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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votes
0answers
121 views

Real Life Chaos Theory Puzzle [on hold]

I accidentally threw a red at my teammate during a Mario Kart Wii organized race, causing someone else to be banned from a MKW lounge for a week as a result. Try to figure out the missing steps. If ...
10
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2answers
829 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 ...
12
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1answer
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 ...
14
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1answer
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 ...
7
votes
1answer
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, ...
40
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7answers
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 ...
6
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2answers
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 ...
18
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4answers
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 ...
13
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5answers
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 ...
7
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5answers
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 ...
4
votes
1answer
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 ...
11
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2answers
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 ...
26
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3answers
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 $2 \times 2$ ...
6
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2answers
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 ...
20
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2answers
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 ...
6
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2answers
509 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 ...
19
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3answers
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 ...
31
votes
1answer
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 ...
4
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1answer
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 ...
12
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2answers
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,...
11
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3answers
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 ...
7
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2answers
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 ...
25
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1answer
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 ...
6
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2answers
682 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 ...
13
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1answer
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 ...
9
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3answers
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 ...
4
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5answers
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 ...
12
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1answer
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 ...
11
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3answers
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 ...
8
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3answers
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 $ ...
7
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5answers
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 ...
2
votes
1answer
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 ...
3
votes
2answers
651 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, ...
20
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3answers
672 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 ...
27
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3answers
1k 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 $...
8
votes
4answers
514 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 $ ...
2
votes
2answers
315 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, ...
5
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2answers
190 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,+...
3
votes
2answers
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 ...
12
votes
1answer
512 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 ...
5
votes
5answers
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....
4
votes
1answer
230 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 ...
13
votes
2answers
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 ...
9
votes
2answers
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 ...
11
votes
4answers
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 ...
0
votes
1answer
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 ...
3
votes
4answers
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 ...
72
votes
12answers
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 ...
16
votes
1answer
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 $k\ge1$ ...
8
votes
3answers
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. ...