Questions tagged [combinatorics]

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

Filter by
Sorted by
Tagged with
3
votes
2answers
379 views

The most probable key

You broke into John Doe's house and found the safe. It has a combination lock with 4 dials ranging from 0 to 9 each. ...
8
votes
3answers
1k views

Possible pawn combinations

This may seem simple, but I have a problem calculating it. It may be because it's Monday morning. How may possible valid combinations of one color pawn (white or black, your choice) positions are ...
1
vote
2answers
451 views

Cover all squares in a square grid by moving to adjacent squares

Admittedly this is a problem I encountered from school but I cannot think of a proper proof solution. I thought about the logic that in order to cover all squares, there must be closed loops of ...
-3
votes
5answers
230 views

Interesting card combinations [closed]

In a normal pack of playing cards (without jokers), the probabilities of the following events are the same: (1) Drawing a set of 4 different cards (irrespective of its color and suit) - let us call ...
12
votes
1answer
372 views

Jigsaw Logic: ?s galore

I am working on a 256 piece jigsaw puzzle, but I am having a lot of trouble. Instead of the picture being a landscape or painting, the final image is just a sixteen by sixteen grid of identical ...
0
votes
1answer
172 views

Arrange six cigarettes in such a way that each cigarette touches every other cigarette [duplicate]

What are some ways to arrange six cigarettes in such a way that each cigarette touches every other cigarette?
5
votes
1answer
194 views

15 pawns on a chessboard

15 pawns are placed on the centers of distinct squares of a chessboard. Prove that there are three pawns which form a right triangle. In the example board below, a couple of right triangles are ...
22
votes
3answers
4k views

bored of eating soup

A man orders spicy noodle and leek soup from a restaurant, but gets bored while eating. When he gets bored, there are exactly 100 noodles in the soup. Because he is bored, he decides to play a game ...
1
vote
1answer
259 views

222 Black and 333 white balls [duplicate]

You have a bag with 222 black balls and 333 white balls in it. You also have an almost infinite supply of black and white balls that are not in the bag. You do the following over and over again: ...
3
votes
1answer
307 views

Simple solution for a scheduling puzzle

An amateur football coach wants to train the players by dividing them into two team and having the teams play each other. However, because of time constraints, each player comes and leaves at a ...
4
votes
4answers
530 views

Find the smallest positive integer that cannot be made from four 8s, or show it doesn't exist

This puzzle was inspired by one recently posted by @Max. What is the smallest positive integer that cannot be made from four 8s, where bracketing is allowed and the only permitted operations are ...
6
votes
2answers
681 views

Crystal Maze totem pole puzzle

The UK game show "The Crystal Maze" features a puzzle based around a totem pole. The contestant has four different coloured blocks which can be stacked up to make a totem pole. The blocks may be ...
1
vote
0answers
200 views

Which string can't be accessed? [closed]

We start from the string $ABCDEF$ and we can edit it with the following rules: 1.$ABCDEF \Rightarrow ADBECF$ 2.$ABCDEF \Rightarrow DAEBFC$ Which string can't be accessed? DBAFEC ...
0
votes
1answer
139 views

Rotating teams without repeating [closed]

I have 24 teams that will rotate to 12 stations (2 teams per station) Teams will not compete against another team more than once.
6
votes
4answers
536 views

Divide $n$-gon into triangles

Find all positive integer $n \geq3$ such that the convex $n$-gon can be divided by its $n-3$ diagonals into $n-2$ triangles so that no two of these diagonals intersect inside the $n$-gon and all the ...
6
votes
2answers
510 views

Determining integral weights

We have a balance scale, where can determine whether two quantities $a$ and $b$ satisfy $a>b$, $a=b$ or $a<b$. Given an integer $w_n$, we want to be able to distinguish between weights of $1,2,\...
5
votes
1answer
518 views

Number of pairs of adjacent cells that have distinct colors

Each cell of an $\;8\times8\;$ table is colored either black or white such that every column has equal number of black cells and no two rows have equal number of black cells. Find the maximum ...
0
votes
2answers
664 views

What is the number of pattern locks possible for n × n grid?

