# 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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### 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 ...
11k 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

### 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 ...
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

### 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 ...
547 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 ...
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

### 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,...
637 views

### 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 turns ...
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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 ...
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 ...
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 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 ...