# 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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### A camel transporting bananas

A somewhat well-known puzzle is described as such: You have a pile of 3,000 bananas. You wish to transport them to a place 1,000 miles away on the back of a camel; however, the camel can only carry ...
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### Dinosaur egg drop

Archeologists discover two dinosaur eggs, and you are given the chance to test the durability of these eggs (bad move on their part). Suppose that these eggs will absorb a specific amount of force ...
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### Knights and jokers

There are $N$ men. $K$ of them are knights, $M$ of them are jokers. $N$ is known, $K$ and $M$ are unknown. You know that: $K + M = N$, $K \gt M$, $M \ge 1$, $N$ is odd. Knights always tell the truth, ...
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### 1 Fake Coin among N Amount of coins

You are given $N$ coins which consists of only $1$ fake coin. You also have a sensitive old-fashioned Pan Balance Scale. You are asked to find the fake coin in totally 5 times weighing on the Pan ...
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### Find a straight tunnel

There is a circular area with radius 1 km. And there is a tunnel, which is just under the surface, but invisible - unless you dig. It is known that the tunnel goes under the area (at least touches it ...
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### Peaceful Encampments

This math puzzle is due to Donald Knuth (as far as I know; maybe he got it from someone else) circa 2014. Consider a plain represented by the unit square. On this plain we want to “peacefully encamp” ...
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### Strategy to beat the casino over the long haul

$A$ and $B$ are attempting to beat the casino again, and the rules are the same: Each round, $A$, $B$, and the casino simultaneously decide to show a $0$ or a $1$. If all three numbers match, $A$ ...
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### Square Spin #1: Sun and Moon

Defining Square Spin Lets play a game! The aim of this game is to transform one image into another using the least amount of moves possible. Here is what a move consists of: ...
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### My Mother's Dish Collection

From every trip she makes, my mother brings as a souvenir a well-decorated dish to hang in a wall. She now has a collection of 12 dishes, all disks, of radii 1, 2, 3, ..., 12 inches respectively. ...
329 views

### Barrel - Part 3

An entry in Fortnightly Topic Challenge #35: Restricted Title 1. Title based on this xkcd. This is a continuation of Barrel - Part 1 and Barrel - Part 2, but this puzzle is still self-contained. ...
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### Odd hours with two timers

Two motorized 24-hour light timers are daisy chained between a power outlet and a light bulb. For these timers, devise schedules and choose initial times that produce the following repeated 9-hour ...
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### Halve time with two timers

Two motorized 24-hour light timers are daisy chained between a power outlet and a light bulb. For these timers, devise schedules and choose initial times that produce the following repeated lighting ...
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### The case of the 4 campers and the rickety bridge! [duplicate]

Four people need to cross a rickety rope bridge to get back to their camp at night. Unfortunately, they only have one flashlight and it only has enough battery left for seventeen minutes. The bridge ...
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### Prove that π > 3

It seems that once upon a time some politicians tried to pass a law fixing the value of π to be exactly 3. The idea being to "make math simpler so that our children can get better at math". ...
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### Eight coins for the fair king

You are responsible for creating new types of coins for the court. King respects the forgetful: he wants you to create 8 coins of different value, no more. King respects the feeble: he wants that any ...
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### 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 - ...
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### Sudoku net that is always solvable

A sudoku net is a 9x9 grid where every cell is either (blank) or has a cross (block). A random solved sudoku is chosen. The net is placed on top of the sudoku (in any of the 4 ways). We can only see ...
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### Worst case scenario in 2048

The game 2048 can be thought as a two player game, where one player shifts the table around to get to the 2048 tile, while the other player tries to hinder them by putting down 2s and 4s on the table. ...
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### Trapped in my Cellar

I have taken Doorknob hostage in my cellar. He is "perfectly" trapped - solid walls, solid floor, solid roof, no windows, etc. The only way out is a steel door and the only way to unlock the door is ...
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### 30 fake coins out of 99 coins

You are given 99 coins which consists of 30 fake ones. You also have a digital balance scale with perfect precision that shows how much difference between weighs you put on. For example, if you put 10 ...
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### 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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### Fastest way to collect an arbitrary army

I am looking for solution of this puzzle: There is a kingdom with a square shape with sides of length 1. The castle is located at the center of the square. At the castle the king lives under the ...
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### Neighboring sums 4x4 game

Here is an interesting game. You start with an empty 4x4 grid. At each turn you can choose an empty cell and place a value in it. The placed value is given by the following rules: If the chosen cell ...
974 views

### Minimize the longest King chain on a 5x5 binary board

Given a grid filled with numbers, let's define a King chain to be a path on the grid such that the path can be traversed with chess King's moves (moving to one of 8 adjacent cells at a time), the ...
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### Largest and smallest hexadecagon with sides 1, 2, 3, ...,16