What is the general formula to find the number of pattern locks possible for n × n grid? Rules are quite identical to what phone lockscreen has. Dots cannot be used more than once. Length of a ...
0
votes
0answers
137 views

Mimimum number of guesses [duplicate]

Given the numbers 1 to 1000, what is the minimum numbers guesses needed to find a specific number if you are given the hint "higher" or "lower" for each guess you make? I realize the answer for this ...
18
votes
4answers
2k views

Consecutive Towers of Hanoi

Consider the following variant of the Towers of Hanoi puzzle. There are six pegs. One of the pegs has a stack of $n$ differently sized disks, sorted by size so the smallest disk is at the top. All ...
2
votes
1answer
135 views

How many moves are at least needed to change the order of cubes to a arbitrary order? [closed]

We have several cubes that are ordered as in the picture below: What is the smallest number of moves with which we are able to reach every possible order? Only the order of the cubes in the ...
2
votes
2answers
123 views

Right angled triangle to all acute angled triangles

What is the minimum number of cuts needed to dissect a right angled triangle into acute-angled triangles ?
11
votes
3answers
392 views

Tiling a hexagonal chessboard with “tribones”

A tribone is a tile made of three hexagons in a line. A hexagonal chessboard is a hexagonal grid of 91 cells in the shape of a larger hexagon. When 30 tribones are placed on a hexagonal chessboard ...
5
votes
3answers
658 views

The coolest checkerboard magic trick. Version 2

Version 1: The coolest checkerboard magic trick You and your friend are imprisoned. Your jailer offers a challenge. If you complete the challenge you are both free to go. The rules are The jailer ...
2
votes
3answers
220 views

Menu subset puzzle creation help

I'm having trouble finding out the best way to create the lists of numbers for the puzzle I'm creating. There are 2 sets of 13 integers (from 1 to 50 inclusive). Call these $A$ and $B$. There exists ...
6
votes
3answers
5k views

Flip one of the 64 coins on chess board [duplicate]

I know the answer of this puzzle. I want to know does this puzzle works perfectly for every n × n chess board? Is there a upper bound to n? What is the upper bound for n, if m coins are allowed to ...
-1
votes
3answers
255 views

Find the missing numbers in this constrained matrix/grid

The numbers 1 to 16 inclusive are arranged in this 4x4 matrix, such that no two numbers that are adjacent (horizontally, vertically or diagonally) that are consecutive, i.e. they must have a ...
13
votes
2answers
846 views

The impossible digital sum

There are 10 digit numbers you are supposed to use shown as below; And there is a very special addition where every digit is used only once. As you see, most of the digital signals (blue squares) are ...
7
votes
6answers
2k views

How will Y lose the game?

X and Y are playing a game. There are eleven coins on the table and each player must pick up at least one coin but not more than five in a turn. The person picking up the last coin loses. X starts. ...
4
votes
3answers
1k views

Total no of squares on a Chess Board

Is there any formula than calculates the total number of squares on chessboard? For example in a $8\times8$ chessboard, there are squares of sizes $1\times1$, $2\times2$, $\ldots$, $8\times8$. So I ...
19
votes
4answers
1k views

Magician's hide and seek with 8 cards

This question is a followup to this question by @Wen1now. After demonstrating the previous trick, the magician decided to make things a bit harder, discarding some cards so that only eight were left. ...
7
votes
1answer
653 views

How many different pentagons in this grid? [duplicate]

The 3x3 grid is given as below; How many pentagons could be drawn by connecting the dots by lines in the grid? Rules: Pentagons could be convex or concave. The lines cannot intersect each other. ...
9
votes
2answers
470 views

Cheating aplenty at Build-a-Die 2017

Each year, 9 affluent acquaintances meet to compete in a unique game of chance: each creates a fair six-sided die and rolls against each opponent 100 times -- the player with the most total wins in ...
4
votes
1answer
309 views

Coffee machine queue

A coffee vending machine which only sells 0.5 € coffee cups, accepts only 1 and 0.5 € coins. At a given moment when it is empty of coins, there are $m+n$ people queuing to buy their coffee cups. $m$ ...
9
votes
1answer
472 views

A Very Dizzy Puzzle

Here's an odd little sliding puzzle: As you can see, there is currently one free space, in the centre circle. That circle can spin freely, even when it is holding a piece in its 'holder'. Note, ...
12
votes
2answers
2k views

How many squares can you make with equal ranged points?

