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Questions tagged [combinatorics]

A puzzle based on combinatorial mathematics, which is the study of finite or countable discrete structures.

1
vote
2answers
64 views

alphanumerical sequence Correct one

I have tried to solve this sequence which, I suppose has been created from the combination of a,b,c,d,e. I've reposted the correct sequence The sequence: (a,c), ?, ?, ?, (a,e), ?, ?, (c,d), ?, (b,...
4
votes
1answer
121 views

Dragons and Dragon Slayers

In the Kingdom, there are 7 dragons and 7 dragon slayers. Each dragon has ($\text{index}\times10000$) hit points (i.e. $10,000\to70,000$). Each dragon slayer does ($\text{index}\times1000$) damage (i....
23
votes
7answers
1k views

Wolves and sheep

All the sheep were living peacefully in the Land of Shewo. But suddenly they were struck by a danger. A few wolves dressed up as sheep entered the territory of Shewo and started killing the sheep one ...
3
votes
2answers
614 views

Bridge building with irregular planks

Imagine you have a big rectangular pond in your back garden. You wish to build a bridge from your house in the lower left corner to the small pagoda in the top right. You have lots of planks of ...
12
votes
1answer
800 views

Hacking a Safe Lock after 3 tries

My friend has a safe lock, with a numeric password of 4 digits (varying between 0-9 each). She lost the password and looked for my help. Fortunately, I was able to hack it. I could try some different ...
3
votes
1answer
187 views

Albrecht Durer Inspired Magic Square

A magic square is an n-dimensional matrix in which each row, column, and main diagonal sums to the same 'magic number' (denoted by s). A normal magic square uses each of the numbers from 1 to n ...
16
votes
3answers
2k views

Four buttons on a table

I was asked lately (in an interview) to solve this puzzle, which is similar to the 4 cups on table puzzle. In a certain room there is a rotating round table, with 4 symmetrically located ...
0
votes
2answers
114 views

Chess tournament winning streaks #2

So I asked a question here, which asks how many ways there are to score 7 points in 7 chess tournament games using the system on lichess.org, outlined both here and in the original question. That was ...
6
votes
7answers
308 views

Find minimum number of meeting periods to reach 2 degrees of separation for a group [closed]

I am running a training course and I want to arrange a set of 1:1 meeting periods between the participants such that at the end of the day there is a maximum of 2 degrees of separation between any of ...
9
votes
1answer
282 views

Chess tournament winning streaks

On lichess.org, they use a points system for keeping track of who is winning in a tournament. A win is worth two points, a draw is worth one point, and a loss worth zero points. Once a player has ...
3
votes
4answers
2k views

Move fast … Or you will lose

Suppose you're on a 4 × 6 grid, and want to go from the bottom left to the top right. How many different paths can you take? Avoid backtracking -- you can only move right or up.
2
votes
2answers
237 views

Unlock an answering machine using minimum number of digits

You know of answering machines with a remote inquiry facility, where you can call the answering machine and enter a four digit passcode into your telephone keypad, so you can listen to your messages ...
7
votes
2answers
2k views

Rectangles in a chess board

How many rectangles can be made from the individual spaces of a chess board?
7
votes
3answers
206 views

Distinct Arrangements of Balls on Tiles

You want to put several balls on $8 \times 8$ tiles, such that all $16$ ball arrangements on its rows and columns are different. What is the minimum number of balls to be put? Two arrangements of ...
3
votes
2answers
244 views

Enumerate the ways of putting six armies of queens on a humongous chessboard

This is a sort of a sub-problem of the open puzzle Peaceful Encampments, for high numbers of armies. Consider a chessboard with an astronomically large number of vanishingly small squares, on which ...
4
votes
2answers
161 views

Proving the count of symmetric configurations of pentagon

In a 3 × 3 dot grid, there are 5 configurations of symmetric pentagons. I am confused about how to prove that it is really just 5. Can anyone enlighten me?
2
votes
0answers
89 views

Telephone Exchange Vandalism [closed]

