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
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 ...
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6 votes
4 answers
2k views

Fair d5 with as few faces as possible

Challenge: design a polyhedral die that will always give one of five outcomes, each with equal probability. While achieving that, minimize the total number of faces of the die. Think of the die as ...
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12 votes
3 answers
1k views

Mixing liquids in bottles

You have 3 identical bottles with a capacity of 4 litres: The first bottle contains 2 litres of liquid A The second bottle contains 3 litres of liquid B The third bottle contains 4 litres of liquid C ...
16 votes
3 answers
1k views

Most polyominoes in an 8x8 grid

What is the most number of distinct free polyominoes you can form by painting an 8x8 grid in two colours? Here a polyomino is a set of orthogonally adjacent cells of the same colour, so polyominoes of ...
4 votes
2 answers
299 views

Boxeslayers to the rescue

This is about layering boxes, not about slaying them. We have 1,830 2×5 boxes to stack safely as 10 alternating contiguous layer patterns of 183 boxes each. Layers have identical silhouettes that fit ...
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30 votes
5 answers
3k views

A Queen and her Pawns

Place a queen and as many pawns as possible on a chessboard so that the queen has just one way of capturing all the pawns in precisely as many moves as there are pawns. Pawns do not move and do not ...
3 votes
1 answer
180 views

Smallest 3x3 Magic Square of different square sums

Consider the follow magic square highlighted in yellow. The sum of its rows and columns are in green and the sum of the diagonals in red. All of its sums are a square number with the sum of the whole ...
  • 529
4 votes
1 answer
231 views

Farthest From Backgammon

This puzzle is part of the Monthly Topic Challenge #7: Board games. The famous Backgammon game can be seen as being played on a number axis with one player on the non-negative side (from 0 to 25) and ...
7 votes
2 answers
381 views

Smallest square that can pack thin digits

If we draw the digits 0 to 9, segmented into squares, across a rectangle of 2x5 (except the 1) they use up 81 total squares. Is it possible to pack them all into a 9x9 grid. What is the smallest n by ...
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3 votes
2 answers
211 views

From white to black

This puzzle is part of the Monthly Topic Challenge #7: Board games. Place a knight on each of the white squares on the 8x8 chessboard. That would make a total of 32 knights. Is it possible to use a ...
2 votes
2 answers
398 views

A bit complicated 20 card games v2

this is follow up question of A bit complicated 20 card games. As the person who determines the order of the cards without changing the numbers on them, you want to make it more difficult for the ...
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3 votes
1 answer
292 views

A bit complicated 20 card games

In the game of 20-card games, the objective is to select a card or pass on it at every turn, with the goal of maximizing the sum of the numbers on the chosen cards. You have knowledge of the numbers ...
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-4 votes
2 answers
132 views

Smallest rectangle that fits the first 10 rectangles [closed]

What is the area of the smallest rectangle that can fit 10 rectangles with areas 1 to 10, inclusive? Rectangles must have integer sides and cannot overlap.
12 votes
4 answers
3k views

11 cyclists in a race

A team of $11$ cyclists begin a race. If each cyclist rides individually then their respective race times will be $10, 11, \ldots, 20$ minutes. However, if three cyclists with individual times $(a, b, ...
5 votes
1 answer
552 views

Sorting five plates

You have a stack of 5 plates of different sizes as shown in the left figure below. In each move you can pick up the top k plates (k is from 1 to 5) and flip them over. For example, if you do this with ...
4 votes
5 answers
579 views

How much can rounding up help you?

I go to a very fair university that likes to make sure that every student gets their best shot. One way they ensure this is by their rounding system. The rules are as follows: All marks for all ...
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0 votes
1 answer
122 views

Maximum and minimum numbers of combinations in a double Latin Square

Let's have a double 6x6 Latin Square (see Figure 1). You can see that this Latin Square has thirteen combinations (see Figure 2). Can you make a double Latin Square that contains the maximum number of ...
4 votes
2 answers
218 views

5-digit numbers with at most one digit in common

5-digit numbers are created using the digits 1, 2, 3, 4, and 5. Any two numbers can have at most one corresponding digit equal to each other. How many different 5-digit numbers can you get in total ...
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8 votes
3 answers
413 views

PSE Advent Calendar 2022 (Day 17) Priority polar problem

This puzzle is part of the Puzzling Stack Exchange Advent Calendar 2022. The accepted answer to this question will be awarded a bounty worth 50 reputation.< Previous Door Next Door > Dear ...
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14 votes
5 answers
3k views

Dividing a square field into 5 equal regions

A farmer has a 10m x 10m field that has fences around the perimeter. What is the least number of 1m fences he needs to add to divide the field into 5 regions of equal area?
4 votes
2 answers
678 views

Bow or lazy checkmate?

