# 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.

52 questions
Filter by
Sorted by
Tagged with
828 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 ...
2k views

### 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 ...
146 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, ...
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 ...
2k 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 ...
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 ...
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 ...
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 ...
121 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 ...
461 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 $2 \times 2$ ...
5k views

### 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 ...
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 ...
508 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 ...
1k 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 ...
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 ...
115 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 ...
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,...
329 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 ...
351 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 ...
2k 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 ...
678 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 ...
403 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 ...
2k views

### 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 ...
1k 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 ...
601 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 ...
1k views

### 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 ...
298 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 $2x + 1$ ...
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 ...
168 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 ...
646 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, ...
671 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 ...
1k views

222 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 ...
511 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 ...
426 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....
230 views

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 ...
2k views

### The last number on the blackboard

The numbers $1, 2, \ldots, 500$ 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 ...
225 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 ...
1k 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 ...
217 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 ...
196 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 ...
10k views

### Turn off all lights in a ring-shaped palace

Not a very difficult question, but one I enjoyed nonetheless and wanted to share with the community. You are a servant in a palace. The palace is in the shape of a circle, and you do not know how ...
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 $k\ge1$ ...
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. ...