Questions tagged [combinatorics]

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

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10
votes
4answers
1k views

Finding the hardest 5x5 grid for a blindfolded robot to solve

This puzzle is based on the framework described here: Controlling a robot blindfolded on a 9x9 grid Here is a quick summary. A robot is located somewhere on a grid, but you cannot see it. You can ...
0
votes
2answers
174 views

Controlling a robot blindfolded on ANY 2x2 grid

This puzzle is based on the framework described here: Controlling a robot blindfolded on a 9x9 grid Here is a quick summary. A robot is located somewhere on a grid, but you cannot see it. You can ...
39
votes
3answers
3k views

Controlling a robot blindfolded on a 9x9 grid

A robot is located somewhere inside a 9x9 grid shown below, but you don't know where it is. You can send commands to the robot to make it move one cell left, right, up or down. Shaded areas and edges ...
2
votes
0answers
110 views

Peaceable Bishops on a 10x10 grid version 2

Can you place 42 bishops with 6 bishops for each of the 7 colors on a 10x10 grid, such that no two bishops of different colors attack each other? Here are some similar questions: Peaceable Bishops ...
7
votes
1answer
230 views

Heavy-duty computing challenge — maximum-density word packing

Caution: This is a challenge for computers only. Humans are advised to stand well clear of the protective safety cage. There may be CPUs on fire before it's over. Introduction Nothing makes a ...
9
votes
2answers
2k views

Peaceable Bishops on an 10x10 grid

Can you place 22 red, 22 white and 22 black bishops on a 10x10 grid, such that no two bishops of different colours attack each other? Here is a similar question for 8x8 grid: Peaceable Bishops on an ...
7
votes
2answers
615 views

Peaceable Bishops on an 8x8 grid

Place an equal number of red, white and black bishops on a 8x8 chess grid, such that no two bishops of different colours attack each other. What is the largest number of bishops you can place? Bonus ...
9
votes
2answers
613 views

Two button calculator part 2

A calculator has only 2 buttons. The first multiplies the current value by 2, the second divides it by 3 without a remainder (so 8 becomes 2). Can you use this calculator to reach every positive ...
27
votes
6answers
4k views

A robot surviving on top of a 3x3 platform

A robot sits in the central square on top of a 3x3 platform. The robot can move up, down, left or right, but if it steps off the platform it will crash and die. You can preprogram the robot to make a ...
6
votes
1answer
950 views

Two super-button calculator

A calculator only has 2 buttons. The buttons are, however, very powerful: they are programmable buttons, i.e. you can pre-set them to be any function (meaning any map from $\mathbb{Z}$ to $\mathbb{Z}$)...
8
votes
3answers
350 views

Two button calculator

A calculator has only 2 buttons. The first multiplies the current value by 2, the second divides it by 3 without a remainder (so 8 becomes 2). Starting with 1 what is the least number of presses you ...
8
votes
3answers
1k views

Three button calculator part 2

A calculator only has 3 buttons. The first multiplies the current value by 3, the second adds 2 and the third subtracts 2. The calculator always starts with 0. What is the smallest positive even ...
9
votes
1answer
2k views

Three button calculator

A calculator has only 3 buttons. The first multiplies the current value by 3, the second adds 2 and the third subtracts 2. Starting with 0 what is the least number of presses you need to reach 100?
11
votes
4answers
2k views

What's the most rewarding path?

Get from the top-left to the bottom-right using only right and down moves. Pick up as much gold as possible. There is only one maximum.
4
votes
1answer
225 views

Prime parallel rows for the first 20 numbers

Two positive integers can be joined with a straight segment if their sum is a prime and the segment doesn't intersect any other segments. What is the most number of pairs you can join if you can place ...
2
votes
1answer
99 views

The first 10 prime butterflies

A prime butterfly is a set of three distinct numbers $a,b,c$, such that $a+b$ and $b+c$ are both primes. Can you divide numbers from 1 to 30 into 10 prime butterflies?
3
votes
3answers
181 views

