Questions tagged [combinatorics]

A puzzle based on combinatorics, which is the study of counting discrete structures. Use with [mathematics]

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3
votes
1answer
54 views

Is it possible to calculate group 3's factor of 3 in Thistlethwaite algorithm?

https://www.jaapsch.net/puzzles/thistle.htm I'm trying to generate 29400 ($8C4^2 * 6$) indices for each one of the cube states in G2. $8C4^2$ = 4900 is for solving the corner and edge pieces (forming ...
3
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3answers
223 views

Not selling 100 pencils

A shop sells pencils only in boxes of fixed size. It cannot sell 100 pencils. It can sell any larger amount of pencils. It has one of each size box on display. question 1: What is the minimum amount ...
6
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1answer
194 views

A 3x3 grid with common factors

A $3 \times 3$ grid $G$ is filled with every number from the set $\{2,3,5,6,7,11,14,15,30\}$. Now a new $3 \times 3$ grid $H$ is formed, such that $H_{ij}$ is the number of neighbors of $G_{ij}$ that ...
11
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4answers
912 views

Deriving a 3x3 grid from another one

A $3 \times 3$ grid $G$ is filled with every number from $1$ to $9$. Now a new $3 \times 3$ grid $H$ is formed, such that $H_{ij}$ is the number of neighbors of $G_{ij}$ that are greater than $G_{ij}$....
1
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1answer
195 views

How to think about permutation puzzles?

I know this is a very general question, but these kinds of puzzles are the only ones I can't figure out on my own. Rubik's cube, Twiddle, 16... (the last two are in Simon Tatham's Puzzle collection.) ...
8
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3answers
666 views

Perfect magic 4x4 square

Can you fill a 4x4 grid with every number from 1 to 16, such that every row, every column and every 2x2 sub-grid of numbers sum to the same value?
8
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5answers
1k views

Minimize 1×3 tiles on a 5×5 table to block any more 1×3 tiles

What is the minimum number of 1×3 tiles that can be put on a 5×5 table so that no more 1×3 tiles can be put on it? Borders of a tiles are parallel to sides of the table. It is 5 but I can not prove ...
0
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1answer
72 views

Unusual 3x3 square

Can you fill a 3x3 grid with every number from 1 to 9, such that the sum of numbers in the first row is equal to the sum of numbers in every 2x2 sub-grid? Can you find multiple solutions?
8
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1answer
301 views

Manual tiling with 8 dodecadudes

Here are 8 dodecadudes. (Drawn from the very numerous dodecadrafters, made from 12 half equilateral triangles, dodecadudes are a subset of 770 pieces with sharp points and narrow necks excluded). ...
9
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1answer
471 views

Plants vs Zombies!

Several plants and zombies (no more than 20 creatures in total) came to the party “Plants VS Zombies”, and it turned out that all the creatures are of different heights. When a plant speaks to a lower ...
17
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5answers
868 views

Maximize the number of paths

You have exactly 990 edges. Assemble them into a simple undirected graph with two distinguished vertices A and B, such that the number of different simple paths from A to B is as large as you can make ...
8
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3answers
653 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! Don't ...
11
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3answers
1k views

Swapping 3 knights in a 4x4 grid

Can you swap black and white knights in this 4x4 grid? There is one important constraint: at no point can a knight be under attack by an opponent knight. Knights move using standard chess moves (L ...
2
votes
2answers
98 views

Surrounding an L-shaped tromino

You are given an L-shaped tromino, shown below. Can you surround it by 10 more L-shaped trominoes? The new trominoes can be rotated and must touch the original tromino at an edge or a corner. This ...
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2answers
334 views

Find the least number of objects from a jar when those have two colors

I've been going in circles with this question which belongs to certainty about something. The original source of this problem is unknown. I found it in a textbook who doesn't have an author but rather ...
10
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5answers
949 views

2 fake coins from a pile of 30 coins

You need to find two fake coins from a pile of 30 coins. You know that a fake coin has a different weight to a real coin, but you don't know whether it is lighter or heavier. You also know that all ...
10
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2answers
582 views