This question is directly related to How Many Squares on the Peg Solitaire. Is it possible to formulate with a given dimension of equal ranged points $m\times n$ where $m,n\geqslant 2$? For example;...
8
votes
3answers
580 views

How Many Squares on the Peg Solitaire

We have a well known peg solitaire which is not played yet as seen below: At most how many squares can you make by joining the points as exemplified below? Note: No ball (point) in the middle! so ...
2
votes
2answers
262 views

How many different results?

$a,b,c,d,e,f,g,h,i$ are real numbers and you would like to calculate the equation below by adding as many paranteses as you want: a-b^c-d/e+f/g^h*i What is the maximum amount of different results ...
8
votes
2answers
500 views

Guess n binary digits

There's a N (1 ≤ N ≤ 1 000; N is pair) digit binary number and you have a maximum of 2050 guesses to find out which is it. With each try, you will know one of the following: You got it right (100%); ...
6
votes
2answers
821 views

Building a pyramid from di-spheres

Suppose you want to build a square pyramid made up of 30 spheres. So the bottom layer is a 4x4 square arrangement of touching spheres, and it has 3x3, 2x2 and 1x1 layers on top. The building blocks ...
7
votes
1answer
270 views

Introducing: Sudoku-Janpu

This is an entry into the 29th fortnightly topic challenge - Retrograde Analysis. Scenario: Alice and Bob were playing a game of Sudoku-Janpu. Alice: I win. Bob: (looking at the board) Oh man, not ...
8
votes
1answer
428 views

How many different kinds of rings are there?

Far far away in the distant future some explorers stumble across the remains of an alien civilization. You are the teams linguist, and it's your job to attempt to translate the hodge podge of symbols ...
3
votes
3answers
588 views

A variation of the poisoned bottle problem

Old story. The King has received X=950 bottles of wine, exactly one of which is poisoned enough to kill a man exactly 24 hours after ingestion. The King wants to feed some N prisoners some mix of the ...
3
votes
0answers
367 views

The King's Routes Problem: How many possibilities? [closed]

On a chessboard, a king is to be allowed to move one square at a time: horizontally to the right, vertically downward, or diagonally to the right and downward. Imagine a reduced 4x4 chessboard, with ...
9
votes
3answers
1k views

Calendar Cubes are Impossible!

An easy one for the middle of the week! Jane and John were discussing a business idea. John wanted to make a little set to keep track of the date. It was to include two cubes with a single digit on ...
6
votes
3answers
417 views

Two knights and two rooks on a 5-by-5 chessboard

In how many different ways can two knights and two rooks be placed on a $5\times5$ chessboard, so that no piece attacks another piece?
4
votes
1answer
196 views

Word-making game

I like to play a game with my family and friends. It goes like this: The players take turns to say a letter. The letters taken in order must be the start of some word, and the person who finishes a ...
12
votes
1answer
284 views

Personally Attacked by Squares in a Bathroom

Well, this is a bit of a mess! Your bathroom has a rectangular floorspace of size $xn$ by $yn$, where $x, y, n$ are all positive integers, and now that you've removed those ghastly yellow and purple ...
6
votes
1answer
166 views

Logical expression puzzle

A through I are all binary variables (so they're either true of false). How much is the binary number 0bABCDEFGH if all of these are true? (A or B or C) and (D or E or F) and (G or H or I) (C and D ...
8
votes
2answers
731 views

A Strange Box of Buttons

Your friend has given you a very unusual birthday present. It is a round box, and inside the lid are $n$ buttons equally spaced in a circle. The box functions like this; you can press two buttons, ...

1 3 4 5 6 7 11