Dots in Squares

What pattern could be placed in the last square to complete the sequence? This image from https://drive.google.com/file/d/1W0A2E2GIwi6mq9b94FzA7PhfXCae-VPB/view originally created by Daniel ...
10
votes
5answers
452 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 ...
5
votes
2answers
276 views

Two equal-sized lists that produce prime sums

Place one or more distinct numbers between 1 and 100 into the lists $𝑃$ and $𝑄$, such that they contain the same number of elements and any number from $𝑃$ added to any number from $𝑄$ gives a ...
15
votes
5answers
955 views

Dividing the first 20 numbers into 3 lists

Place every number from 1 to 20 into one of three lists $P$, $Q$ or $O$, such that any number from $P$ added to any number from $Q$ gives a prime. What is the fewest number of elements that can be in $...
14
votes
1answer
1k views

Prime tree game

Let's play a game. On the first step you place the number 1. On the $n$-th step starting from $n=2$ you place the number $n$ such that: It is adjacent (horizontally or vertically) to one or more ...
2
votes
3answers
242 views

Place 28 dominoes into a 7x8 rectangle

A standard set of double-six dominoes has 28 tiles with 2 numbers on each side from 0 to 6. Tiles can be placed next to each other if all the touching numbers match (from all 3 adjacent sides). Can ...
4
votes
1answer
134 views

Place 28 dominoes in a loop

A standard set of double-six dominoes has 28 tiles with 2 numbers on each side from 0 to 6. Tiles can be placed next to each other if the numbers at each end match. Can you place all the 28 tiles such ...
1
vote
2answers
230 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 ...
5
votes
3answers
306 views

land of light bulbs

Leuks is the land of light bulbs, where the bulbs are leading a peaceful life, with no humans at all. Presently there are 100 residents . (Of course, they all are light bulbs!). Let $a_{1}, a_{2}, . ....
3
votes
2answers
190 views

SAME DIGIT NUMBER PUZZLE

You have 5 numbers 1, 9, 3, 5, and 7 you need to arrange these number like Rules create 2 pairs from numbers above keep one of the number in the middle then apply this law ...
3
votes
1answer
115 views

A subset of a subset of a subset of a subset of a set of $\{1,2,\cdots,10\}$

There is a set of $10$ first natural numbers, $S = \{1,2,\cdots,10\}$. Alice picks a subset of it, say $A \subseteq S$. Bob picks a subset of it, say $B \subseteq A$. Charlie picks a ...
7
votes
2answers
702 views

Paint numbers from 1 to 23 with three colours

Can you paint every number from 1 to 23 with three colours, such that there are no distinct numbers $𝑎,𝑏,𝑐$ of the same colour with $𝑎+𝑏=𝑐$? For example, you cannot have 2, 3 and 5 of the same ...
5
votes
1answer
406 views

Paint numbers from 1 to 8 with two colours

Can you paint every number from 1 to 8 with two colours, such that there are no distinct numbers $a, b, c$ of the same colour with $a+b=c$? For example, you cannot have 2, 3 and 5 of the same colour ...
1
vote
1answer
103 views

Last Person Remaining Avoids Death [duplicate]

There are 1600 people sitting around a circular table. The first person (person 1) has a sword and kills the second person then hands it to the next alive person (in this case person 3). Person 3 ...
8
votes
2answers
247 views

Hand tiling puzzle demonstrating Eisenstein triple $c^2 = a^2 -ab + b^2$

An Eisenstein triple is related to 60 degree triangles and a special case of the cosine law. But we need not worry about that except to note that a specific example of an Eisenstein triple is $7^2 = 5^...
4
votes
2answers
170 views

Covering an 8x8 grid with W pentominoes

What is the minimum number of W pentominoes you need to cover every cell of an 8x8 grid? Pentominoes may overlap each other and sit outside the boundary of the grid. They can also be rotated in any ...
14
votes
6answers
2k views

Covering an 8x8 grid with X pentominoes

What is the minimum number of X pentominoes you need to cover every cell of an 8x8 grid? Pentominoes may overlap each other and sit outside the boundary of the grid. An X pentomino looks like this:
0
votes
4answers
128 views