Our telephone exchange has 1000 wire-based connections. You dial a number, you get connected to the exchange and they put you through. Yesterday, vandals broke in and disconnected all the wires. Your ...
3
votes
1answer
105 views

Find a specific path on an n x n grid [duplicate]

Given a puzzle of the following form: Find a path between the top left corner to the bottom right corner, visiting each spot (.) exactly once. You can only move horizontally or vertically. ...
5
votes
2answers
186 views

Rotating 12 players on 2 tables

I'm stuck trying to find a good way to rotate 12 poker players on 2 tables. Assumptions: There are 12 players There are 2 tables We play for 4 hours Playing with sb means sitting at the same table ...
3
votes
3answers
1k views

Three coins for the fair king [duplicate]

Based on the question Eight coins for the fair king: I saw a comment saying "There isn't a good solution known even with three coins in all cases". So the challenge here is to try to solve the same ...
14
votes
4answers
565 views

Balancing Balls

I have a disk with 6 equally spaced dents around the edge. The disk balances on the center point. I want to place marbles around the edge so that it stays balanced. There are four ways that this can ...
12
votes
4answers
787 views

Crosses and Circles

Place two crosses on two cells of each row and column of this 9×9 board, and circles elsewhere, so that the number on the right of each row indicates the number of circles between its two crosses, ...
8
votes
2answers
661 views

Four Magic Ellipses

These four ellipses represent four sets and all the possible ways they can intersect (a Venn diagram, in other words). There are 8 regions inside each ellipse, and 15 regions altogether. Is it ...
9
votes
1answer
226 views

Puzzles like Sokoban?

I am looking for some puzzles like Sokoban or 15-puzzle but more difficult to solve and satisfy the following requirements: The number of possible moves at each step should be limited, let's say < ...
7
votes
1answer
158 views

A Christmas Tennis Tournament

Three friends, Charles, Hugh, and Freddy, played a one-month tennis tournament, just one match between two of them every single day during last December. During the tournament, whoever lost the day's ...
4
votes
2answers
149 views

Twelve friends and their birthdays

Twelve friends: Anna, Bill, Deb, Dory, Eliza, Gaby, Jan, John, Judy, Mary, Otto and Sam, were talking about their birthdays, and much to their surprise discovered that they were all born on different ...
9
votes
3answers
1k views

How many hexagonal paths?

Here is a hexagonal tiling, borrowed from Wikipedia. I start in any hexagon on the left hand side. I end at any hexagon on the right hand side. I can only travel to the right, not up, down or ...
14
votes
9answers
7k views

Salesman's claim for mechanical keypad lock - 5 buttons and 545 combinations!

A company makes mechanical keypad locks. The keypad is a set of five buttons arranged vertically. O O O O O The buttons are quite close together. Once a ...
3
votes
1answer
86 views

Creating my climbing wall

[based on a true story] I have here some climbing holds that I've made. There are two relevant parameters: The angle on the top, and The thickness, as shown. Now it is definitely the case that $...
4
votes
1answer
203 views

Making transmitting data safe - double reading

Hopefully in the spirit of the Fortnightly Challenge 3rd Sept 2018 - Reusing Information If we are given a binary string, say: 1100010101100011 and we wish to transmit it safely, we might use '...
1
vote
1answer
101 views

How to divide them into groups

You have found $13$ gold coins and strangely their weights are from $1$ to $13$ grams (such as $1,2,3,...$). You are bored and out of the blue you decided to divide golds into groups such as the sums ...
7
votes
1answer
159 views

Class Seating Arrangement

There are $25$ students with distinct heights in a class. The seats are arranged in the class like a square array ($5$x$5$) and students are seated such a way that each person will be taller than both ...
0
votes
0answers
48 views

Help me visit my friend through his new digital key lock [duplicate]

A friend of mine whom I have repeatedly tried to visit has upgraded his house‘s security system. Instead of using doorkeys which were quite easy to borrow, he‘s using one of these fancy new electric ...
7
votes
3answers
124 views

