# 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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### Is my bank going to be bankrupt?

Today I heard a rumor that my bank will go bankrupt soon. So I went to my bank, and indeed I found out that they have introduced some weird new rules that clearly indicate that there is something ...
2k views

### 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 ...
10k views

### Placing 2x1 dominoes on a chessboard with two corners removed

Suppose you have a checkerboard with two opposite corner squares removed, like this: Is it possible to place 31 dominoes of size 2x1 so as to cover all of these squares?
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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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
1k views

### Turn 16 into 25. You have 4 magic spells. Can you do it?

You have been given the number 16. You also have 4 magic spells. You have to use the spells to turn 16 into 25. Each spell changed your number a specific way, but the objective is never altered, 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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### 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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### Martin Gardner - Persistence

A number's persistence is : The number of steps required to reduce it to a single digit by multiplying all its digits to obtain a second number Then multiplying all the digits of that number to ...
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### Pathfinding with disappearing platforms - is it solvable?

This is a puzzle but might not have a solution. So "is it possible?" is a proper question. The puzzle idea was inspired by the question logic problem/puzzle solving, and if the puzzle here ...
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 ...
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 ...
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 ...
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### Cutting three sticks

Cosmo plays the following (single-player) game with three wooden sticks: He first checks whether he can form a triangle from the three sticks (which means: whether the longest stick is at most as ...
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, ...
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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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### 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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### 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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### 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 ...
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 ...
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 $2 \times 2$ ...
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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 ...
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 ...
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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,...
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 ...
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 ...
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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 ...
491 views

### It's twelve o'clock!

The twelve numbers on a clock are each either colored red or colored black. You are allowed to make several moves, where a move consists in picking a black number, and in flipping the colors of its ...
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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 ...
733 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 ...
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### The juggling magician

In the magical circus, today a magician juggles with balls, cones, rings and umbrellas. While performing his art, he sometimes applies a powerful magic word: Ebrecedebre transforms one ball ...
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 $2x + 1$ ...
1 vote
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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 ...
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, ...
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 ...
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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 ...
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....