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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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 ...
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4 votes
3 answers
237 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 ...
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0 votes
2 answers
137 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 ...
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19 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 ...
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4 votes
2 answers
188 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 ...
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12 votes
2 answers
511 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 ...
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2 votes
1 answer
292 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 ...
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  • 1,044
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 ...
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  • 141
13 votes
2 answers
438 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?
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4 votes
4 answers
962 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&...
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8 votes
0 answers
294 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 ...
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  • 181
2 votes
2 answers
339 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 ...
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2 votes
1 answer
145 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 $n$ ...
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  • 383
6 votes
2 answers
386 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,...
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  • 309
5 votes
1 answer
285 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 ...
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22 votes
3 answers
1k 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 ...
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  • 3,979
7 votes
3 answers
259 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 ...
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3 votes
2 answers
255 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 ...
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0 votes
0 answers
71 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 ...
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4 votes
2 answers
770 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, ...
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1 vote
2 answers
243 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 ...
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6 votes
2 answers
492 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 ...
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2 votes
0 answers
110 views

Maximize my flags - 2x2 version

Because Maximize my flags was not solved to optimality by the community, perhaps because the coding required was too harsh, I present you Maximize my four flags. The rules are exactly the same as in ...
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  • 5,497
6 votes
4 answers
798 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 ...
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  • 5,497
2 votes
1 answer
226 views

4x4 words 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 ...
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2 votes
1 answer
184 views

Progressive Daedalian Opus

The 1990 Game Boy game Daedalian Opus is essentially a series of 36 pentomino puzzles. In the first level, however, you only have three pentominos; the rest are introduced in the levels after that, ...
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4 votes
1 answer
242 views

Move and Remove

From the initial position Black makes a regular move to an unoccupied square, and removes any piece from the board. Then repeats, alternately making a move and a removal. The objective is for Black to ...
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  • 6,596
21 votes
2 answers
2k views

A Tetris puzzle made with love

I love designing perfect clear puzzles for my dear friend who loves Tetris. Here's a lovely puzzle I crafted today. Original Puzzle (Warm-Up) Starting with this field, place this exact sequence of ...
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4 votes
2 answers
291 views

Merging knights and blocking rooks

A chess grid is filled with knights and rooks as shown in the following diagram. Each turn you can issue a move in one of 8 directions available to a chess knight. This will move all the knights in ...
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8 votes
2 answers
570 views

Merging knights

A standard 8x8 chess grid is filled with knights. Each turn you can issue a move in one of 8 directions available to a chess knight. This will move all the knights in that direction. If a knight would ...
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5 votes
2 answers
309 views

Board game: Risk (two players)

Consider a game of two players: Player A and Player B. Each of them is assigned with the same number of soldiers. There is a battlefield (like a board game) with 7 tiles numbered 1 through 7 (a single ...
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  • 51
5 votes
2 answers
520 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 ...
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11 votes
3 answers
2k views

Longest infinite loop of 5 states

This is based on a question I posed in The Nineteenth Byte: What group of 5 states have the longest total name, under the constraint that you must be able to travel from one state to another in the ...
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7 votes
2 answers
416 views

How many squares can a limp queen move to?

Consider a large chessboard. A limp rook is a chess piece that moves one step orthogonally, but it turns $90$ degrees after every move. The limp rook makes some moves, not crossing over its own path, ...
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  • 333
2 votes
2 answers
212 views

Quickest chess stalemate with Queens exchange

Continuing my previous puzzle. If both players cooperate, what is the quickest stalemate in chess that includes a Queens exchange, in a legal game?
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  • 5,497
3 votes
2 answers
233 views

Quickest mate with Queens exchange

If both players cooperate, what is the quickest mate in chess that includes a Queens exchange, in a legal game?
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5 votes
1 answer
195 views

Insert Plus Signs and Add

If you take any integer (in base 10) and insert plus signs, "+", in between its digits (as few or as many as you like), and carry out the indicated sum, you will end up with a smaller number ...
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-1 votes
1 answer
133 views

A Complex Dash To Stalemate

For today's contribution to PSE, I present a sliding block puzzle! I have a series-helpstalemate in 26 for you all with a relevant question attached. The illegal position is intentional. Objective #1: ...
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7 votes
0 answers
140 views

What's the best path through a garden that maximises the ground you can reach but minimises the steps taken?

I'm messing around with this toy problem I came up with. Say you have a yard that's 16 x 16 feet. Each square foot of the grid can either be a paving square for the path or a soil square for plants. A ...
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15 votes
2 answers
1k views

An overcomplicated Boat Puzzle

Based on Xkcd's Boat Puzzle At the riverbank, a succession of people have given you a large lump of objects to transport across the river. These are: 101 cabbages 2 goats, one of which eats wolves 4 ...
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12 votes
2 answers
1k views

Laser and mirrors on a 4x4 grid

You are given an empty 4x4 grid. You can place some diagonal mirrors into the cells of the grid. You then fire a laser from some location outside of the grid. The laser travels in a straight line. ...
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0 votes
2 answers
142 views

How can we allocate when we have 150 open slots every day (5 days a week) for those 200 arrivals every day

My question is to solve a very basic problem related to the allocation of slots. Say there are 20 teams with 10 persons in each team. I have 150 open slots every day (5 days a week) for those 20 teams ...
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12 votes
2 answers
680 views

Introducing S-sequences: which is the shortest to contain all integers 1 to 20?

Consider a sequence (finite or infinite) of different positive integers, such as the following, in which the first term is 1, and thereafter the nth term is either the previous term plus n, minus n, ...
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6 votes
1 answer
190 views

42 lines on a chessboard with associated numbers

The 64 squares of a chessboard can be associated with 42 lines as follows: the 8 rows the 8 columns 13 diagonals from north-west to south-east 13 diagonals from north-east to south-west Those ...
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  • 11.5k
12 votes
1 answer
614 views

All distances different on a chess board

Here is a simple formulation for, I believe, a quite difficult problem. I have played with it, I don't have the answer yet. The question: How many pawns can you put on a standard 8x8 chess board in ...
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  • 19.3k
2 votes
2 answers
372 views

Minimum number of questions?

This is an interesting puzzle which I've been racking my brain to solve for some time. I've found different variations of this puzzle but it does not seem like this one has been asked before. Here it ...
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1 vote
2 answers
121 views

Alice and Bob play neighboring sums game version 2

Alice and Bob are playing the neighboring game which is originally a single person game with the aim of getting the highest points at the end. You start with an empty 4x4 grid. At each turn, you can ...
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8 votes
2 answers
3k views

Social distancing in a 5x5 room [duplicate]

I have booked a square meeting room that is 5 by 5 meters. Our Covid-19 policy says that each person must be at least 1.5 meters away from any other person. What is the highest number of people that ...
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3 votes
1 answer
183 views

Some blissfully ignorant math?

I was playing around with the code for my game the other day in an effort to create some unique effects. One thing I created was what I called an "ignorant assignment" in which I applied ...
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5 votes
2 answers
542 views

A piece of paper repeatedly cut into 8 or 12 pieces

You are given a piece of paper. It will be cut into 8 or 12 pieces. Each of those new pieces can be cut again into 8 or 12 pieces or left uncut. This process is (theoretically) repeated as often as ...
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