All Questions
Tagged with combinatorics puzzle
566
questions
2
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0
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48
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Dividing $N$ coins into at most $K$ groups such that I can get any number of coins by selecting whole groups
Problem
Inspired from Dividing $100$ coins into $7$ groups such that I can choose any number of coins by selecting whole groups . I am interested in the number of possible ways we can get such a split....
- 1,181
1
vote
1
answer
70
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Dividing $100$ coins into $7$ groups such that I can choose any number of coins by selecting whole groups
The problem goes somewhat like this:
Let's say that we have 100 coins. Now, I have to split these 100 coins into seven different groups such that I can choose any number of coins only by selecting ...
user1319345
4
votes
0
answers
98
views
Which abelian groups and odd integers lead to a well-posed weights puzzle?
Consider the following puzzle (which I quote from here):
In a collection of 101 balls, each ball weighs a whole number of pounds. If any one is removed from the collection, the remaining balls can be ...
- 2,243
1
vote
0
answers
40
views
Number of paths with $m$ good pairs of moves [duplicate]
Consider the set $S$ of all paths from $(0, 0)$ to $(n, n)$, formed by sequences of $2n$ moves of the form $(a, b) \rightarrow (a, b + 1)$ or $(a, b) \rightarrow (a + 1, b)$, such that at any point $(...
- 378
2
votes
0
answers
75
views
Smallest number of groups
Eighty-four developers sign up to contribute to a public open-source project. You need to divide the developers into $n$ subteams such that each contributor is on exactly one team. Their personalities ...
- 378
0
votes
1
answer
84
views
Largest collections of subsets [closed]
I need to find largest collection of subsets of $\{1,\ldots, 84\}$ such that each subset has size 5 and any two distinct subsets have exactly one element in common.
Any help is appreciated, Thanks
- 378
0
votes
0
answers
52
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Cinema Hall Seating Problem [duplicate]
There are n people in line to enter a cinema that own n seats. Each of the n people have an allotted seat in the cinema hall where they are supposed to sit. The first person forgets his/her seat ...
0
votes
0
answers
34
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Maximum Line Segments in an n × n Grid Without Loop formation
Exploring Proof for Maximum Line Segments in an (n * n) Grid Without Loop Formation
Hello Math SE community,
I am investigating how to maximize the number of line segments in an $(n * n)$ grid ...
3
votes
1
answer
82
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You pick $N$ positive integers between $1$ and $M$ without replacement. If you add another number, what is the probability the maximum hasn't changed?
You initially start with all the integers between $1$ and $M$. You then pick $N$ of them randomly, without replacement, to generate a new set of $N$ non-repeating numbers. The maximum of this set is $...
3
votes
1
answer
124
views
Given 99 bags of red and blue sweets, is there a selection of 50 bags containing at least half of each type of sweet?
Assume you have 99 bags containing sweets of two kinds, say blue and red.
Is it always possible to pick out 50 bags such that you have at least half the total of red sweets and half the total of blue ...
- 41
1
vote
2
answers
145
views
Simple solution to random walk
The final score of a football match was 4:3 in favour of the home team. How many ways could the result have gone if there was a period of the match when the away team was leading? I have learnt of a ...
- 53
4
votes
1
answer
146
views
Formalising the problem and create a proof for the game "Waffle"
Waffle is an online game at https://wafflegame.net/daily.
It consists in moving letters (swapping them) to recreate the original words. While you have 15 moves, it can be done in 10. I usually try to ...
- 1,125
1
vote
2
answers
92
views
$2n$ knights around a table with namecards, is it possible that for every rotation there is exactly one person with a correct namecard?
I need help with the following puzzle:
Consider a round table which hosts $2n$ knights. At each seat, there is a namecard of one knight. The knights don't necessarily sit in front of their own ...
- 184
1
vote
1
answer
92
views
A combinatorics and divisibility puzzle [closed]
I've recently encountered the following puzzle that seems a bit too "mathy" for Puzzling Stack Exchange so I decided to ask here. This is not a homework.
The numbers from $1$ to $15$ must be ...
- 2,183
2
votes
2
answers
111
views
Smallest number of absent workers in a factory
I am trying to solve a brainteaser and would like some direction if my line of thought is correct...
At a worker plant, 25 workers were absent on Monday, 22 absent on Tuesday and 19 absent on ...
- 77
4
votes
0
answers
144
views
No two adjacent bulbs on
The problem is to count number of configuration of $9$ bulbs on a $3\times 3$ grid, where no two bulbs that are adjacent are switched on.
I solved this problem in a very ad-hoc kind of manner, the ...
- 378
1
vote
0
answers
79
views
Solving formulas for Coupon Collector's problem variant
I considered a modified version of the Coupon Collector's problem where there are $m$ transparent balls, $k$ different colors and $c$ balls of each color for a total of $n=ck+m$ balls in the urn.
