Questions tagged [reachability]
A puzzle on a discrete system where one has to decide whether a certain system state can be reached through a finite number of steps.
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Labyrinth of Teleporters
You find yourself in an empty room, with a few distinctly numbered elevated platforms on the floor; your only possession is a pebble that can easily be picked up and placed down. You step on one of ...
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Sliding balls and stars on a 4x4 grid
You are playing a game on the following 4x4 grid. It contains balls, stars, empty cells, walls (solid blue squares) and target cells (T). Each turn you can slide all the balls and all the stars into ...
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Sliding balls on a 5x5 grid
You are playing a game on the following 5x5 grid. Each turn you can slide all the orange balls into one of four directions: left, up, right or down. A ball will continue sliding along a direction ...
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Sliding balls on a 4x4 grid version 2
You are playing a game on the following 4x4 grid. Each turn you can slide all the orange balls into one of four directions: left, up, right or down. A ball will continue sliding along a direction ...
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Sliding balls on a 4x4 grid
You are playing a game on the following 4x4 grid. Each turn you can slide all the orange balls into one of four directions: left, up, right or down. A ball will continue sliding along a direction ...
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votes
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answer
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Exterminating blobs on a grid
On an infinite square grid, some of the squares are occupied by little creatures called blobs. Cute as they are, it is your mission to exterminate all of them! You only have two methods at your ...
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Will you be the first to get free?
It is your first day in prison and you are approached by a guard having a hunch for puzzles.
He tells you that he gives every new prisoner the chance to be freed if they can present him with a version ...
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Swapping registers in an old calculator
I came up with this problem inspired by the limitations of an old non-scientific calculator I owned years ago (the two registers were the display, and an internal memory for an additional number).
We ...
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Permuting rows and columns to switch white rooks with black rooks
An adversary places eight white rooks and eight black rooks on sixteen squares of a chessboard, subject to these rules:
In any row, there must be exactly two rooks, one of each color.
In any column, ...
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Amnesiac in a ring shaped palace
Related: Turn off all lights in a ring-shaped palace
Your boss has trapped you inside a ring-shaped palace, and all you know about the palace is that there are some number* of identical rooms, each ...
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Consecutive Towers of Hanoi
Consider the following variant of the Towers of Hanoi puzzle. There are six pegs. One of the pegs has a stack of $n$ differently sized disks, sorted by size so the smallest disk is at the top. All ...
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Is it possible that the last piece the ant has eaten is the central one?
A cube of dimension $3×3×3$ is made of sugar and consists of $27$ small cubical sugar pieces arranged in the $3×3×3$ pattern. An ant is eating the sugar in such a way that it starts at one of the ...
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Is it possible (for some configuration of initial 9 flowers) to get all red flowers after finitely many years?
An isolated garden has the shape of a circle. Initially, there are 9 flowers on the circumference of the garden: 5 of the flowers are red and the other 4 are yellow. During the summer, 9 new flowers ...
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Professor Halfbrain and the sum of the digits of all divisors
Yesterday I met professor Halfbrain in the city. The professor looked tired and somewhat exhausted. He told me that he had spent his nights and days with adding up digits of divisors of positive ...
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Aatif averages numbers on the blackboard
Aatif has averaged numbers and made the final number $2$:
Averaging numbers on the blackboard
Today Aatif once again sees the numbers $ 1 , 2 , 3 , .... , 2016 $ written on the blackboard. In one ...
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Averaging numbers on the blackboard
Aatif sees the numbers $ 1 , 2 , 3 , .... , 2016 $ written on the blackboard. In a move Aatif can pick any two numbers on the blackboard, erase them and write instead once their average. As an example,...
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Blackboard problem with polynomial
At the beginning the blackboard contains $n$ real numbers, one of which is $0$. In every step, we may take any polynomial such that all its coefficients are currently on the blackboard, we compute all ...
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Four indeed is cosmic!
This puzzle deals with positive integers in decimal representation.
From every integer you can move to one or two or three other integers.
The allowed moves for integer $n\ge1$ are as follows:
You ...
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Which is larger? $\sqrt{15} - \sqrt{7} + \sqrt{5} + \sqrt{2}$ versus $5$
Which side is larger?
$$ \sqrt{15} - \sqrt{7} + \sqrt{5} + \sqrt{2} \stackrel{?}{\lessgtr} 5 $$
Without using a calculator, computer, or estimating square roots, please
determine which side has the ...
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Can a Rubik's cube be put in a position not solvable by Fridrich's algorithm?
I think today I have faced a bug on rubik's cube. As it should be, I followed the steps Cross and then F2L, instead of OLL algorithms, I used R'FRF' and three corner swapping moves to complete last ...
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Blackboard problem with 2016
At the beginning the blackboard contains a single integer $N$. If the blackboard contains some number $x$, then we may additionally write the two numbers $\displaystyle\frac x {x+2} $ and $ 2x + 1 $ ...
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Making all stones black [duplicate]
1000 white stones and 1 black stone are arranged in a row. A move consists in selecting one black stone and changing the color of the 2 neighbouring stones (or changing the color of 1 neighboring ...
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Professor Halfbrain and the 52 cards
Professor Halfbrain has spent his entire weekend by analyzing stacks of $52$ cards that are numbered by $1,2,\ldots,52$.
Halfbrain always started with a stack having the cards face-up and in ...
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Put the colours back in order
Background
I wrote up a web-based sketchpad that lets the user pick from a series of colours, and tracks the recently selected colours for ease of use. I randomly had this idea as a personal puzzle, ...
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One hundred tiles
One hundred tiles are arranged in a $10 \times 10$ square. Each tile is black on one side and white on the other side. Two types of move are allowed:
Flip over all four tiles in any $2 \times 2$ ...
