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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8
votes
3answers
173 views

A Perfect Diamond of Numbers

A diamond of numbers is an arrangement of circles in the shape of a trapezoid (see figure) in which the number in any circle above its central (longest) row is the sum of the two numbers in the ...
10
votes
5answers
1k views

Spiders on a cube

Two spiders are trying to catch an ant. All are constrained to move along the edges of a transparent cube. The speed of the ant is $1$. The speeds of the spiders are $v_1$ and $v_2$ respectively. What'...
3
votes
2answers
713 views

Most points on a circle

What is the most number of integer lattice points that lie on the circumference of a single circle whose radius is 80 or less? Please no computer computations.
5
votes
2answers
204 views

Most polyominoes on a Rubik's cube

What is the most number of distinct free polyominoes you can form on the faces of a standard 3x3x3 Rubik's cube? Here a polyomino is considered as a set of orthogonally-adjacent cells of the same ...
7
votes
1answer
783 views

Cutting off one's nose to spite one's eyes

Disclaimer: to keep graphic depiction of gratuitous violence to a minimum the face to be spited has been deliberately kept abstract. You are required to further reduce any distress this puzzle may ...
3
votes
1answer
347 views

Using squares to prove e > 2.7

Edited to replace $\exp(-x)$ with $\exp(x)$. My apologies. I loved this puzzle, so thought I'd submit a similar one: The definite integral $\int_{−\infty}^{1} \exp(x)dx$ is equal to $e$ . Using ...
5
votes
2answers
251 views

Longest sequence of two-digit sum replacements

You are given a two-digit number. You can replace one of its digits with the sum of its digits modulo 10. For example, if the starting number is 58 then you can change it to 38 or 53. You can continue ...
96
votes
2answers
9k views

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". ...
12
votes
2answers
410 views

Square Crayon Sticks

Ten sets of crayons forming all the digits from 0 to 9 can be moved, flipped, rotate and intersect without changing their digital forms. What is the maximum number of 1x1 stick squares that can be ...
3
votes
1answer
185 views

A solitaire Blokus problem on a rectangular board

Rules As in Blokus, you have a total of 21 pieces (every piece from monomino to pentomino) in hand: All of these polyominoes are free, this means that you can rotate or flip them as you wish before ...
5
votes
2answers
257 views

Five friends and two motorcyclists

Five friends Alice, Bob, Carole, Dylan and Emma are heading to a common destination 100 unit distance away. They start together. Grandma Alice walks at a speed of 1. Bob and Carole walk at speeds 4 ...
8
votes
2answers
251 views

What pieces does White need to beat Black's veto?

Consider the following chess variant: before each of White's moves, Black chooses one move which White is not allowed to make. (The rules for which positions constitute checkmate are unchanged - Black ...
3
votes
1answer
116 views

Exchanging stones on a 8x8 board with sum of two adjacent numbers not being prime

You are given 64 stones labelled with number 1 to 64 each. All those stones are randomly placed on the squares of a 8x8 chess board such that each square is occupied with exactly one stone. A move is ...
3
votes
1answer
246 views

Is there a minimum number of clues that every sudoku puzzle has?

I've seen that the fewest clues on a Sudoku board has been proven to be 17 but I'm wondering if it's possible for every board to have some combination of 17 clues or, if not, if there is a proven ...
3
votes
3answers
158 views

Rock climbing higher and faster

The Olympic rock climbing competition has 20 climbers. Each climber competes in 3 separate events, where they rank from 1st to 20th. The final score of a climber is the product of their rankings from ...
25
votes
3answers
4k views

Rock climbing at the Tokyo Olympics

The idea for this puzzle came from my friend Jan. The puzzle is based on real world events from the Tokyo Olympics. The Olympic rock climbing preliminary round has 20 climbers. Each climber competes ...
1
vote
2answers
258 views

Elementals and Aliens

Rules/Story You have been kidnapped and locked in the maze under the bathhouse you work at by 5 aliens: a Mercurian, Venusian, Martian, Jovian, and Saturnian. Currently, the aliens are spread ...
3
votes
3answers
464 views

How old is Grandfather?