I ...
- 31
3
votes
1
answer
187
views
How to prove whether this grid puzzle is unsolvable/solvable?
[Disclaimer: I have researched a bit before this post and I have found no other questions that address my problem specifically, hence this post]
So I have been made a puzzle concept in a python ...
- 33
0
votes
1
answer
107
views
Light and bulb problem from an old maths contest
In a room there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you put on the $n$ -th switch the $n$ -th bulb will light up. There is a group of men who are ...
- 626
39
votes
1
answer
3k
views
Is it possible to assemble copies of this shape into a cube?
A couple of friends of mine were discussing a problem concerning this shape:
Is it possible to assemble enough of these to form a cube?
I have discovered a lot of impossible positions but was not ...
- 507
2
votes
1
answer
3k
views
Compute the number of ways to reach a point without overshooting puzzle
I am trying to solve the following puzzle:
An ant is trying to get from the origin (0, 0) to X = (13, 4) without overshooting, but he can only move up or right. He always alternates his step sizes ...
- 1,027
2
votes
1
answer
277
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How many ways can I pick a size $3$ set from $\{1, 2, 3, 4, 5, 6, 7, 7, 7\}$ such that at least one is odd?
I am trying to solve the following puzzle:
I have nine cards, where six of them are labelled $1$ through $6$ and the remaining three are indistinguishable and labelled with 7. Calculate the number ...
- 1,027
0
votes
0
answers
56
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k-cycles and permutation of remaining elements in the 100 prisoner problem formula
The original problem can be found here, 100 prisoners problem , where $n=100$ and $n/2 \lt k \le n$. Letting $L(n,k)$ equal the number of permutation on $n$ distinct objects with greatest cycle length ...
- 43
0
votes
0
answers
42
views
How many locks and keys: combinatorics problem [duplicate]
A village keep all their most precious belongings in a vault. The vault has a certain number of locks, each lock with an individual and specific key. The people in the village want to make sure that ...
3
votes
0
answers
70
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Puzzle of an ant rearranging stacks of seeds in a line [duplicate]
Interesting puzzle that I haven't been able to solve or find a solution to.
An ant rearranges a line of stacks of seeds as follows:
With each iteration, the ant goes to each stack in order and grabs ...
- 99
3
votes
2
answers
427
views
Infected Dinner Brainteaser
I came across this brainteaser online that I found quite confusing:
There are $1000$ people having dinner at a grand hall. One of them is known to be sick, while the other $999$ are healthy. Each ...
- 31
3
votes
0
answers
92
views
Maximum tiling by Y Hexomino
"Y Hexomino" has a shape as shown in the picture.
What is the maximum number of Y Hexomino that can be placed on a $13\times 13$ chessboard, where each Hexomino does not overlap?
From the ...
- 183
1
vote
1
answer
210
views
The toys problem: Probability of getting two matching good item and a different third Item
I've encountered an intriguing probability problem. I just registered to ask this, so this will be my first post.
Disclaimer: I met this problem in a real setting that's it too convoluted to explain (...
- 13
6
votes
1
answer
438
views
Placing the $21$ two-digit primes into a grid, such that primes in adjacent squares have either the same tens digit or ones digit
This is USAMTS round 3, problem 1 of the 2020-2021 Academic Year.
Place the 21 2-digit prime numbers in the white squares of the grid on the right so that each two-digit prime is used exactly once. ...
- 1,837
6
votes
1
answer
335
views
Minimum swaps to put an array into desired order, where some elements are identical/repeated
Inspired by a word game Waffle, see footnotes if interested. The abstracted problem:
You're given an input array of letters, some of which might be identical (i.e. repeated), e.g. ...
- 15.5k
1
vote
1
answer
141
views
100 prisoners riddle - Dependency of probabilities
I am referring to the well-known riddle of the title (if you don't know what I am talking about here it is: https://en.wikipedia.org/wiki/100_prisoners_problem - See sections "Problem" and &...
- 13
0
votes
4
answers
237
views
A candle burns for an hour, and $M$ burnt candles can make a new candle. For how long can $N$ candles keep the room lit?
There is a solution to a puzzle about candles, which I can't follow.
Here is the puzzle:
Imagine there are $N$ candles. Each of the candles takes $1$ hour to burn. Out of burnt $M$ candles you can ...
user1197011
4
votes
1
answer
322
views
Gym Locker Combination Puzzle
Here's a problem I've been trying to solve, but I can't reach the correct answer (39 minutes).
Before I can open my gym locker, I must remember the combination. Two
of the numbers of this three-term ...
25
votes
1
answer
635
views
+500
Number of Chess Games Possible: Parity Discussion
Chess is an incredibly intricate game, offering an immense number of moves and combinations. Due to this complexity, determining the precise count of legal chess games poses a significant challenge. ...
