Questions tagged [combinatorics]

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

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6
votes
2answers
226 views

A very tiny probability puzzle [closed]

Imagine there are six football teams (A,B,C,D,E,F). They play each other in a league once only (on neutral ground). To determine the score, two $d4-1$, i.e. the outcome is $\{0,1,2,3\}$ are rolled, ...
17
votes
4answers
2k views

Paint 21 Squares of a 7×7 Board Without Forming a Rectangle

Got a nice puzzle from my friend, when he was competing in IWYMIC/IMC 2011. Paint $21$ of the $49$ squares of a $7 \times 7$ board so that no four painted squares form the four corners of a rectangle....
4
votes
4answers
241 views

Rotating numbers in a 3x3 grid

We have a 3x3 grid numbers like so 9 8 7 6 5 4 3 2 1 Four numbers in any 2x2 sub-grid can be rotated clockwise or anti-clockwise. For example a b c d rotated clockwise becomes c a d b Is it ...
2
votes
1answer
59 views

Rotating numbers in a 2x3 grid

We have a 2x3 grid numbers like so 6 5 4 3 2 1 Four numbers in any 2x2 sub-grid can be rotated clockwise or anti-clockwise. For example a b c d rotated clockwise becomes c a d b Can you obtain ...
8
votes
4answers
475 views

Running Out of Digits, level 2

The challenge idea, and images are credited to Andrew. You initially have 100 of each digit from 0 to 9. This means you have 1000 digits in total. This count for each digit is shown in the table ...
10
votes
3answers
957 views

Math puzzle - Running Out of Digits

You initially have 100 of each digit from 0 to 9. This means you have 1000 digits in total. This count for each digit is shown in the table below. Now start counting by ones, from 1. Each time you ...
-6
votes
1answer
196 views

Puzzling set of numbers [closed]

Let's have the following numbers $23, 40, 42, 44, \sqrt{43},\ 128i, 130, \sqrt{172}\ $. What is the relationship between these numbers, taken four at a time? There are only two combinations when you ...
-3
votes
1answer
250 views

Traps on Sudoku grids [closed]

Let's have a 9x9 Sudoku grid. Where the dots are shown above, you are allowed to put the numbers 2, 4, 6, 8. Each of these appears seven, seven, seven, and six times. It does not matter which number ...
14
votes
2answers
1k views

Make a hexiamond star by hand

Using some or all of the hexiamonds (pictured), make a star. You may flip pieces. The usual tiling rules apply, no overlaps, no gaps. Use only one or none of each piece. Answer is unique. Target shape ...
-1
votes
1answer
66 views

Painting a grid with 3 colours such that there are no right-angled triangles of one colour

What is the largest rectangular NxM grid (by area) that can be painted with 3 colours, such that no three cells of the same colour form a right-angled triangle. N and M must be 4 or greater. We only ...
3
votes
3answers
292 views

A 3x3 grid of numbers with unique row and column medians

Can you place every number from 1 to 9 into a 3x3 grid such that the median of every row and column is a unique value? The median of a row is the number that is greater than one number and smaller ...
7
votes
2answers
258 views

A 4x4 grid of numbers with unique row, column and diagonal ranges

Can you place every number from 1 to 16 into a 4x4 grid such that the range of every row, column and two main diagonals is a unique value? The range of a row is the difference between its maximum and ...
5
votes
2answers
131 views

A 3x3 grid of numbers with unique row and column ranges

Can you place every number from 1 to 9 into a 3x3 grid such that the range of every row and column is a unique value? The range of a row is the difference between its maximum and minimum values (...
3
votes
1answer
160 views

Two dice with same probability for each sum mk2

Inspired by Two dice with the same probability for each sum? To cheat in a game of sums, you get yourself a pair of magic dice. That pair behaves in a wonderful way where each individual die is fair (...
5
votes
6answers
1k views

Two dice with the same probability for each sum? [duplicate]

A friend invites you to play a game. The game is using two standard six-sided dice with faces numbered 1, 2, 3, 4, 5, 6 each. As usual, the dice are considered distinguishable, i.e. throwing a 1 with ...
6
votes
1answer
221 views

12 Distinct Weights' Sorting

There are 12 balls with all different weights. You want to sort these balls by weight. You have a friend who will help you with the ordering process. In each step, you give any 4 balls to your friend ...
1
vote
2answers
77 views

How many different ways can be read a number without repeating the same digit in an arrangement which resembles a triangle?

The problem is as follows: The figure from belows shows a triangular arrangement where there is a set of numbers. The condition is that in each reading you cannot repeat the same digit and the ...
20
votes
10answers
5k views

Creating 2020 in the fewest number of steps

You start with the number 1. You can create a new number by applying an operation on two existing numbers (can be the same). The operations are +, - and *. What is the fewest number of steps needed to ...
11
votes
3answers
538 views

Sorting marbles based on weightings

You have 50 marbles which look all the same. However, 25 marbles have weight l and the other 25 marbles weight h with l < h. A person creates two piles and claims he has sorted all 50 marbles by ...
11
votes
5answers
2k 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
196 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 ...
41
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 ...
3
votes
1answer
167 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
256 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
643 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
626 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 ...
29
votes
6answers
5k 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
963 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
362 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?
10
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
229 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
103 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
209 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 ...
9
votes
5answers
543 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
289 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
968 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
244 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
137 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
236 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
326 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}, . ....
2
votes
2answers
212 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
123 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
729 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
429 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
107 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
277 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^...

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