1
vote
2answers
60 views

“Relatively Close” Soccer Game

A soccer game between 2 teams is "relatively close" if the scores never differ by more than 2. In how many ways can the game be "relatively close" for the first 12 goals? Just to clear it up, the ...
3
votes
0answers
50 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 ...
6
votes
0answers
63 views

Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
3
votes
1answer
48 views

For which chess boards do solutions exist for this generalised Knight's Tour problem?

We know from a theorem by Schwenk that for any (m x n) chess board with $m \leq n$ it is always possible to create a knight's tour unless one or more of these three conditions are met: m and n are ...
13
votes
2answers
677 views

The Best Strategy and Highest Possible Score for the “Threes!” Game.

[There's still the strategy to go . . . ] Here's my description of the game: There's a $4\times 4$ grid with some random, numbered cards on. The numbers are either one, two, or multiples of three. ...
0
votes
0answers
65 views

Describing the sequence A224239.

I've been trying to describe mathematically the $n$th term $a_n$ of the sequence A224239. We get $a_n$ by counting the distinct ways to fill an $n\times n$ grid with squares of smaller integer size, ...
2
votes
1answer
46 views

Guests at a table

Fifteen chairs are evenly placed around a circular table. On the table are the name cards of fifteen guests. After the guests sit down, it turns out that none of them is sitting in front of his own ...
2
votes
1answer
52 views

Ping Pong players

A and B play ping pong game multiple times. The person serving first has a probability p of winning that game. A serves the first game and thereafter the loser serves first. If p(n) = pbt that A ...
2
votes
1answer
188 views

Sequences, sets and element position in the set.

I have a sequence Q with the length of N. This is the fragment of this sequence: 68 70 72 74 76 78 80 The sequence has been divided into the sets of 4 elements ...
10
votes
2answers
188 views

Is it possible to shuffle a 3x3 Rubik's cube so that there's no more than 2 pieces of the same color in every face?

I'm not sure if this question belongs here but I see lots of Rubik Cube's questions around so here it goes: Can I take a standard 3x3 Rubik's Cube and shuffle it so that, for every face, there are no ...
5
votes
1answer
79 views

Hockey Classics at Matheletics '13

I'm trying to solve a challenge from Matheletics '13: Micheal Nobbs is organizing a training camp for identifying new talents in Indian Hockey. The camp witnessed a total of ($3K+1$) players. Each of ...
4
votes
1answer
67 views

Expected number of clusters on chessboard

N distinct squares are selected uniformly at random on an MxM chessboard, what is the expected number of clusters? A cluster is a collection of squares which are connected sideways, not cornerwise.
1
vote
1answer
51 views

Articles on matchstick puzzles

There are many ingenious puzzles involving matchsticks that are arranged as squares, rectangles or triangles, and can be moved under some restrictions (for a lot of examples see ...
0
votes
1answer
139 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.
0
votes
1answer
53 views

number of ways to fill a 2D grid

We have a 2D grid with n rows and m columns, we can fill it with numbers between 1 and k (both inclusive). Only condition is that for each r such that 1<=r<=k ,no two rows must have exactly the ...
2
votes
2answers
49 views

how many ways to go from place a to place b through 9 squares

Please see the image. How many ways are there from M to N without passing through the sqaure more than once... I counted upto 6 ways...is it the right answer??
0
votes
0answers
79 views

How many melodies are there?

Clearly if we assume only 12 chromatic notes to the scale, not all of which sound good next to each other, a melody of length $N$ chooses from less than $12^N$ potential melodies. Allowing melodies to ...
0
votes
1answer
26 views

Sorting $N$-ary Gray codes into a plane/grid

Is there a formal algorithm to arrange a set of "numbers" on a grid/plane such that each adjacent set differs from the other by only one value. Something similar to the Grey code but further extended ...
0
votes
3answers
59 views

How many ways can you ascend a stairway of any number of steps?

I wrote out by hand every way from 1 to 6 steps and came up with the formula $f(x) = 2^{x-2}$. Is that correct? I then tried to solve the problem recursively but could not. So I wanted to know if my ...
4
votes
1answer
75 views

999 coins in 3-by-3 piles

999 coins are organized in a 9 piles in a $3\times 3$ grid. There number of coins in each column is the same (333). We are allowed to take the 3 piles in a single row, but only if we manage to ...
6
votes
1answer
256 views

Counting Valid Strings in Will Shortz 3D-Word Hunt

So I was reading the NY Times Magazine (only the puzzle section of course) and I came across a puzzle I had never seen before. Titled the "3D-Word Hunt", the goal is to find as many five letter words ...
7
votes
2answers
217 views

A Nim-like game with conditions and strategies

The game: Given $S = \{ a_1,..., a_n \}$ of positive integers ($n \ge 2$). The game is played by two people. At each of their turns, the player chooses two different non-zero numbers and subtracts ...
0
votes
3answers
37 views

Different answers from different formulations of combinatorics problem

A certain men's club has sixty members; thirty are business men and thirty are professors. In how many ways can a committee of eight be selected if at least three must be business men and at least ...
2
votes
1answer
90 views

Very tricky probability

I just came by this "easy" question but i am abit worried, ill tell you why. ...
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 ...
4
votes
0answers
100 views

Card game probability

Suppose the following solitaire with a standard deck. I turn four cards visible on the board and on each turn, I remove those suits that appears more than once in the board. Then I fill the board such ...
5
votes
3answers
115 views

How to minimize $|z_1 - z_2|^2 + |z_1 - z_4|^2 + |z_2 - z_3|^2 + |z_3 - z_4|^2$?