- 344
0
votes
0
answers
171
views
Where to find hard logic problems
I was searching for hard logic problems like for example counterfeit coin problem or the problems about merging two sorted sequences of weighted balls. Whenever I was searching for serious logical ...
- 23
7
votes
1
answer
170
views
Maximum possible number of 1012-element subsets of {1,2,...,2024} such that no three intersect at more than one element
I came across the following problem:
At most how many $1012$-element subsets of $\lbrace 1,2,\dots,2024 \rbrace$ may be chosen such that the intersection of any three subsets has at most one element?
...
- 155
3
votes
3
answers
121
views
Coloring the faces of n^3 unit cubes s.t., for each color j between 1 and n, the cubes can be arranged to form nxnxn cube with j-colored outer faces
I encountered the following problem in Paul Zeitz's The Art and Craft of Problem Solving (problem 2.4.16 on page 56 of third edition):
Is it possible to color the faces of 27 identical $1 \times 1 \...
- 155
0
votes
2
answers
90
views
Count the number of unique permutations of this puzzle
I have a puzzle that has 9 square slots. It looks like a rubik's cube face. In each square (except the middle one) you can put a coin in the middle.
The goal is to count the number of unique puzzles ...
1
vote
1
answer
207
views
Finding distinct paths for an NxN chessboard. Is there a pattern? [duplicate]
I decided to come up with a puzzle for my friends at lunch today, and it seems to be much more complicated then I would have ever imagined. Here's the puzzle:
A rook is located at the top left corner ...
- 13
2
votes
2
answers
198
views
Shapes made of concave and canvex curves combination problem
I have been working on this particular problem for a considerable amount of time. The problem is as follows:
[[ Image of problem ]]
In the $7$-by-$7$ grid above, one can draw a simple closed curve ...
- 23
6
votes
5
answers
533
views
Math competition question about ways to spell BANANA in a square
This is a math competition question I did. Essentially, starting at B, a move consists of moving to a non-diagonal adjacent square and noting the letter you land on (with the exception of the starting ...
- 728
4
votes
0
answers
109
views
What percent of lighted grids are walkable: a trick-or-treating problem
I am a math teacher that likes to invent fun math problems to explore. Here is one I have been investigating for a little while and have made little progress on because the number of possible $n \...
- 53
1
vote
1
answer
122
views
Permuting the rows in ascending order first and then the columns of any Young tableau gives a standard Young tableau
Show that if you take any Young tableau and permute the rows in ascending order first and then the columns in ascending order (or columns first and then row), then you get a standard Young tableau.
I ...
- 9,592
4
votes
1
answer
127
views
Word and number ladder puzzles
Introduction
$
\begin{array}{}
\begin{array}{c|c|c}
\text{1} & \text{SIZE}\\
\hline
2 & \\
3 & \\
4 & \\
5 & \\
\hline
6 & \text{RANK}
\end{array}
&
\begin{array}{c|c|...
- 3,075
4
votes
0
answers
44
views
Arrangements of "skyscrapers" in which $n$ are visible [duplicate]
There is a logic puzzle called "Skyscrapers", in which you must place all numbers from $1$ to $n$ in each row and column of a square grid. Each number represents the height of a building at ...
- 1,569
5
votes
2
answers
189
views
How to count - probability puzzle
A $3 \times 3 \times 3$ big cube consists of $1 \times 1 \times 1$ smaller cubes. The big cube is painted black on the outside. Suppose we disassemble the cube and randomly put it back together. What ...
- 5,450
6
votes
3
answers
600
views
Why does the number of solutions for the $n$-Queen problem drop at $n = 6$ specifically?
I was looking into determining the number of solutions to the $n$-Queens problem, admittedly to study DFS for a course.
During this study, I naturally found OEIS A000170, which lists the number of ...
0
votes
1
answer
69
views
Determine if you are inside or outside a closed region
You wake up in a desert and you find yourself next to a very, very long wall. All you know is that the wall forms a closed region. You are only allowed to walk in the space, and to put "flags&...
- 2,510
5
votes
2
answers
194
views
Covering a $kn+1\times kn+1$ region on a $(k+1)n-1\times (k+1)n-1$ square grid
We are given a $\left((k+1)n - 1\right)\times \left((k+1)n-1\right)$ square grid and tiles of size $1\times n$. We can place the tiles anywhere on the board, provided that they never cover the same ...
- 337
1
vote
1
answer
78
views
Probability that a person does not receive a gift
There are two groups $G_1, G_2$ of $n\in\mathbb N$ persons each. Each member of $G_1$ independently gives a gift to a random member of $G_2$. What is the expected number $\mathbb E[L_n]$ of members of ...
- 661