Of all hexadecagons lying in the cartesian plane, all of whose vertices are lattice points, and whose sides are of length $1,2,3,\dots,16$ in some order, which two have the largest and smallest area? ...
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### Day and night of the two timers

Make a non - 24 - hour day /  night cycle       &...
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### 99 chips into more chips

You are at a casino in Vegas and you have earned 99 chips by playing poker! While you are checking out a slot machine, someone comes to you and congratulates you that you have a chance to make more ...
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### Third timer's a charm

Three motorized 24-hour light timers are arranged between a power outlet and a light bulb, with timer A parallel to a daisy chain of timers B and  C. For these timers, devise schedules that ...
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### What is the minimum number of steps to solve this cloning particles on chessboard puzzle?

I'm creating a puzzle based on the idea from this numberphile video: pebbling a chessboard. You can see a working (yet unfinished) version here. I would like to give the user a feedback of how well ...
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### Square Spin #2: Climb the Mountain!

Square Spin History: #1>#2 New Rules This puzzle introduces two new square types: Unmovable squares (Un) Replaceable squares (Re) Plus the concept of ambiguity! Ambiguity of squares Certain ...
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### Dinosaur Egg Drop v2

This question is a little different and kinda follow-up version of Dinosaur Egg Drop 1- You have 100 story building and 6 Apatosaurus dinosaur eggs. 2- You have 81 story building and 3 ...
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### Knights covering a 10x10 chess board

What is the minimum number of knights you need to place on a 10x10 chess board, such that every empty cell is attacked by at least one knight? Good luck!
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### Four queens on public transport

The enlightened but, sadly, fictional country of Switzerland has it all: A strong democracy that resists centralism, a reliable public transport network and its own version of chess. Swiss chess ...
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### 8x8 grid with no unpainted pentominoes

What is the smallest number of cells you need to paint in an 8x8 grid, such that it contains no unpainted pentominoes? Can you find multiple solutions? Note that a pentomino is a set of 5 adjacent ...
582 views

### Infinitely simple polygon solipsism

Solipsism — The self is all that can be known to exist. Above is a simple polygonal region divided into infinitely many different-sized copies of itself.   Each copy is √2 = 1.414... times as large ...
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### Two genies and their kind of chess

While playing chess Parcly and Tori Taxel, best friends and genies, got bored and transformed all the pieces into pawns to make pretty patterns. They found this 22-pawn arrangement where every 3×3 ...
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### How can 3 queens control the white squares?

It is well known that there is no way of arranging 4 queens on a checker board in such a way that every square is occupied or threatened. Now consider a slight variation where we only need to cover ...
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### Two spies throwing stones into a river

There is a puzzle about two spies: Two spies must pass each other two secret numbers (one number per spy), unnoticed by their enemies. They have agreed on a method for doing this using only 26 ...
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### 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 ...
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### A COVID-19 puzzle: How large a class do you need to fit 30 pupils?

Some countries are proposing to reopen high schools soon. To ensure safety, they want to make sure that all pupils in a class are at least 2 m apart. To help them find the smallest room that can ...
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### 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 ...
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### From Puzzling to StackExchange

This is a variation of a game I used to play with my friends in my childhood. You are given a starting word, and using the following operations, you have to reach the target word. The solution that ...
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### Can you irrigate your lawn with 23 sprinklers?

You have a perfectly circular lawn with radius exactly 4 metres. Lately the grass has been turning yellow and quite rough, so you go to Stiv's Diabolical Instruments and describe your problem. "...
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### How much water do you need to cross the desert?

This question is inspired by Terry Pratchett's "Small Gods," in which an army crosses a vast desert by making multiple trips and caching water along the way. 1. Provide an answer. 2. I doubt I'm the ...
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### PSE Advent Calendar 2023 (Day 8): A Quilt for Santa

This puzzle is part of the Puzzling Stack Exchange Advent Calendar 2023. The accepted answer to this question will be awarded a bounty worth 50 reputation.< Previous Door Next Door > Mrs. Claus ...
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### Universal dissection

Alice has a squared paper 8 by 8. She cuts out one 1x1 square from it, at row N, column M. Bob cuts the rest of the paper into pieces. Once he is done, Alice asks Bob to put the pieces together in a ...
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### Largest and smallest dodecagon with sides $1, 2, 3, \dots,12$

Of all dodecagons laying in the cartesian plane, all of whose vertices are lattice points, and whose sides are of length $1, 2, 3, \dots,$ and $12$ in some order, which two have the largest and ...
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### 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 ...
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### Connect four towers by roads

Four guard towers are situated in a square formation of side length 1km. A general wants to build roads to connect the towers so that one can walk from any tower to any tower along the roads, possibly ...
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