Questions tagged [optimization]

A puzzle where you have to optimize a certain objective function (maximize profit, minimize cost). There should ideally be a provable best answer, to avoid making the puzzle into an [open-ended] game.

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Shortest algorithm to rotate 2 corners on a Rubik's Cube

I'm looking for the shortest algorithm to rotate 2 corners without rearranging any other pieces location or orientation. For example, turning the Top-Front-Right corner clockwise and the Top-Front-...
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  • 27
1 vote
0 answers
95 views

Shortest Algorithm to flip 2 edges on a Rubik's cube

I'm looking for the shortest algorithm to flip 2 edges without rearranging any other pieces location or orientation. For example, flipping the Top-Front edge and the Top-Right edge. My current best is ...
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  • 27
3 votes
2 answers
311 views

What's the minimum number of airplanes needed for one to make a round-the-world trip?

Inspired by this puzzle. You have many airplanes starting at the same airport. Each plane has a fuel tank that holds just enough fuel to allow the plane to travel $\frac{1}{10}$ the distance around ...
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  • 5,145
11 votes
2 answers
3k views

Sharing 41 sticks among 42 people

Question: Ali-Baba is a prisoner of the $42$ thieves who have just stolen $41$ identical magic incense sticks. The thieves want share this loot in such a way that everyone has exactly the same pieces ...
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  • 113
11 votes
4 answers
1k views

Prime lights out

You start with a 4x4 grid filled with zeroes. If you press a cell then the cell and all its neighboring (horizontally and vertically) cells will have their numbers increased by 1. What is the most ...
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1 vote
1 answer
108 views

Diversified Rubik's Cube [duplicate]

I have, cleverly I hope, shuffled my Rubik's Cube for a while. I could not find a position where, for each faces, all colors appear 3 times or less. I always had a or several faces with at least 4 of ...
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  • 5,807
0 votes
1 answer
145 views

Puzzle regarding 100 random objects of different sizes, and choosing one at random one at a time to get largest

Basically, the riddle at hand was used to demonstrate a principle (of which I forgot), and it was asked thusly; I know there are other forms of it, but here goes: You are fishing in a pond of 100 fish ...
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  • 3
2 votes
1 answer
343 views

Largest word tree

I was inspired by this awesome puzzle. Here is an image of a word tree borrowed from there: In a word tree every path from the root to the leaves must form a distinct word. The size of the tree is ...
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0 votes
1 answer
150 views

Number of 6-person events so all groups of 3/10 people have dined together [closed]

Assume 10 people numbered 1-10 have to be invited for dinner events. However, the hotel can accommodate only 6 at a time. Therefore, they will be invited in batches until all groups of 3 people have ...
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  • 103
9 votes
11 answers
1k views

Ten Pills in Ten Countries [closed]

You're about to leave for an international vacation where you'll be visiting ten different foreign countries. To make sure you don't catch any local diseases, your doctor has given you ten pills and ...
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8 votes
3 answers
924 views

Chess lead change

Given the following Chess piece relative values and such that both players cooperate, what is the fastest way such that White first has an advantage of at least +10 piece value then secondly, Blacks ...
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  • 5,807
2 votes
1 answer
101 views

What is the name of this problem about maximizing the accepted offer from consecutive offers?

There was a problem I have read more than 15 years ago. I just remember the problem and the rough outline of the solution, and I would like to know if it has a canonical name to look up the ...
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17 votes
7 answers
1k views

Efficient Mowing at PSE

Your task: Find the most efficient mowing path around the dark green bushes that mows (passes over) all of the grass (light green). For those who cannot view the image above, there are 9 rows of 16, ...
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  • 16.9k
5 votes
2 answers
204 views

Robot painting a $K_5$

A robot starts at a node of a fully connected graph of 5 nodes (shown below). Each turn the robot can move across an edge and paint it in one of two colours - blue for odd turns and red for even turns....
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8 votes
4 answers
872 views

Yet Another Card Payoff Game

From A Practical Guide to Quant Interviews: A casino offers yet another card game with the standard 52 cards (26 red, 26 black). The cards are thoroughly shuffled and the dealer draws cards one by ...
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  • 603
7 votes
1 answer
192 views

13 scales and a ... fake scale

You are given 14(*) scales that look and feel identical. The scales are 2-pan scales. When you put stuff on each pan, the scale indicates whether both sides balance, and if not, which side is ...
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4 votes
0 answers
167 views