If $z_1,z_2,z_3,z_4 \in \mathbb{C}$ satisfy $z_1 + z_2 + z_3 + z_4 = 0$ and $|z_1|^2 + |z_2|^2 + |z_3|^2 + |z_4|^2 = 1$, then the least value of $|z_1 - z_2|^2 + |z_1 - z_4|^2 + |z_2 - z_3|^2 + ...
9
votes
1answer
169 views

A congruence in the number of certain ternary strings

Let $a_n$ be the number of ternary strings of length $n$ which do not contain three consecutive symbols that are all different. That is, $$a_n = \Bigl|\bigl\{\,(b_k)_{1\leq k\leq n}\in ...
16
votes
2answers
293 views

Counting the number of polygons in a figure

I often come across figures like this on the net, or as contest problems, asking to find the number of a specific type of polygon in the figure (triangles, in this case). But I've never really found ...
3
votes
1answer
100 views

Number theory: 2 numbers within a set with same difference

You have the numbers 1,2,3...,99,100. From that set you have to choose 55 different numbers. Show that: There are 2 numbers with a difference 9,10,12,13 Show that there aren't ...
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
93 views

Monotonically increasing path in a complete graph

Given a complete graph with n vertices such that all edge weights are distinct. Prove that we can find a monotonically increasing path of length n-1. I tried finding such a path by sorting the edges ...
1
vote
0answers
93 views

K non-intersecting diagonals in a polygon

Given a regular N-sided polygon, how many ways can you draw K non-intersecting diagonals? Any pair of diagonals must not intersect strictly inside the polygon. For e.g. N = 4 and K = 2 -> 2 ways ...
0
votes
0answers
21 views

Number Of Triangles of All Sizes in an Equilateral Triangle [duplicate]

https://mail.google.com/mail/u/0/?ui=2&ik=4622e6803e&view=att&th=1422d3806080ed0d&attid=0.1&disp=emb&realattid=ii_1422415e014f71c5&zw&atsh=1 Consider an ...
1
vote
0answers
49 views

How many Hamiltonian loop are there in a big rectangle?

Suppose I have some big rectangle made of $n \times m$ squares, and I want to place tiles on it in a manner that makes a picture of a hamiltonian loop. I can transform this problem into a problem ...
2
votes
1answer
75 views

Throw dice, what does this mathematical expression mean in real life?

Assuming we have a dice and the event that if we throw dice for the k-th time and get a 6 is given by $A_k$, is there an actual explanation what $A:= \cap_{i=1}^{\infty} \cup_{j=i}^{\infty} A_j$ is?
0
votes
1answer
60 views

Partition of circumference into $3k$ arcs

The following problem is from 1982 Russian Mathematical Olympiad. If you go to this link, and scroll down to the section Russian Math Olympiad, then this is Problem 333 in that text-file. Let $k$ ...
3
votes
2answers
67 views

A bishop on a grid

Suppose that we have an $n\times m$ chessboard and bishop on the square $(1,1)$. It starts to move diagonally with the following rules: If bishop is in any corner square except $(1,1)$, it stops ...
5
votes
1answer
142 views

Surprising limit (probability of no two coinciding pairs)

I stumbled upon this question by random chance. The motivation is kind of long, the question is pretty short; if you're just here for the limits, feel free to skip to the break. I'm taking five ...
0
votes
1answer
60 views

finding the total number of ways of arranging bulbs!

In a grid of size 3*n,each tile is 1 unit long and 1 unit wide.There are bulbs beneath the tiles.Each bulb is responsible for lighting exactly two tiles.No tile should be lighted by more than 1 bulb ...
-4
votes
3answers
179 views

In how many ways can $16 be divided among 4 people? [closed]

In how many ways can $16 be divided among 4 people, assuming that each person has to get something and there are 5 cent coins and up
0
votes
2answers
74 views

Selection minimum out of $n$ different objects!

Suppose we have $n$ differnt persons and $n$ different objects.They have to select from the the objects such that every pair of person has at least one uncommon object. What is the minimum number of ...
5
votes
0answers
66 views

Generalization of Gray codes

A friend of mine asked me if it was possible - physical difficulties aside - to generate all 32 combinations of raised/lowered fingers by changing status of a fixed number of fingers at every step. ...
0
votes
1answer
82 views

Variation on Birthday Problem - Probability that 47 of 191 students have birthdays on two conditions.

It's my birthday, and I figured I will create a problem based on birthdays that I myself am unable to solve! Assuming time is denoted by HH:MM:SS, MM/DD/YYYY, what is the probability that in a class ...
4
votes
2answers
171 views

Filling a bag with fruits of four types, with constraints

Here's a diabolical math problem that I found. In how many ways can we fill a bag with n fruits subject to the following constraints? • The number of apples must be even. • The number of bananas ...
1
vote
0answers
66 views

Probability of occurrence of games in a football league

This question just came to me as I was watching a football game. There is a football league with 20 teams. Each team has to play every other team at home and away, which means each team will play a ...
2
votes
3answers
378 views

There are 81 trees with {1,2,3,…81} apples on them respectively. Distribute among 9 people. Each get equal apples

It is clear. There are 81 apple trees. 1st tree has 1 apple, 2nd tree has 2 apples, 81st tree has 81 apples. Distribute the "trees" (not apples) among 9 people so everyone gets equal amount of apples. ...
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 $n$. ...
0
votes
1answer
128 views

Discrete mathematics for someone from a non-mathematical background

I have been a software programmer for over six years and I'm from a non-mathematical background. Though I had some limited exposure to discrete mathematics in my college years it didn't leave any ...