Two football teams

Twenty two football players have agreed to split every week into two teams and play a match against each other. Every week, teams will be chosen differently, 11 players in each team, and they will ...
45
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4answers
8k views

The coolest checkerboard magic trick

In the small town of Terni (Italy), there's a couple of young friends named Marco and Leonardo, who like to perform magic tricks to a restricted audience of common friends and relatives. They like to ...
13
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2answers
902 views

Placing 9 cars into a 4x4 carpark

A carpark is arranged in a 4x4 grid and has a single entry/exit as shown in the diagram. A car has the size of a single cell of the grid. Cars can move through adjacent empty cells of the carpark ...
13
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3answers
1k views

Ten distinct numbers in the table

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9
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1answer
325 views

Four-by-four table with equal row and column products

Is there a way of filling a 4×4 table with 16 distinct integers from 1, 2, ..., 100 such that the products of the numbers in every row and in every column are all equal to each other?
1
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1answer
115 views

Probability of a successful DT Cannon

In multiplayer Tetris, there are a number of opening setups that let you send powerful attacks very quickly. One popular opening is the DT Cannon, which allows the player to very quickly send a T-Spin ...
3
votes
1answer
147 views

Swapping eggs puzzle

You can try this with pencil and paper, or make it a physical puzzle you can try out with your kids if you have one of those large 3x6 egg cartons laying around. Imagine you have 3 rows of 6 spots. ...
4
votes
1answer
150 views

Non-increasing arrangement

We are arranging the numbers from 1 to 8 in an order so that three consecutive terms cannot be increasing. For example, 12345678 isn’t allowed but 81436572 is. How many ways are there to do it? Please ...
15
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5answers
644 views

Domino tiling on 8x8 checkerboard with four squares removed

I once posted this problem on the (now deleted) Area 51 Math Puzzles proposal. It was well-received there, but obviously I didn't get an answer. I still don't know the answer, and I'm not even sure if ...
9
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5answers
3k views

How many different non congruent polygons can you make on a 3x3 dot grid?

There is a 3×3 dot grid. How many different non-congruent polygons can you make on the grid? Rules: All vertices of the polygon must be on the grid Only non self intersecting polygons Only polygons ...
2
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0answers
68 views

Fair and square island hopping [duplicate]

If amateur fiction is not your thing skip to the bottom. As IP (Implausible Physics) expert for DREAM, the Department for Reckless Engineering and Advanced Megalomania you have been tasked by sheikh ...
1
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2answers
228 views

Minimize the longest King chain on a 6x6 ternary grid

This puzzle is an extension of this one: Minimize the longest King chain on a 5x5 binary board Given a grid filled with numbers, we define a King chain to be a path on the grid such that: The path ...
0
votes
1answer
125 views

Minimize the longest King chain on a 7x7 binary grid

This puzzle is an extension of this one: Minimize the longest King chain on a 5x5 binary board Given a grid filled with numbers, we define a King chain to be a path on the grid such that: The path ...
4
votes
1answer
336 views

Two knight tours on a 4x4 grid

Two knights are placed on opposite corners of a 4x4 grid. Can you move* each knight 7 times, such that each cell of the grid is visited exactly once by exactly one of the knights? *Note that a knight ...
13
votes
7answers
838 views

Minimize the longest King chain on a 5x5 binary board

Given a grid filled with numbers, let's define a King chain to be a path on the grid such that the path can be traversed with chess King's moves (moving to one of 8 adjacent cells at a time), the ...
5
votes
0answers
192 views

Choosing squares on a square board [closed]

I have an $8 \times 8$ board. On the board, I want to choose 2 unit squares in each column and row such that none of the chosen squares are touching. This means they cannot share a side or a corner. ...
3
votes
2answers
91 views

Dividing the first 10 numbers into two groups with similar product

Can you paint all numbers from 2 to 10 with red and blue colour, such that the product of all red numbers is as close as possible to the product of all blue numbers?
9
votes
1answer
1k views

On an 8x8 board, a knight is on the second square of the last row. Only moving upwards, how many routes to the top?