Lots of Parallelepiped

A,B,C,D are four points which are not on the same plane. How many different parallelepiped can be constructed whose vertices are these points? Parallelepiped is a solid figure with six faces ...
12
votes
3answers
752 views

Sticky sticky stick stick

You are given a long enough stick. Your task is to create a new type of foldable ruler something like shown below with it by cutting the stick into pieces and folding them at one point: You need to ...
7
votes
3answers
406 views

Rectangular Prisms

Eight corner bricks are taken out from a 5x5x5 block, which is something like below: How many rectangular prisms of all sizes can be counted in this block? Source: Oyun 2018 Final Exam Question
6
votes
3answers
234 views

Oddy Chessboard

On a standard chessboard, What is the number of different arrangements of pawns such that every square has an odd number of pawns on its neighbor (horizontally or vertically) squares? Note: ...
6
votes
2answers
177 views

Mathematical formulation for Dr. Eureka

I have to prepare an algorithm to solve the puzzle part of Dr. Eureka, a multiplayer game from Blue Orange Games. This is part of a research project that also involves computer vision and robotics. ...
3
votes
2answers
333 views

Rectangles and Diagonals

A 4×4 table has 18 lines, consisting of the 4 rows, the 4 columns, 5 diagonals running from southwest to northeast, and 5 diagonals running from northwest to southeast. A diagonal may have 2, 3 or 4 ...
7
votes
1answer
142 views

Permuting rows and columns to switch white rooks with black rooks

An adversary places eight white rooks and eight black rooks on sixteen squares of a chessboard, subject to these rules: In any row, there must be exactly two rooks, one of each color. In any column, ...
11
votes
1answer
428 views

6 nails String Art

String art is an arrangement of thread strung between nails to form geometric patterns or representational designs such as a ship's sails, sometimes with other artist material comprising the remainder ...
10
votes
2answers
439 views

A simple puzzle about moving students

You have n students sitting in a line and you want to move them so that no student is sitting next to anyone they were originally sitting next to. What is the ...
15
votes
1answer
375 views

Covering an 8×8 grid with trominoes

I tried to find non-trivial solutions on a deficient 8×8 grid covered with trominoes and I regret to say that after extensive efforts I found only two non-trivial solutions: The conditions of ...
9
votes
3answers
975 views

Lego brick towers

Disclaimer: I'm honestly not sure whether this question is best placed at Puzzling, Maths, or Programming SE, but I'm interested in the best solution, and I'm sure mods will shift the question around, ...
5
votes
3answers
348 views

Magic-preserving Permutations on a 4x4 Magic Square

Messing around with some magic-square puzzles, I faced the problem of deciding whether some two magical squares are, in fact, the one and same square wearing a different hat. It seemed to me, that for ...
13
votes
3answers
531 views

The $1 question: Tiling a triangle with trapezoids (the hard way)

Take a triangular grid consisting of 64 equilateral triangular cells in the shape of a larger triangle, and remove a single triangle at one of the tips. Can you tile this shape with 21 trapezoidal ...
3
votes
2answers
388 views

Lots of ships in the arbitrarily large battleship

This question is inspired by Oray's puzzle Lots of ships in the battleship. You have an $n\times n$ grid (a battleship board) and a certain number of $2\times2$ squares (ships) to place in the grid. ...
22
votes
7answers
5k views

Hacking an electronic keypad

You are a spy trying to break into an enemy facility. The back door is protected by an electronic keypad lock. You know that this particular lock is opened by a four digit code. Any stream of button ...
2
votes
5answers
499 views

Word game - possible to play all words?

You're trapped in a chamber and the only way out is to beat the chamber guardian on his own game: Word88. Both of you each take a turn to play a word. You can play one of the following: Add Play - ...
2
votes
1answer
239 views

Make lots of squares with only 6 squares

You are going to draw $6$ congruent squares to make as many squares as you can! What is the maximum amount of squares (except the original squares) you can create by drawing 6 congruent squares? ...