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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16 votes
4 answers
940 views

Make the three products equal and as large as possible

Beginner puzzle This puzzle is intended to be suitable for people who are new to puzzle solving. Clarification: Both experienced solvers and new solvers are welcome to post solutions to this puzzle. ...
6 votes
2 answers
745 views

Make a square table top with the minimal needed amount of straight cuts

inspired by : Make a square table top with six (or fewer) pieces A carpenter has three pieces of beautiful wood, measuring 12 inches, 15 inches, and 16 inches square, respectively. They want to use ...
10 votes
2 answers
444 views

Make a square table top with six (or fewer) pieces

A man had three pieces of beautiful wood, measuring 12 inches, 15 inches, and 16 inches square respectively. He wanted to cut these into the fewest pieces possible that would fit together and form a ...
4 votes
1 answer
227 views

Dishonest dungeon staff

You are faced with the difficult task to set up a dungeon for adventurers. However you made a deal with the guild: any adventurer brave enough to beat the dungeon and reach the treasure room will be ...
6 votes
5 answers
635 views

Circles crossing every cell of an NxN grid

What is the least number of circles you need to draw, such that every cell of an NxN grid is crossed? A circle crosses a grid cell if one of the points on its circumference lies completely inside the ...
44 votes
8 answers
7k views

How many Chess Pieces are needed to control every square on the board? No Piece Restriction

You may use as many pieces as you like. Pieces do not control the square they occupy. Accepted Answer goes to the person that has the least score. Piece cost: Pawn - 1 Knight - 3 Bishop - 3 Rook - ...
15 votes
5 answers
5k views

How many Chess Pieces are needed to control every square on the board?

You may only use pieces in the original set, and all your pieces must be the same color. The king is not allowed. Pawns do not get promoted. Pieces do not control the square they occupy. Accepted ...
18 votes
3 answers
1k views

Winning chance in coins game with fixing

A player plays the following solitaire game. The game consists of as many rounds as are needed to produce a result. They have 20 fair coins, which may in any round be live or fixed; each coin starts ...
13 votes
3 answers
2k views

Circles crossing every cell of an 8x8 grid

What is the least number of circles you need to draw, such that every cell of an 8x8 grid is crossed? A circle crosses a grid cell if one of the points on its circumference lies completely inside the ...
5 votes
1 answer
293 views

Number operation problem - find minimum number of operations

On the blackboard, there are nine numbers from 1 to 9. In each operation, two of the numbers are chosen, erased, and replaced with their sum and difference. If two identical numbers appear on the ...
6 votes
4 answers
864 views

Flip counters in a grid so that they alternate in color

Beginner puzzle This puzzle is intended to be suitable for people who are new to puzzle solving. Clarification: Both experienced solvers and new solvers are welcome to post solutions to this puzzle. ...
3 votes
2 answers
712 views

Connecting 3 circular power timers to lights so that they turn for one day on Christmas

My friend gave a question to me. I am not sure if it is an original puzzle, but he has 2 lines of Christmas lights and 3 circular timers with 96 switches apiece. Each switch turns on power for 15 ...
22 votes
7 answers
3k views

Two prisoners and twenty marbles

Two prisoners are planning their escape. Their cells are locked with a padlock that must be opened with a 5-digit code (numbers 00000 to 99999). Each prisoner knows the code of the other prisoner's ...
9 votes
1 answer
448 views

Tile a square as small as possible using two different sizes of square tiles

There are two types of square tiles. One type has a side length of 1 cm and the other has a side length of 2 cm. What is the smallest square that can be made with equal numbers of each type of tile? ...
4 votes
1 answer
437 views

75 integers are squared or cubed: minimum distinct results? [closed]

75 different integer numbers are written on a blackboard. Each is erased and replaced with either its square or its cube, the operation being random for each. What is the minimum quantity of different ...
21 votes
5 answers
2k views

6 Water Glasses Upside Down

There are 6 water glasses as shown in the picture below: You need to turn all of them upside down with the rules below: You have to choose any 5 of them at every turn. Chosen ones need to be turned ...
10 votes
5 answers
2k views