This position has been used in hundreds of chess problems for various reasons by a plethora of problemists over many, many decades. It is known as "Vielväterstellung" in German, or "The ...
10
votes
2answers
444 views

Transform a pile of cards in reverse order

There are 52 cards on a table and numbered in order from 1 to 52 with number 1 on top. The following operation can be repeatedly done: split the pile into two piles without shuffling and take cards ...
8
votes
2answers
917 views

Five positive integers in a row, each being the sum of the digit sum of its neighbours

Five positive integers should be put in a row such that each integer is the sum of the digit sum of its neighbours. The integers at the beginning and at the end have only one neighbour, i.e the first ...
5
votes
3answers
429 views

100 prisoners and a secret number (sequel)

Inspired by this brilliant puzzle by @DmitryKamenetsky! Much to the annoyance of the guards, the sly prisoners bribed a guard and learnt about the number in the envelope in advance, and the first ...
11
votes
3answers
1k views

Find an integer where each sum of 5 consecutive digits is prime

Find the largest positive integer with the following properties: every sum of 5 neighboring digits in its decimal representation is a prime number. those prime numbers get smaller and smaller from ...
12
votes
6answers
1k views

How many bottles can you drink?

You have 120 bottles of Cola and 120 bottles of Sprite. You can exchange 3 empty Cola bottles for a new bottle of Sprite. You can exchange 4 empty Sprite bottles for a new bottle of Cola. You can ...
1
vote
3answers
491 views

The Third Hardest Logic Puzzle in the World

This puzzle is heavily derived from the Second Hardest Logic Puzzle in the World, which I also created. The current puzzle and the aforementioned are not identical, however. You have 5 guards: one ...
22
votes
7answers
5k views

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. "...
3
votes
3answers
305 views

Chess Construction Challenge #9: Two Eggs & A Flood

I have an odd idea for this challenge. It involves a very specific proof game construction. So pay attention! Given that: White and Black cannot move more than one pawn each throughout the entire ...
5
votes
2answers
601 views

A school needs a talented student (covering the hamming graph H(3,15) with closed balls of radius 5)

Assume a standardized test has $15$ multiple choice questions, with $3$ options each. A school has some students and wishes for at least one of the students to get $10$ correct answers. Since no one ...
3
votes
1answer
126 views

Chess Construction Challenge #8: Three Times The Charm

On a chessboard, a piece has a set number of legal moves. It can range from 0 to 27. However, this amount can also restricted. My previous questions have covered n=1 and n=2, it is time for n=3! Given ...
6
votes
3answers
425 views

Generalization of the two-surgeons-two-patients-and-two-gloves puzzle

This is the original puzzle with $n=2$. I recommend solving it before this one to get acquainted with the mechanisms. There are $n$ patients in an hospital (let's call them $p_1 \dots p_n$), each of ...
8
votes
2answers
218 views

Making 100 starting with two 1s

You have an empty blackboard. At each step, you can either write two ones on the blackboard, or erase two copies of a number n and replace them with n−1 and n+1. What is the fewest number of steps it ...
7
votes
3answers
499 views

Chess Construction Challenge #7: Double The Fun

Many thanks goes to @Retudin in a comment on my last question for this idea. In a chess position, pieces can be restricted as to how many moves they can legally make in theory. Since my previous ...
9
votes
5answers
1k views

Chess Construction Challenge #6: The One Move Royale

In chess, it is possible for a piece to have only one legal move. For example, in this position, the Black king can move only to h2. Given that: All 32 pieces are available for use. Construct: A ...
8
votes
1answer
447 views

Jumping token on a grid

A token starts at location (0,0) on an infinite grid. On turn $n$ it must jump $n$ units horizontally or vertically, in one of four directions. What is the least number of turns needed for it to reach ...
10
votes
2answers
334 views

Gentrification in Chessshire

You can skip the back story and directly jump to the question. Capitalism has arrived in suburban Chesster the community most famed for The Game, and with a vengeance. Rents have trebled in less than ...
6
votes
1answer
376 views

Discounts in a shop

I came across this sign in a shop and thought it could make a nice puzzle. So you can buy items and get discounts depending on how many items you got. You can combine discounts and use as many as you ...
23
votes
7answers
2k views

What is the minimum number of helpers that an explorer need to cross the desert?