In chess, a knight moves in L-shaped jumps consisting of two squares along a row or column plus one square at a right angle. If it can only move up the board, how many routes can it take to reach any ...
3
votes
2answers
196 views

Determine minimal number of moves to find cells on a square table 10×10 in which a treasure is hidden

In a 10x10 square table, two neigbouring 1x1 cells contain a hidden treasure. John needs to guess these cells. In one move he can choose some cell of the table and can get information whether there is ...
10
votes
2answers
458 views

Sixteen chess pieces on a square board

It is well known that the eight main chess pieces cannot cover a chess board. Suppose I have two sets of the eight main pieces. What is the size of the largest chess-like square board all of whose ...
6
votes
1answer
394 views

Eight distinct numbers in the table

While I was working on Ten distinct numbers in the table, I found that odd-sized tables will all work. Additionally, the technique @xnor used to prove a 10x10 doesn't work does not automatically rule ...
13
votes
7answers
975 views

Three Stacks N cards

You have n cards that have been numbered from 1 to n. You will divide these cards into three stacks such that, the sum of the numbers of any pair of cards taken from any stack will not be a perfect ...
7
votes
2answers
900 views

21 knights covering a 11x11 chess board

Can you place 21 knights on a 11x11 chess board, such that every empty cell is under attack? Good luck! Here is a similar question for 10x10: Knights covering a 10x10 chess board
6
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2answers
894 views

Knights covering a 10x10 chess board

What is the minimum number of knights you need to place on a 10x10 chess board, such that every empty cell is attacked by at least one knight? Good luck!
5
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2answers
190 views

Running Out of Digits, Level 3

The challenge idea is credited to HelloWorld1337. You initially have x of each digit from 0 to 9. This means you have x * 10 digits in total. This count for each digit is shown in the table below. ...
19
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7answers
5k views

How many digits can you create with only one seven segment display?

Let say you have a digital indicator which consists of seven lights as you see below. You are trying to make as many distinct digits as possible but there are two requirements for that: At least one ...
14
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2answers
2k views

Three queens and two rooks covering the chess board… again!

Three queens and two rooks can be placed on a chess board so that all empty squares are under attack, as has been shown here: 3 queens and 2 rooks covering a 8x8 chess board. What if we require that ...
5
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1answer
827 views

3 queens and 2 rooks covering a 8x8 chess board

Can you place 3 queens and 2 rooks on a 8x8 chess board, such that every empty cell is under attack? Good luck!
2
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2answers
491 views

Lightbulbs in a 3×3 square

Suppose we have a $3\times 3$ arrangement of lightbulbs and we switch them on/off randomly (probability $½$). What is the probability the no adjacent bulbs are on? My attempt was: Let $1= $ on and $0 =...
1
vote
2answers
81 views

No four cells forming a rectangle

You are given a 5x5 square grid with 25 cells. Can you paint 12 cells, such that no 4 painted cells form the corners of a rectangle with sides parallel to the edges of the grid? Good luck!
62
votes
8answers
105k views

Twelve balls and a scale

You are given twelve identical-looking balls and a two-sided scale. One of the balls is of a different weight, although you don't know whether it's lighter or heavier. How can you use just three ...
9
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3answers
1k views

No three points in a line

You are given a 4x4 square grid. It has 16 cells and 25 grid intersections. Can you place 10 points at grid intersections, such that no three points lie on the same straight line? Lines can be ...
2
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0answers
455 views

A robot moving on a grid. Part 2

This is an extension of the discussion A robot is placed on a grid point. At each move the robot must take three steps along the edge of the grid. After each step the robot must turn right. Lengths of ...
14
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2answers
826 views

Counting numbers with 3 dice

Yesterday I saw a pair of calendar dice that can be rotated and swapped to show all days of a month between 01 and 31. The dice have a digit painted on each face. First die: [0, 1, 2, 3, 4, 5] ...

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