Make 38 using the least possible digits 8

This is a follow up question to this one, which was: Make the number 1998 using the minimum amount of digits 8. Your allowed operations are +, -, *, /, ^, % (percent). You need not use only integers ...
5 votes
6 answers
1k views

Minimum number of questions for real treasure

There are 2000 treasures, among which only one is real and the others are fake. There are also 2000 knights, among whom one always tells the truth while the others always lie. You can ask the knights ...
4 votes
1 answer
179 views

Approximate ln(2) out of small numbers

Given numbers 1,2,...n, the goal of this puzzle is to make a number as close to the mathematical constant $\ln(2) ≈ 0.69314718055$ as possible. Rules You can only use the four mathematical ...
13 votes
7 answers
2k views

Fewest number of prisoners needed

You are an assistant to the king, who has scheduled a party tomorrow. There are 1000 wine bottles in the king's possession, and unfortunately, 3 of them are poisonous. Consuming from any of these ...
24 votes
2 answers
1k views

Hexominos from pentominos, heptominos from hexominos

All twelve pentominoes can be obtained by attaching a single unit square (edge to edge) to one of the squares that make up one (or more) of the following four tetrominoes: a) What is the least number ...
5 votes
5 answers
461 views

cheapskate jigsaw inc

Jigsaw puzzles are awesome and making them is an easy way to get rich. So I bought a cutting machine to make puzzles. Unfortunately I could only afford the cheapest machine available. All pieces are ...
13 votes
5 answers
2k views

General attacking chessboard squares

This is a general version of this beautiful puzzle. Place any number of standard chess pieces on a 8x8 chessboard, such that there is at least 1 empty square attacked by exactly 1 piece, at least 1 ...
11 votes
3 answers
600 views

New record for minimally clued 5x5 Hidato

CrSb0001 has been studying the minimum number of clues to force a unique solution in Hidato for variously sized boards. They came up with a 5x5 Hidato with 5 clues but it was found that it has ...
13 votes
3 answers
1k views

What is the minimum amount of numbers needed to create a 4x4 Hidato with a unique solution?

Goal of Hidato: Fill in a grid with a series of consecutive numbers that connect each other horizontally, vertically, or diagonally. So I have been working on figuring out the minimum number of ...
1 vote
1 answer
192 views

Maximum filled days

I have two types of items, $i_1$ and $i_2$. $i_1$ items can be used at most $50$ times and $i_2$ items can be used at most $120$ times. I have $7000$ items $i_1$ and $800$ items $i_2$. Each item $i\in ...
38 votes
4 answers
3k views

Pythagorean pentagons

To follow up on the theme of so called "pythagorean" dissections, here is one more for you to chew on. I hope you don't get bored. The pentagons above have sides respectively 3, 4 and 5. ...
1 vote
1 answer
175 views

How long will it take for the horse to be captured?

Source: https://open.kattis.com/problems/ambush Problem: When Farmer Oskar doesn’t watch his cows closely enough, they tend to wander off into the forest to hunt for horse spies. To catch an enemy ...
0 votes
1 answer
2k views

The swap puzzle

The swap puzzle Place small coins heads up on the squares marked H and tails up on the squares marked T. Swap the positions of the Heads for the Tails in as few moves as possible. There are two ways ...
16 votes
4 answers
1k views

Covering a 15x15 grid with rectangles

You are given a 15x15 grid and asked to cover it with rectangles whose dimensions are a power of 2. For example you can use rectangles 8x1 and 4x4, but not 2x3. The rectangles must cover every cell of ...
19 votes
1 answer
1k views

An Amazing Configuration

Ed Pegg found in December 2019 this amazing configuration consisting of 22 points in 28 lines of 4. On those points place 22 different positive integers such that the sum of any of the four points in ...
9 votes
3 answers
887 views

Minimum number of flips needed to fully set a binary string

Source: My school teacher, it was just given as a problem statement. Problem: You are given a binary sequence $S$ of up to $10^6$ bits. You wish to set all the bits in the string to $1$, but the catch ...
8 votes
5 answers
821 views