Suppose that both players cooperate, find the shortest, in terms of number of moves, black checkmate such that the game is valid and white pieces never go on rows 4 to 8. e.g. ...
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0 votes
0 answers
181 views

Is Connect X a solved game?

Connect 4 is known to be a solved game. After Connect 4 is Excellent, Connect 3 is Great being trivialised by @Trenin, I am introducing the final puzzle I wanted to share to puzzling.SE :) Connect X, ...
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5 votes
1 answer
461 views

Is "Connect 4 is Excellent, Connect 3 is Great" a solved game?

Connect 4 is known to be a solved game. I am introducing the Connect 4 is Excellent, Connect 3 is Great, variant of Connect 4. Both players play like connect 4 and fill the grid i.e. they don't stop ...
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5 votes
3 answers
404 views

Laziest Checkmate

Consider a game of chess where White delivers checkmate in their $X^{th}$ move. For all the pieces that remain on the board at the end of the game, consider their Manhattan Distances from their ...
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4 votes
1 answer
162 views

Packing the Primes

Here is a 5x4 rectangular wordsearch (area 20) containing the primes between 1 and 100, {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97} (numbers in any ...
  • 18.2k
38 votes
8 answers
4k views

The Bee Lover's Hotel

Dr Bees likes bees. Dr Bees is not his real name, but that is what everybody calls him. I should actually say that he is obsessed with bees. Bees are his whole life. He knows everything about them. ...
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12 votes
1 answer
357 views

25 distinct numbers in a 6x10 Minesweeper grid

You are making a new Minesweeper game. You can place 1, 2 or 3 mines in a cell. As usual the number in an empty cell is the number of mines surrounding it (in 8 directions). Can you place mines into a ...
3 votes
1 answer
160 views

The longest shortest sequence of series moves

Being inspired, I concoct a question. It starts from a set chess board with only the White pieces. Chess problems in which only one side moves are called seriesmovers. What is the longest shortest ...
2 votes
1 answer
241 views

Cow Training Farm

Other than ordinary farms like Animal Farm, there is a farm called 'Cows Training Farm', which is a farm that trains $15$ very orz cows. Each year, six cows are chosen to join a competition. Mason is ...
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13 votes
1 answer
1k views

A robot that doesn't repeat move triplets

A robot is placed on an empty infinite grid. Each turn it can move in one of four directions: left (L), up (U), right (R) or down (D). It cannot move to a cell that it already visited. Also, it cannot ...
13 votes
1 answer
323 views

What is the shortest chess game with 30 distinct captures (PxP, PxN, PxR, etc...)?

Progress So Far So far, the shortest known solution takes 87 turns (9 turns from optimal), and was found by Retudin. Progress over time: Date Moves Found by 2022-09-29 87 Retudin (initial) 97 ...
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8 votes
1 answer
218 views

Diagonal pattern for 4x4x4 Rubik Cube with all colors on each side?

I love creating patterns on my 4x4x4 cube, so I came up with the idea to create the following pattern: All sides have a diagonal pattern, meaning something like: ...
  • 183
5 votes
1 answer
219 views

Near Magic Squares with the First 25 Primes

There are 25 primes smaller than 100. What is the closest to a 5 x 5 magic square I can construct with them? By "closest" I mean the one with the most columns, rows, and diagonals (12 in all)...
37 votes
6 answers
2k views

Packing pentominoes in a circle

You want to prepare a pizza of 12 flavors. You have 12 oddly-shaped pieces of cheese that you decide to use for the pizza. The shapes happen to be ... Oh, well, forget it! This isn't going to be ...
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17 votes
2 answers
1k views

He Hauls Hydra Heads across Hilbert's Hexagonal Hotel

(Special thanks to greenturtle3141 for providing an intriguing narrative backstory to my puzzle.) To aesthetically appeal to his customers, Hilbert has recently redesigned his hotel. You see, all ...
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2 votes
1 answer
555 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-...
  • 59
2 votes
0 answers
203 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 ...
  • 59
3 votes
2 answers
394 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 ...
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11 votes
2 answers
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 ...
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11 votes
4 answers
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 ...
1 vote
1 answer
122 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 ...
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0 votes
1 answer
163 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 ...
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3 votes
1 answer
364 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
160 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 ...
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9 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 ...
8 votes
3 answers
949 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 ...
  • 5,923
2 votes
1 answer
118 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 ...
17 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, ...
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5 votes
2 answers
229 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....
8 votes
4 answers
1k 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 ...
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