# 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
2k 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 ...
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 ...
• 6,033
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 ...
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 ...
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 ...
• 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 ...
• 695
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 ...
• 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 ...
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, ...
• 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....
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 ...
• 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 ...
• 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 ...
• 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 ...
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 ...
• 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?
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, ...
• 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-...
• 1,484
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. ...
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, ...
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 ...
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 (...
• 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 ...
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 ...
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 ...
• 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 ...
• 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 ...
• 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 ...
• 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 ...
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 ...
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 ...
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 ...
• 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 ...
• 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 ...
• 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?
• 6,498
5 votes
4 answers
1k views

• 10.6k
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 ...
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 ...
• 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 ...
• 6,033