# Tagged Questions

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### total number of combinations?

Patient Age ---> Avg Visits / Year <1 year ---> 7.5 1-4 years ---> 3.0 5-14 years ---> 1.8 15-24 years ---> 1.7 25-44 years ...
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### How to calculate the number of combinations of getting a pair in a deck of 52 cards?

I am confused over calculating the number of ways in which I can select a pair out of a deck of 52 cards, this is how I go about solving the problem, following the definition of a pair in card games, ...
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### How many orientations are there for pawns (for a single player) on a chess board?

How many orientations are there for pawns on a chess board? Pawns can only move forward or diagonally forward. No two pawns may exist on the same square. Pawns start on the second row, and ...
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### Reverse Hex board game winning strategy

I just wanted to know the winning strategy to this question: In a reverse Hex board game I know it means where the player who first forms a path between his/her edges loses. Find a winning ...
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### 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 ...
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### Fusible numbers — can we prove fuse(4) is finite?

Fusible numbers have been discussed here before. Other links: fusible numbers. OEIS A188545. You have an unlimited number of irregularly burning fuses that will nevertheless burn for exactly 1 minute. ...
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### Winning or Non-losing strategy for A or B

Find a winning or a non-losing strategy for the following game: Consider $25$ sticks arranged in a $5$ x $5$ square. Players alternately take any number of sticks from a single row or column. At ...
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### 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?
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### Marriage puzzle

I need some insight for the following problem: There are 100 girls and 100 guys. The girls each have their individual preference list for which guy they’d like to marry. Each list contains all 100 ...
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### Paint a cube with 6 colors

I have a unit cube and $6$ colors to paint its sides in. How many different cubes to I get if I use one color per side? I think I know the answer, but just want to be sure regarding my solution. Let ...
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### Question About the Solutions to the Eight Queens Problem [closed]

How is $a_{15}n_8e_9k_5f_{10}d_7b_4m_6$ a solution to the Eight Queens problem? J. W. L. Glaisher, On the Problem of the Eight Queens, Philosophical Magazine, 1847 says that each one of these terms ...
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### Derangements and the “other” secretary problem

I just found out that the name "Secretary problem" is given to two different problems. The first one talks about a secretary who mixes letters and envelopes, and ask for the probability that no letter ...
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### “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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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.
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### 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 ...
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### 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??
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Very tricky probability

I just came by this "easy" question but i am abit worried, ill tell you why. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
### 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 $k$ ...
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 ...