Rectangles formed from every tetromino, tromino and domino

Can you form a 4x7 rectangle from every tetromino, tromino and domino? There are 5 different tetrominoes, 2 trominoes and 1 domino. Can you find different arrangements that are not mirrors/rotations ...
15
votes
5answers
4k views

Princesses covering an 8x8 chess board

What is the minimum number of princesses you need to place on an 8x8 chessboard so that every empty square is attacked by at least one princess? A princess is a piece from fairy chess that can move ...
7
votes
2answers
2k views

Knights covering a 9x9 chess board

What is the minimum number of knights you need to place on a 9x9 chess board, such that every empty cell is attacked by at least one knight? Here is a similar question for a 10x10 chess board: ...
5
votes
2answers
590 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!
10
votes
3answers
289 views

Four hand tiled squares demonstrating a Pythagorean Quadruple

Demonstrating the Pythagorean Quadruple $6\times6 + 6\times6 + 7\times7 = 11\times11$ Using the pieces shown in the $11\times11$ square: The objective: Arrange the pink pieces (four enneominoes) ...
6
votes
2answers
479 views

10x10 grid with no unpainted hexominoes

What is the smallest number of cells you need to paint in an 10x10 grid, such that it contains no unpainted hexominoes? Note that a hexomino is a set of 6 adjacent cells (horizontally or vertically). ...
4
votes
2answers
370 views

8x8 grid with no unpainted pentominoes

What is the smallest number of cells you need to paint in an 8x8 grid, such that it contains no unpainted pentominoes? Can you find multiple solutions? Note that a pentomino is a set of 5 adjacent ...
4
votes
1answer
150 views

Prime magic star

Can you replace the letters with 10 consecutive primes such that the sum of numbers on each line is equal? I expect this to be solved with a computer. Good luck!
3
votes
1answer
150 views

Partition a 3x3 square into rectangles [closed]

Yesterday I watched "The man who knew infinity" about the amazing Ramanujan. Inspired by the partitions problem from the movie I came up with a puzzle: In how many ways can you partition a 3x3 grid ...
5
votes
3answers
204 views

Rawrdon Mamsay pays a visit

Now, I should warn you, this is one of my practical problems; meaning I don't know the solution and the answer's probably anticlimactic (like this or that). Still... My old pal Rawrdon Mamsay is soon ...
11
votes
3answers
2k views

Find number 8 with the least number of tries

You and your friend plays a game. In this game, there are numbers written on the cards from 1 to 12 on one side (so 12 cards in total). The other side is blank. You can write anything on the blank ...
8
votes
1answer
646 views

A curious 5x5 square

Can you fill a 5x5 grid with numbers from 1 to 5, such that every number occurs exactly once in each row, exactly once in each column and exactly once in each broken diagonal (in both directions)? ...
0
votes
2answers
98 views

Painting edges of a 3x3 grid with 4 colours

Can you paint the edges of a 3x3 grid with 4 colours, such that: The colours of edges of every 1x1 square are different. The colours of edges adjacent to every vertex are different. Here is a ...
0
votes
1answer
62 views

Painting edges of a 2x2 grid with 4 colours

Can you paint the edges of a 2x2 grid with 4 colours, such that: The colours of edges of every 1x1 square are different. The colours of edges adjacent to every vertex are different. Good luck!
8
votes
0answers
142 views

The Flippin' Magician's 7-card Grand Finale

This question is a followup to this question by @ais523, which itself was a followup to this question by @Wen1now. After touring the globe to accolades when performing his 10-card trick and 8-card ...
3
votes
2answers
185 views

Cross the pond, but there's a catch!

There is a square pond, conveniently divided into segments, with coordinate $(0,0)$ in the bottom left and $(10,10)$ is the top right. You have planks length $2$ and $3$. You start at $(0,0)$ and ...
15
votes
3answers
1k views

Transferring 9 pegs on a 9x9 grid

You are given a 9x9 grid with a set of 9 pegs (red circles) arranged in a 3x3 pattern in the corner, as shown below: A peg can jump over another adjacent peg in any direction (horizontal, vertical or ...