Buying weight plates for a home gym, what is the fewest number I need

I'm buying a home gym set-up for my son (really!) and I want to know the fewest number of weight plates I need to buy to be able to go up in 5kg increments to 170kg. The bar weighs 20kg and the ...
12 votes
1 answer
2k views

Add rules to Minesweeper until you can start with nothing

Think of classic Minesweeper game, with the following list of restrictions on the placement of mines: C (Connected): All mines are connected via 8-way neighborhood (orthogonal and diagonal). Q (Quad):...
10 votes
3 answers
751 views

Optimal refueling strategy

On the way home from work, I want to refuel my car. There are 5 petrol stations along the way that all have a different price for petrol. I know that the prices will be in the range [A, B], chosen ...
5 votes
1 answer
948 views

Anna and Boris play the Red Blue game

Anna and Boris play a game on a 9x9 chessboard. Anna goes first and turns alternate thereafter. In each move, Anna puts a red counter on a vacant square while Boris puts a blue counter on a vacant ...
4 votes
3 answers
1k views

Two 10-letter words consisting of 20 unique letters?

I'm looking for two words, 10 letters each, 20 unique letters. Below are my by-hand attempts. https://www.wordunscrambler.net/word-list/10-letter-words-with-no-repeating-letters Find words with 3 or ...
13 votes
2 answers
3k views

What's the most distant chess position?

It's well known that the combinatorial explosion means that there are many, many, possible chess games. And yet most games are decided within 100 moves, and Wikipedia says that the longest tournament ...
9 votes
1 answer
311 views

Designing a four-pan scale

You are an ancient merchant, and you need to weigh out many different items of many different weights. To do so, you'll design a scale to weigh objects. Unlike a standard scale, your scale will have ...
18 votes
5 answers
1k views

Maximize the number of paths

You have exactly 990 edges. Assemble them into a simple undirected graph with two distinguished vertices A and B, such that the number of different simple paths from A to B is as large as you can make ...
0 votes
1 answer
382 views

Rooks covering Dark Squares on a Chessboard

How many rooks are required such that all dark squares on the chessboard are covered by at least one rook.
26 votes
7 answers
2k views

Approximate pi out of small numbers

Given the numbers 1, 2, ..., n the puzzle is to try to make a number as close as possible to pi using only the four mathematical operations +, -, *, / and parentheses (brackets). You don't have to use ...
23 votes
7 answers
3k views

You are at a crossroads with five roads leading away from it

To allow new users to solve this puzzle and earn reputation points, I encourage all users whose reputation is 200 or more to not post an answer until 48 hours after this question is posted. Thank you! ...
28 votes
7 answers
5k views

A Special Parking Lot

There is a special kind of parking lot that stores special 1x1 cars that only can move to adjacent squares not including diagonals if they are empty (provided no car occupies the square). Every car ...
6 votes
2 answers
190 views

River crossing cats and dogs

A farmer and his $n$ cats and $m$ dogs are on one side of a river. There is a boat that can carry the farmer and at most $k$ animals at once. The boat doesn't move without a person operating it. The ...
11 votes
1 answer
565 views

List Roman numerals to minimize total distance

This is based on a discussion with @mousetail. List the numbers from 1-100 in Roman numerals in the order that minimizes the total sum of the Levenshtein distances between consecutive numbers.
15 votes
2 answers
2k views

Maximum number of pieces on a board

Suppose we have a 3x3 board. On this board, several squares are empty and several squares are filled with pieces. A legal move on the board is made using the same idea as taking in checkers: one piece ...
3 votes
2 answers
359 views

Approximate e out of small numbers

Given the numbers 1, 2, ..., n the puzzle is to try to make a number as close as possible to the mathematical constant $e$ using only the four mathematical operations +, -, *, / and parentheses (...
12 votes
2 answers
638 views

Arithmetic-progression-free sequences

This question Sudokus with no 3-term arithmetic progressions within made me wonder about the following question. What is the longest sequence composed of the integers 1..N (or 0..N-1 if you have a ...
7 votes
8 answers
6k views

Divide 3 cakes into 4 equal parts?

You have three cakes of diameter 20cm, 16cm and 12cm respectively as shown in the figure. They all have the same height. Now your task is to divide these 3 cakes into 4 equal parts and you are ...

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