \cup A_3 = ...
4
votes
1answer
95 views

Classrooms and students puzzle

My school has many classes. Any two students share exactly one class. Any two classes share exactly one student. A class must have at minimum $3$ students, and there is at least one class with $17$ ...
3
votes
1answer
170 views

Safes and keys probability puzzle [duplicate]

I have $100$ keys and $100$ safes. Each key opens only one safe, and each safe is opened only by one key. Every safe contains a random key. 98 of these safes are locked. What's the probability that I ...
2
votes
1answer
111 views

puzzle: A spy and the keypad

A spy encounters a keypad that requires a 4 digit PIN. He uses a fine dust to find which keys are used in the combination. He does not know the sequence of keys, nor which ones repeat if any. ...
10
votes
6answers
284 views

When two voters meet, they switch allegiance; might they all ally with the same candidate?

Let's assume that there are three candidates running in an election. Right before the elections (when there is no more propaganda), it is forbidden to gather in groups of more than two people ...
11
votes
1answer
149 views

How can one determine the chess configuration that maximizes the number of possible moves?

To clarify, what is the chess-board configuration that would maximize the number of valid moves one player could make on his or her turn? I thought of this question while playing chess, how apropos. I ...
0
votes
1answer
97 views

How to calculate Team Strength for future prediction?

You are given with $4$ players name, namely Player $A$, Player $B$, Player $C$ and Player $D$. These players are grouped into two teams with two players each. A Game is played between the two team.For ...