There is a desert which can be crossed only by walking and this takes six days. The explorers of the desert can carry at most four days' water and food. If they want, they can take helpers with them. ...
3
votes
1answer
124 views

Mix a 2x2 Rubik’s cube

What is the minimum number of 1/4 turns of a solved 2x2 Rubik’s cube, such that no face will have two tiles of the same color? (I do not have this puzzle’s solution.)
1
vote
1answer
190 views

Catch the fugitive

The fugitive is at the origin. He moves at a speed of 100. You have a guard at every integer coordinate except the origin. A guard's speed is 1. The fugitive and your guards move simultaneously and ...
5
votes
3answers
283 views

The vaccine distribution conundrum

The context is that of a pandemic that is spreading wildly and requiring global vaccination of the population. You are working in a distribution center for the vaccines. One day you have been ...
20
votes
2answers
1k views

Beans under the chessboard

Under every grid cell of a chessboard, I put either one bean or nothing. Now if you choose a (grid) rectangular area on the chessboard, then I will tell you the parity of the number of beans under ...
2
votes
1answer
110 views

Fetching Alchemist, Excavation I

This is a puzzle in the Fetching Alchemist series. It has been generated especially for Puzzling Stack Exchange. Please note that, in my opinion, imperfect solutions should be up-voted so long as they ...
0
votes
1answer
74 views

Fetching Alchemist, Grand Potion I

This is a puzzle in the Fetching Alchemist series. There's no selling in this puzzle, just one potion to brew, but with a lot of ingredients. Please note that, in my opinion, imperfect solutions ...
2
votes
3answers
196 views

Advanced Fetching Alchemist II

This is a puzzle in the Fetching Alchemist series. From now on, you complete quests at the place you start at as well. Please note that, in my opinion, imperfect solutions should be up-voted so long ...
2
votes
1answer
102 views

Advanced Fetching Alchemist I

This follows the same rules as previous Fetching Alchemist puzzles, except you choose where you start, and you may now return to your starting place after leaving it. How to Play You are looking for ...
10
votes
2answers
579 views

Adding coins inside a ring of coins

17 identical coins with diameter 1 are lying flat on a table, such that their midpoints build the vertices of a regular 17-gon (regular heptadecagon) and adjacent coins touch each other. What is the ...
0
votes
0answers
68 views

Grids with tiles of two colors and connectors

Make a square grid with an even side. Make a continuous trail which passes through all small squares; the trail ends where it started. Where there is an angle you can put a red or a blue tile. On ...
5
votes
3answers
180 views

Numbers with minimal sum at the vertices of a cube

The eight vertices of a cube are marked with numbers from 1 to 8 such that the sum of any three numbers on any face is not less than 10. What is the minimum sum of the four numbers on a face?
4
votes
1answer
75 views

A 4x6 grid with adjacent integers with gcd > 1

You are given a 4x6 square grid. Each square of the grid should be filled with different positive integers. The gcd (greatest common divisor) of any two adjacent (horizontally or vertically) squares ...
3
votes
3answers
270 views

Not selling 100 pencils

A shop sells pencils only in boxes of fixed size. It cannot sell 100 pencils. It can sell any larger amount of pencils. It has one of each size box on display. question 1: What is the minimum amount ...
3
votes
1answer
77 views

Divisors ending with digits 0-9 each

What is the smallest positive integer, which has - for each of the digit 0-9 - a divisor ending with this digit?

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