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Concentrating tokens on an infinite board
One token is placed on each square of an infinite checkerboard. One square is marked with an X. You want to get as many tokens on the marked square as possible. To do this, you may make any finite ...
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Create an impossible knight transformation
Some examples: Desegregate the Knights and Switch The Knights
You must give two 8x8 chess board positions that have any number of black and white knights.
Both boards must have the same number of ...
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Desegregate the Knights
You are given a 3 by 3 chessboard with a knight on each corner, where the knights in the top row are black and in the bottom row are white.
On each turn, you may move a knight of either color (the ...
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A row of 2015 red and white chips
There is a row of 2015 chips, of which 2014 are white and one is red.
You are allowed to make moves of the following type:
"Choose one red chip, and flip the colors of its two neighboring chips (from ...
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votes
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Blackboard problem with polynomial
The blackboard contains the two numbers 0 and 120. Each minute you are allowed to write an additional number $ x $ on the blackboard if $ x$ has not been written before on the blackboard and if $ x $ ...
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A final incident in the flea circus: Part 2
This puzzle is the continuation and second part of: "A final incident in the flea circus: Part 1"
The ringmaster of a flea circus draws a square $ABCD$ with corners
$A=(+1,+1)$, $B=(+1,-1)$, $C=(-1,+...
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votes
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A final incident in the flea circus: Part 1
The ringmaster of a flea circus draws a square $ABCD$ with corners
$A=(+1,+1)$, $B=(+1,-1)$, $C=(-1,+1)$, $D=(-1,-1)$ in the Euclidean
plane and picks a point $P$ with integer coordinates outside ...
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4
votes
1
answer
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Flipping coin quadruples
Eighteen coins are arranged in a circle. In the beginning, all eighteen coins show tails. The following two moves are allowed:
Simultaneously flipping over four consecutive coins (so that tails ...
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votes
5
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The smallest value on the blackboard
The square numbers $ 1^2, 2^2, 3^2, 4^2, \cdots, 100^2, 101^2 $ are written on the blackboard.
Each minute any two numbers are wiped out, and the absolute value of their difference is written instead....
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answers
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Yet another curious incident in the flea circus
The ringmaster of a flea circus puts four fleas $A$, $B$, $C$, $D$ on
four different points in the plane that form the corners of a square.
Whenever the ringmaster shouts "Hop!", one of the four ...
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The infinite flea circus
Based on Another curious incident in the flea circus and A curious incident in the flea circus by @Gamow
There is a $n$ dimensional cube in an $n$ dimensional world. There is a flea on each vertex of ...
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Another curious incident in the flea circus
The ringmaster of a flea circus puts four fleas $A$, $B$, $C$, $D$ on four different points in the plane that form the corners of a square.
Whenever the ringmaster shouts "Hop!", one of the four ...
- 45.3k
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votes
4
answers
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A curious incident in the flea circus
The ringmaster of a flea circus puts three fleas $A$, $B$, $C$ on three different numbers on the real number line, so that flea $B$ sits exactly in the middle between $A$ and $C$.
Whenever the ...
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answer
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Professor Halfbrain and the right-angled triangles
Today I met professor Halfbrain at the tea house.
The professor looked very tired, and apparently had not slept for the last couple of days. He told me that he had been spending his time with cutting ...
- 45.3k
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Numbers on the blackboard: From 2-2015 to 1-2014
The numbers $ 2, \ldots, 2015 $ are written on a blackboard. Each minute any two numbers $ x $ and $ y $
are wiped out and are replaced by two numbers $\displaystyle \frac { 4x + 3y } { 5 } $ and $...
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The last number on the blackboard
The numbers 1, 2, ..., 500 are written on a blackboard. Each minute any two numbers
are wiped out and their positive difference is written instead. At the end only one number remains.
Which values can ...
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Closing taps (generalisation) [closed]
You have a tanker that can hold any amount of water. You have $n$ taps that either let in or let out water. The rate of flow of water (in $ml/min$) for each tap is given by $f_n$ (positive for inflow, ...
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The Erasmus 0-1 game
Professor Erasmus told me that today he has proved another fascinating theorem about strings made of 0s and 1s. He takes an arbitrary such string and repeats the following step on it:
if the leftmost ...
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Hopping from 81 to 82
A grasshopper is hopping around on the integers and starts its journey on the number $81$. In a jump starting from the integer $m$, the grasshopper
may jump to any integer $m^k$ with integer $k\ge1$
...
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Four is Cosmic!
This is a little puzzle I heard a while back from one of my mathematically inclined friends- I get the sense that it's bounced around a little, so forgive me if you've heard it.
There is a sort of ...
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Minimum moves to have all coins face Heads up
Given a circular list of coins, that all have Tails facing up.
In each move, if we flip the coin at position $i$, then the coins at positions $i-1$ and $i+1$ get flipped as well. That is, consider: H ...
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Enlarge the Square?
There are four stones, positioned on the ground at the vertices of a square. At any time, you may pick up a stone and "hop" it over another one so that it lands an equal distance beyond the hopped ...
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Coin inverting game
Header and Tailer play the following game.
At the beginning, the juror lays out a row of $n\ge5$ coins on the table that alternately show heads and tails, with the leftmost coin showing heads.
...
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Coin flipping game
An $8\times8$ checkerboard is filled with two-sided coins (that are blue on one side and red on the other side).
The following picture shows three examples of a cross (multiplication sign):
the five ...
- 984
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answers
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Checkerboard piece inverting game
Marco and Leonardo decided to play a game on a checkerbard of 4×4 squares.
The board is initially filled with two-sided identical coins.
The game notes that these two players play turns alternatively ...
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