Mr. Bean's Ridiculous Way to Work

If you saw Mr. Bean as a kid, or ever, you know how crazy and unbelievable he is. In this scenario, Mr. Bean lives in the bottom left house and works for the school at the left, but he always takes ...
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  • 1,133
9 votes
2 answers
276 views

Largest set of factorials whose product is a perfect cube

Just one number must be removed from the set of integers between 1 and 100 (inclusive) so that the product of the factorials of the remaining numbers is a perfect square. What is the least number of ...
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3 votes
3 answers
301 views

Checkmate N Kings with M Knights Perfectly

We have this Existing Puzzle which has got the valid and invalid cases listed ; the Accepted Answer is along the lines of what is invalid and what is valid. But there was a flaw or fault in the Puzzle ...
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  • 4,688
9 votes
2 answers
938 views

Factorials whose product is a perfect cube

Find two or more different positive integers the product of whose factorials is a perfect cube. How small can the largest of these be? How few can they be?
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14 votes
3 answers
2k views

Checkmate N Kings with M Knights

There are N White Kings on the Chess Board. There are M Black Knights. There is no Black King and no other Piece. Only M+N Squares are occupied. Each White King is attacked by atleast 1 Black Knight, ...
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  • 4,688
19 votes
3 answers
1k views

The universal ticket

I am submitting a very interesting problem from a French mathematical recreation site: http://www.diophante.fr/problemes-par-themes/g-probabilites/g2-combinatoire-denombrements/1434-g248-le-billet-...
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3 votes
2 answers
353 views

The longest path of edges on a 3x3 grid

A robot is placed on some vertex of a 3x3 grid. At each move the robot can take one step (up, down, left or right) along the edge of the grid to the adjacent vertex, but it cannot go outside the grid. ...
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26 votes
6 answers
5k views

Checkmate 30 kings with rooks

Using only kings, and as few rooks as possible, set up a position where 30 of black's kings are checkmated. Checkmate: The king is in check The king has no moves out of check Black only has kings, ...
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3 votes
3 answers
493 views

Multi-colored polyominoes inside a 7x7 grid

Can you place four red trominoes, four green tetrominoes and four blue pentominoes inside an 7x7 grid, such that: No two polyominoes overlap No two polyominoes of the same color touch each other ...
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8 votes
1 answer
189 views

Tetromino in a Pentomino Lair

Inspired by this question: Can you fit twelve pentominoes (not necessarily distinct) and one tetromino inside a 10 x 10 grid such that they do not overlap or touch each other orthogonally (...
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  • 124k
10 votes
3 answers
978 views

Fitting pentominoes inside a 10x10 grid

What is the most number of pentominoes that you can fit inside a 10x10 grid, such that they do not overlap or touch each other orthogonally (horizontally or vertically)? Bonus: what is the most number ...
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4 votes
2 answers
323 views

Four pipes on a 8x8 grid

You are managing the construction of 4 water pipes on a 8x8 grid. The rules are the following: Each section of a pipe uses a whole grid cell. Pipes are composed of multiple sections connected ...
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13 votes
4 answers
1k views

The Game of Golden Squares

On a magic chessboard of infinite size, the squares are either wooden or golden. If 4 or more of its 8 neighbors (a king's move away) are golden, a wooden square becomes golden the next day. Golden ...
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  • 5,145
9 votes
3 answers
541 views

Coloring the squares

You can choose from 4 colors to color every square of the following $10\times 10$ grid. After you finish, I'm going to take a connected block with at most three colors away. Your goal is to minimize ...
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  • 5,145
4 votes
2 answers
255 views

Helpmate part 2

This is the second part of Fastest way to helpmate You have been given the task to checkmate/helpmate black in fastest possible way in following ways: Pawn promoting to only queen Pawn promoting to ...
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  • 1,095
0 votes
0 answers
73 views

Disguising a Rubik's Cube rotation

I'm wondering if there's a way to disguise the total rotation of a Rubik's Cube? For example, if I wanted to rotate the (solved) cube I could obviously just apply a 90 degree yaw rotation. However, I ...
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4 votes
3 answers
275 views

8x8 Grid with no parallels

In the 8x8 grid graph shown below; you can put points to the edge of grid as shown below (blue dots). The example above has 4 points and you construct a line between two points as shown below; so ...
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  • 28.9k
0 votes
2 answers
146 views

