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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12 votes
4 answers
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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 ...
Dmitry Kamenetsky's user avatar
2 votes
1 answer
130 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 ...
JKHA's user avatar
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0 votes
1 answer
173 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 ...
Qwert's user avatar
  • 3
3 votes
1 answer
419 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 ...
Dmitry Kamenetsky's user avatar
0 votes
1 answer
168 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 ...
2FaceMan's user avatar
  • 103
10 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 ...
Admiral Jota's user avatar
8 votes
3 answers
962 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 ...
JKHA's user avatar
  • 6,033
2 votes
1 answer
133 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 ...
papparafurifuri's user avatar
18 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, ...
JLee's user avatar
  • 18.2k
5 votes
2 answers
247 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....
Dmitry Kamenetsky's user avatar
9 votes
4 answers
3k 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 ...
Charlie's user avatar
  • 635
8 votes
1 answer
233 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 ...
Florian F's user avatar
  • 29.8k
4 votes
0 answers
200 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 ...
Varun W.'s user avatar
  • 1,313
9 votes
2 answers
551 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 ...
Bernardo Recamán Santos's user avatar
3 votes
3 answers
319 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 ...
Prem's user avatar
  • 5,144
9 votes
2 answers
1k 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?
Bernardo Recamán Santos's user avatar
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, ...
Prem's user avatar
  • 5,144
20 votes
3 answers
2k 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-...
franck vivien's user avatar
3 votes
2 answers
2k 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. ...
Dmitry Kamenetsky's user avatar
27 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, ...
SE - stop firing the good guys's user avatar
4 votes
3 answers
549 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 ...
Dmitry Kamenetsky's user avatar
8 votes
1 answer
220 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 (...
hexomino's user avatar
  • 136k
11 votes
3 answers
1k 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 ...
Dmitry Kamenetsky's user avatar
4 votes
2 answers
387 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 ...
Dmitry Kamenetsky's user avatar
13 votes
4 answers
2k 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 ...
Eric's user avatar
  • 6,498
9 votes
3 answers
583 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 ...
Eric's user avatar
  • 6,498
4 votes
2 answers
292 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 ...
I'm Nobody's user avatar
  • 1,334
4 votes
3 answers
448 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 ...
Oray's user avatar
  • 30.3k
1 vote
2 answers
242 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 ...
Dmitry Kamenetsky's user avatar
21 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 ...
Dmitry Kamenetsky's user avatar
6 votes
2 answers
766 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 ...
Bernardo Recamán Santos's user avatar
14 votes
4 answers
1k 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 ...
Eric's user avatar
  • 6,498
3 votes
1 answer
354 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 ...
user21820's user avatar
  • 1,137
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 ...
Ignac's user avatar
  • 141
13 votes
2 answers
515 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?
Eric's user avatar
  • 6,498
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&...
Bernardo Recamán Santos's user avatar
9 votes
0 answers
1k 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 ...
Dave R's user avatar
  • 191
2 votes
2 answers
404 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 ...
FirstName LastName's user avatar
2 votes
1 answer
172 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 $n$ ...
fgrieu's user avatar
  • 383
6 votes
2 answers
412 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,...
TMK's user avatar
  • 319
5 votes
2 answers
683 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 ...
SQLnoob's user avatar
  • 8,540
22 votes
3 answers
2k 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 ...
Eric's user avatar
  • 6,498
7 votes
3 answers
394 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 ...
Florian F's user avatar
  • 29.8k
-1 votes
1 answer
219 views

How to arrange the colored cells in game grid?

Puzzle: In a game grid some cells are missing. Each line has only one colored cell with a label (a number greater than zero). This is an example grid and the number of columns/rows can be less than ...
Nick's user avatar
  • 1,695
3 votes
2 answers
290 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 ...
Oray's user avatar
  • 30.3k
0 votes
0 answers
75 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 ...
Technical Gaming's user avatar
4 votes
2 answers
828 views

Tic-Tac-Collatz

Have you ever heard of the Collatz conjecture? Just in case you haven't, I'll summarize it for you! Take any positive integer $n$, if it is even then simply divide it by $2$; however, if it is odd, ...
Hazel へいぜる's user avatar
1 vote
2 answers
366 views

4x4 grid with the shortest longest path

This is an extension of this beautiful puzzle. This time your task is to find the hardest 4x4 grid. In particular, find a 4x4 grid containing every number from 0 to 9 at least once, such that the ...
Dmitry Kamenetsky's user avatar
6 votes
2 answers
523 views

Find out the longest path being alive [closed]

Start from any integer. Move horizontally or vertically (not diagonally), and if you come across the same integer more than once you will die. Moving diagonally is not allowed. What is the longest ...
Tips's user avatar
  • 163
3 votes
1 answer
271 views

Maximize my flags - 2x2 version

As Maximize my flags was not solved to optimality by the community (perhaps because the coding required was too harsh), I present to you Maximize my four flags. The rules are exactly the same as in ...
JKHA's user avatar
  • 6,033

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