# 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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### Cross-road optimization - what is the proper way to solve this type of puzzle?

This puzzle has 3 levels of increasing complexity. Each "level" is separate and complete, so feel free to post partial solutions to the individual levels only. I'm most interested in the ...
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 ...
1 vote
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 ...
77 views

### 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-...
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 ...
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 ...
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 ...
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 ...
1 vote
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 ...
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 ...
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 ...
312 views

### 4x4 word grid optimization

Given that each letter in the English alphabet has a position: $$a = 1, b = 2, ..., z = 26$$ Can you place 16 different letters such that: Each row, column and diagonal forms a 4-letter valid English ...
924 views

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 ...
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, ...
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 ...
292 views

### Quickly, From B To H, Sextupled Pawns

A couple of years ago, a question was posed about the minimum number of moves to obtain sextupled pawns on the edge of a chess board. The accepted, and proven optimal, answer found 31.0 plies to be ...
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....
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 ...
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 ...
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 ...
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 ...
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 ...
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?
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, ...
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-...
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. ...
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, ...
611 views

### A Complicated Exercise In Addition

Another day, another walk down to the cafe. I was waiting in line for my coffee, wondering what the barista could do this time to make my name look whack. But as I waddled in line, a curious site ...
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 ...
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 ...
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 (...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
810 views

### Paris and Wife Matchstick

Here are two matches dates that I hold with love in my heart: The current sum is 1970 + 1997 = 3967. You must requisition at most 10 matches so that the sum is "as big as possible". We will ...
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 ...
2k views

### Can this be solved without brute force?

The picture above shows a map of 9 train stations. These stations are connected by a single track running from station 1 to 9 as represented by the solid line (the trains runs both ways from 1->9 and ...
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 ...
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 ...
3k views

### Chess solitaire: The King's longest walk

Challenge: maximize the number of moves white needs for its king to reach a square of your choosing, adhering to the following rules: Black does not move. Like normal: The king may not be in a ...
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 ...
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 ...
2k views

### Minimum cells to fill grid without consecutive neighbours

Imagine you have a m x n grid which is initially colored white. you can fill in a cell with black color if and only if there are no immediately neighboring black cells (no black cells to the left/...
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&...
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?