Two triangles in a circle

This puzzle is inspired by this great puzzle. You are given a circle. You can draw two non-overlapping triangles of any size and shape inside that circle. What is the highest percentage of the circle ...
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20 votes
4 answers
2k views

Moving around a plane

A small plane went through some heavy turbulence and all its passengers ended up in the wrong seat. Now they need to get back to their assigned seats. The image below shows the map of the plane. The ...
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6 votes
2 answers
204 views

Splitting the integers 1 to 36

Split the integers 1 to 36 into two sets, A and B, such that any number in set A has a common divisor greater than 1 with no more than two other numbers in A, but for every number in B there are at ...
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12 votes
2 answers
547 views

The mower's challenge

Weeds have taken over the roads. If mowed, they don't grow back, but unmowed weeds spread at speed 1 along the road. What's the minimum speed of the mower to get rid of all weeds? Roads are connected ...
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  • 5,145
3 votes
1 answer
315 views

How many distinct pentominos can be placed on a 8×9 board?

Upon proving optimality of an 8-pentomino solution for an 8×8 board, I was curious to see whether there is a 9-pentomino solution for an 8×9 board, namely a way to arrange 9 distinct pentominos within ...
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  • 1,086
14 votes
3 answers
2k views

How many distinct pentominoes are possible to place on an 8 x 8 board?

Rules Place some pentominoes into an 8 x 8 grid. They do not touch each other. They can touch only diagonally (with corner). Pentominoes cannot repeat in the grid. Rotations and reflections of a ...
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  • 141
13 votes
2 answers
455 views

Put three pieces of cake into a round box

You're about to cut three pieces from a large cake to put in a round box of radius 1. If the pieces must be congruent triangles, and cannot overlap, what shape gives you the maximum amount of cake?
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  • 5,145
5 votes
4 answers
1k views

Six positive integers

Find six different numbers (positive integers) such that each of them has a common divisor with precisely three of the other numbers. How small can the largest of the six numbers be? What if $2n$, $n&...
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8 votes
0 answers
484 views

What is the longest Wordle game?

If you play Wordle in hard mode, with unlimited guesses, then you must solve any puzzle eventually. This is true because there are only a finite number of valid five-letter words, and it is not ...
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  • 181
2 votes
2 answers
363 views

Choose the wine

I based this on a problem from a mathematics presentation, adding a small twist. I did not readily find it here. Your friend comes to dinner and you know he loves to drink Beaujolais. You have 'Cote ...
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2 votes
1 answer
154 views

Hitting twice with different choices

This game of two players has public parameters an integer $n\ge2$, and a probability $p$ with $1/n<p\le1$. E.g. $n=4$, $p=1/3$. In the first phase of a game, a player secretly decides $n$ ...
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  • 383
6 votes
2 answers
388 views

Multiplication puzzle with trios of numbers

I created a puzzle that I was curious what its properties are, and how it could be determined it is solvable or not. It consists of 6 rows of 3 of the same numbers, which in each row in order are 1, 2,...
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  • 309
5 votes
1 answer
346 views

The Median Game - For Money

Inspired by this interesting puzzle which was quickly solved. Five friends play a simple game with the following rules: Players play consecutively one after the other. Each player must call out a ...
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  • 7,162
22 votes
3 answers
1k views

Dividing a piece of land

Alice and Bob try to divide a piece of land $D$, shaped in a perfect closed disk of radius 1. Alice moves first to mark some finite (at least one) number of points in $D$. Bob then draws any number of ...
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  • 5,145
7 votes
3 answers
272 views

How many stickers can a Rubik's Cube lose?

I have an old Rubik's Cube, the kind with stickers. The stickers tend to fall off. That got me wondering: How many stickers can you remove from a Rubik's Cube at most, while preserving the property ...
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  • 20.9k
3 votes
2 answers
263 views

Finding the treasure on a square island

Some treasure is hidden underground in a small square-shaped island of area $64 km^2$. You have no idea where the treasure is exactly, and no time to dig the whole island anyway. But, luckily, you do ...
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  • 28.9k
0 votes
0 answers
72 views

A house with 100 lights and 100 switches [duplicate]

There is a house with 100 lights. In the basement there are 100 switches for the lights. Sadly, you have forgotten which switch is connected to which light. Currently they are all on. You go down and ...
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