# Questions tagged [construction]

A puzzle that requires an example to be built that fits certain criteria.

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### Can you color the 8x8 grid red and blue?

Consider an 8x8 grid made up of 64 unit squares. The goal is to color the 64 squares red or blue so that the following two constraints are satisfied: ROWS: For every pair of adjacent rows, exactly ...
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### Make exactly 101 squares using as few lines as possible

With how few straight lines can you make exactly 101 squares? The squares don’t necessarily have be the same size. Clarification 1: By lines, I mean you can use mathematical line segments and/or ...
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### Are circles required on the edge of the grid? Part 2

This question is a follow up to a previous question: Are circles required on the edge of the grid? The image below is a puzzle from the FlowFree app. Notice there are no colored circles on the edge of ...
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### Are circles required on the edge of the grid?

The image below is a puzzle from the FlowFree app: The image below is my solution to the above puzzle: The rules stated in the app are: Drag to connect matching colors with pipe, creating a flow. ...
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### The ultimate backgammon

The following position, where one player has borne off all their checkers while all of the other player's checkers are on the bar (thus losing a backgammon), is reachable by a legal sequence of dice ...
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### Binary Grid Challenge and X & O Conundrum

You are given a 5x5 grid with some cells filled with either "X" or "O". Your goal is to fill the remaining cells with "X" or "O" following these rules: Each row ...
438 views

### Pawns and a chessboard with no three aligned

This little problem crossed my mind and appeared to be not quite trivial. How can you place P pawns on a chessboard with the constraint that no pawn is exactly midway between two other pawns? Sure ...
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### Mishustin's circle problem

This problem was given to high school students by the Russian prime minister Mishustin. We have a circle. We are given some point on the circle and its diameter, as shown below. We are given a ...
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### A balanced banquet

As your king's fitness adviser you have been tasked with optimising His Majesty's diet. This entails making sure that the king eats equal amounts of carbs (rooks), fat (bishops) and protein (knights). ...
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### 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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### Colour the positive integers without making a blue equation

This puzzle is related to How do we find the numbers? but has a slightly more striking solution in my opinion. It is also based on one of my MathsSE answers. What is the least number of colours you ...
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1 vote
252 views

### A position where both sides have no moves at all, not even to put yourself in check

Inspired by this: No moves at all, not even to put yourself in check Similar to the question linked above, find an arrangement using the fewest chess pieces (total from both sides), except for this ...
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### Can you construct a nonagon with 47 rods?

Stiv's Diabolical Instruments now offers a bundle of exactly 47 equal-length rods that can be joined by hinges at their ends – and only the ends – to form planar linkages (i.e. all hinge axes are ...
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### - Pandora's Box -

After Prometheus had stolen fire from heaven and bestowed it upon mortals, Zeus, the king of the gods, determined to counteract this blessing. Zeus commissioned Hephaestus, the god of fire, to fashion ...
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### Logical Deduction Persistence Test

In the grid below, create a path that starts at cell 1 and ends at cell 49, moving horizontally and vertically only. The path must touch each of the 49 cells exactly once and contain a: A. Maximum of ...
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### A pentagon that can measure the first 7 integer distances

A pentagon can be used to measure 10 distances - one distance between each pair of its 5 vertices. Can you find a pentagon that can measure every integer distance from 1 to 7, inclusive?
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### 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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### Symmetrical Chess Position With No Legal Moves

Your task is to set up a symmetrical (both vertically and horizontally) position on the first 7 ranks of a chess board that meet all of the criteria below. For this puzzle, there is no eighth rank. It ...
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### The Game of Golden Squares

On a magic chessboard of infinite size, the squares are either wooden or golden. If 4 or more of its 8 neighbors (a king's move away) are golden, a wooden square becomes golden the next day. Golden ...
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### 4x4 word 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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### 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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### Permutations of first 10 natural numbers such that all the prefix sums are distinct

I posted this question on Math SE as well. Did not receive any help. This is a question that I was asked in a Quant Interview. I would like you all to have a crack at this. I could not find a problem ...
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### 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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### Smallest number of moves to reach a "reverse checkers" position

