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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14 votes
4 answers
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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 ...
Christian E. Ramirez's user avatar
4 votes
1 answer
162 views

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 ...
Dmitry Kamenetsky's user avatar
11 votes
2 answers
884 views

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 ...
Dmitry Kamenetsky's user avatar
6 votes
3 answers
782 views

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 ...
Dmitry Kamenetsky's user avatar
2 votes
1 answer
174 views

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 ...
Dmitry Kamenetsky's user avatar
16 votes
1 answer
838 views

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 ...
Mike Earnest's user avatar
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5 votes
3 answers
866 views

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 ...
Léreau's user avatar
  • 311
10 votes
2 answers
879 views

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 ...
mlerma54's user avatar
  • 245
8 votes
1 answer
207 views

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, ...
Mike Earnest's user avatar
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41 votes
7 answers
5k views

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 ...
boboquack's user avatar
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18 votes
4 answers
2k views

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 ...
Mike Earnest's user avatar
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7 votes
5 answers
2k views

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 ...
idknuttin's user avatar
  • 239
4 votes
1 answer
127 views

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 ...
idknuttin's user avatar
  • 239
33 votes
1 answer
2k views

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 ...
Gamow's user avatar
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5 votes
1 answer
144 views

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 ...
D.A.G.'s user avatar
  • 2,356
13 votes
2 answers
2k views

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,...
D.A.G.'s user avatar
  • 2,356
11 votes
3 answers
393 views

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 ...
Haobin's user avatar
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7 votes
2 answers
433 views

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 ...
Gamow's user avatar
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25 votes
4 answers
2k views

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 ...
Olive Stemforn's user avatar
7 votes
2 answers
2k views

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 ...
rekt's user avatar
  • 123
8 votes
3 answers
323 views

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 $ ...
D.A.G.'s user avatar
  • 2,356
1 vote
1 answer
185 views

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 ...
D.A.G.'s user avatar
  • 2,356
12 votes
2 answers
602 views

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 ...
Gamow's user avatar
  • 45.4k
3 votes
2 answers
2k views

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, ...
Carcigenicate's user avatar
30 votes
3 answers
2k views

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 $2 \times 2$ ...
rnaylor's user avatar
  • 2,611
23 votes
3 answers
791 views

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 ...
f'''s user avatar
  • 33.7k
7 votes
2 answers
629 views

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 ...
ghosts_in_the_code's user avatar
25 votes
2 answers
1k views

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 ...
Mike Earnest's user avatar
  • 32.3k
14 votes
5 answers
1k views

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 ...
Gamow's user avatar
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8 votes
4 answers
591 views

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 $ ...
Haobin's user avatar
  • 11.7k
5 votes
2 answers
223 views

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,+...
Gamow's user avatar
  • 45.4k
4 votes
2 answers
283 views

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 ...
Gamow's user avatar
  • 45.4k
4 votes
1 answer
260 views

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 ...
Gamow's user avatar
  • 45.4k
5 votes
5 answers
461 views

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....
Haobin's user avatar
  • 11.7k
9 votes
2 answers
244 views

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 ...
Gamow's user avatar
  • 45.4k
0 votes
1 answer
250 views

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 ...
ghosts_in_the_code's user avatar
11 votes
4 answers
2k views

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 ...
Gamow's user avatar
  • 45.4k
3 votes
4 answers
262 views

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 ...
Gamow's user avatar
  • 45.4k
13 votes
1 answer
732 views

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 ...
Gamow's user avatar
  • 45.4k
29 votes
2 answers
2k views

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 $...
Haobin's user avatar
  • 11.7k
20 votes
2 answers
2k views

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 ...
Haobin's user avatar
  • 11.7k
2 votes
2 answers
325 views

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, ...
ghosts_in_the_code's user avatar
13 votes
1 answer
1k views

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 ...
Gamow's user avatar
  • 45.4k
17 votes
1 answer
1k views

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 $k\ge1$ ...
Gamow's user avatar
  • 45.4k
25 votes
1 answer
3k views

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 ...
Patrick N's user avatar
  • 1,790
6 votes
2 answers
3k views

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 ...
rayu's user avatar
  • 163
12 votes
1 answer
711 views

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 ...
Mike Earnest's user avatar
  • 32.3k
8 votes
3 answers
1k views

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. ...
Abr001am's user avatar
  • 1,004
5 votes
5 answers
2k views

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 ...
Abr001am's user avatar
  • 1,004
-7 votes
2 answers
426 views

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 ...
Abr001am's user avatar
  • 1,004