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Questions tagged [combinatorics]

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

7
votes
1answer
465 views

Pattern recognition using a, b, c, d, and e

I need to find the solution of this pattern: {a,b,c}, {c,d,e}, ?, ?, ?, {b,c,e}, {a,c,d}, {b,c,d}, {a,c,e}, ? The solution should be something like this: {a,b,c}, {c,d,e} {x,x,x} {x,x,x} {x,x,x}...
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
552 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
763 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
631 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
189 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
151 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
147 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
84 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
196 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
98 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
153 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
43 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
119 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
749 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
401 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
222 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
173 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
332 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
140 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
419 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
438 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
353 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
944 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
288 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
527 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
384 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
450 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
232 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? ...
12
votes
4answers
486 views

2-Palindromic DNA

There are two bacteria. Both of their DNA sequences are only one letter long:  Bacteria #1 has the DNA sequence 'A'  Bacteria #2 has the DNA sequence 'T' Every minute, two things happen: ...
8
votes
4answers
429 views

A Treacherous Hobnob of Snobby Nobs

You are a lowly party planner, tasked with inviting some Nobs to a wedding dinner, and then seating them appropriately. Now, Nobs are very picky about their seating arrangement: they insist on having ...
14
votes
2answers
741 views

The Stubborn Tenant

A hotel is in the form of a grid that extends infinitely to the right and above, and a strange alien lives in the corner room. The proprietor wishes to vacate the six rooms closest to the corner, but ...
5
votes
2answers
184 views

Move 10 sheep on another shore [duplicate]

We have 10 sheep: 5 black and 5 white, two shores and a bridge through a river. 5 black sheep on a shore and 5 white sheep on another one. We have to move sheep on the opposite side. Sheep can move ...
16
votes
7answers
2k views

Soldiers in the Parade Ground

Twenty-five soldiers are standing in a parade ground consisting of a five-by-five grid of large concrete slabs, laid out in a neat north-to-south, east-to-west square array. Each soldier is standing ...
1
vote
1answer
146 views

Can all sums of a 5*5 matrix with the numbers {-2,-1,0,1,2} be different?

Is it possible to construct a matrix with 5*5 cells, where the value in each cell is taken from the set {-2,-1,0,1,2}, such that all of the 12 sums of the cells in each row/column/main diagonal have ...
8
votes
1answer
413 views

How many friends does Tiffany have?

Tiffany has 14 classmates; all of her classmates have a different number of friends in the class. How many of them are friends with Tiffany? (If A is a friend of B, then B is a friend of A.)
6
votes
2answers
221 views

Challenging 15 rectangle tiling problem

This will test you, a computer will definitely help. Just one set of $1:2$ aspect ratio rectangles this time, but $15$ of them. Short side only is listed. The challenge is to arrange them into a ...
5
votes
1answer
368 views

Unfair tiling puzzle

Your goal is to make two squares of the same size from a set of rectangles. Each of the rectangles has an aspect ratio of $1:2$. Select two sets of rectangles from the list: ...
4
votes
1answer
317 views

Hard tiling puzzle

Your goal is to make two squares of the same size from a set of rectangles. Each of the rectangles has an aspect ratio of 1:2. Select two sets of rectangles from the list: ...
4
votes
1answer
163 views

Slightly bigger 1:2 hand tiling, getting a little harder

Pretty sure an expert hand tiler would be able to find this. Please post a spoiler only if you hand tile it... Tile a $60\times60$ square with $1:2$ rectangles of sizes ...
4
votes
2answers
261 views

Trickier hand tiling

This should be a little trickier. Using $20$ rectangles of aspect ratio $1:2$, make two squares. Use sizes 2,3,7,12,16 in both squares. And split this list among ...
3
votes
1answer
107 views

Dissect a square into 1:2 non-congruent integer-sided rectangles

(Similar to the recent 3:1 and my 3:2 rectangle question) Tile a square completely with rectangles which have aspect ratio 1:2, integral side lengths and all different sizes. In other words selected ...
6
votes
3answers
247 views

Gimped Knight on a Torus

Place a knight on a toroidal $100 \times 100$ board (i.e. the edges wrap). Restrict his movement to a particular 2-square pattern (gimped from his usual 8-square pattern; for example, he may only be ...
7
votes
2answers
237 views

Dissect a square into 3:2 non-congruent integer-sided rectangles

(Similar to the recent 3:1 rectangle question) Tile a square completely with rectangles which have aspect ratio 3:2, integral side lengths and all different sizes. In other words selected from 2x3, ...
4
votes
3answers
486 views

Lottery in Minesweeper

The question is inspired by this scenario: In this case, we had 2 tiles remaining, and each had the equal probability of being a mine, based on the already deduced information. Thus it was necessary ...
3
votes
1answer
552 views

Is there a system to brute force a combination padlock?

Our escape room recently acquired some old lockers with a couple of these combination padlocks on them. Unfortunately, the seller didn't know the code. Is there a mathematical system in cycling ...
2
votes
1answer
152 views

Seven light bulbs in a circle, switch three adjacent ones at a time [duplicate]

You have seven light bulbs in a circle. All the lights are off and you want to turn them all on. You are allowed to switch the state of any three adjacent light bulbs at time. What is the minimum ...