A "reverse checkers" position is a position where every piece for one player is on one colour and every piece for the other player is on the other colour. Consider this position, for example:...
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### Rigid regular nonagon from 21 Meccano strips

You are given 21 Meccano strips, where the distance between adjacent holes is 1 unit: 9 strips of length 10 (hence having 11 holes) 6 strips of length 18 (19 holes) 6 strips of length 19 (20 holes) ...
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### Make a topological torus-with-a-hole out of congruent squares that may share an edge or a vertex with other squares

Suppose we arrange, in 3-dimensional space, 8 identical solid cubes in space so they form a square-shaped ring (using a 3x3 arrangement of squares except for the one in the middle). Its surface will ...
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### Clash of the Robinsons

"Ridiculous!" you think "What can be the odds? Either I'm hallucinating or the amateur writing this story plunged to new depths of incompetence." Both being equally likely you don'...
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### Longest chain of checks and captures

On a standard size chessboard, with white to move, make a configuration of chess pieces and moves, so that with every move by white the black king repeatedly becomes checked. With every move black ...
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### Doubling the cube with rational Meccano strips

In three monographs published in 2006, 2008 and 2014 Gerard 't Hooft considered "Meccano mathematics": how to construct specified distances and regular polygons by a rigid system of ideal ...
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### 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 ...
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### Chess Construction Challenge #D: 8 moves

Since I "burnt" the complete question elsewhere, let's start with "D" (I counted up in hex) and 8 moves. The position in the answer there can be improved by far! OK, to ...
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### 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 ...
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### Flat share share

This is a sequel to Gentrification in Chessshire. Due to the febrile state of the Chesster housing market you and your flat mates are forced to rent out half your place to another group of sharers. ...
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### 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 ...
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### 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 ...
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### 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 ...
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1 vote
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### Self-indulgent numbers

Let's call a positive integer N self-indulgent of degree K>2 if for every positive integer k<K the following is true: More than half of the first k multiples N,2N,...,kN of N contain with ...
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### Infinite beauty

This is a follow-up to Puzzle about 6 infinite cylinders in space Question: Given six identical infinite (no caps) cylinders is there a beautiful arrangement in space such that each touches each ...
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### Never kill your king ... without first seeking proper legal advice!

This is a follow-up to (Almost) all hands on check by @loopy walt. The task (mate in one with as many essential pieces as possible) stays the same, but you are to go about it more cunningly. Recap of ...
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### (Almost) all hands on check

Historical note (skip if you like): In this interesting puzzle @TSLF asks for the "maximum number of black and white pieces that are involved in a checkmate position". Originally, they left ...
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1 vote
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### Four pacifist queens

Earlier, I introduced the beautiful game of Swiss chess. These are animated gifs. You may have to click on them to make them move. Unlike the peace-loving Swiss, we are interested in "domination&...
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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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### Universal bisectors

A bisector is something that cuts some other thing into two equal pieces. More concretely, assume we are given a reasonably well-behaved (for example, compact) 3D object and we are looking for planes ...
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### Anti-Chess Squares

Playing regular rules for both white and black try to eliminate all the square formations by black and/or white. From the above starting setup all rooks form a square. Also white's queen rook pawn, ...
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### A rectangle with non-integer side lengths [duplicate]

Is it possible to build a gapless rectangle with non-integer side lengths using rectangles each with two integer side length and two non-integer side length? The rectangles are not required to be the ...
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### Piercing Bullets

Under 30-secs. bullet time control the above position results after n moves. What is the fewest number of moves possible?
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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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### Triangles, rectangles, nonagons

Which is the nonagon with the least area and which fulfills the following conditions. The nonagon has to be made from 7 triangles and 3 rectangles, all having side-lengths that are integer numbers. ...
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### Experts at failure

Construction challenge: Find a position with the longest sequence of unique losing moves, i.e. white to move has one and only one move that will lead to a lost (for white) position. White makes this ...
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