Keeping a ball lost foreverFour stones on a Go-boardBlock the snake from reaching pointsNumber swapping gameLeast amount of moves is requiredHnefatafl - a lost ArtJonMark Perry's Grid Logic PuzzleGuide the dots to land on the portals at the same timeGuide dots to land on any pair of matching portals at the same time20 cards facing downThe No-Straight Maze
Mimic lecturing on blackboard, facing audience
Yosemite Fire Rings - What to Expect?
Lowest total scrabble score
How to hide some fields of struct in C?
Calculate sum of polynomial roots
How could a planet have erratic days?
Biological Blimps: Propulsion
Does the UK parliament need to pass secondary legislation to accept the Article 50 extension
What should you do when eye contact makes your subordinate uncomfortable?
Can a Canadian Travel to the USA twice, less than 180 days each time?
PTIJ: Haman's bad computer
Pre-mixing cryogenic fuels and using only one fuel tank
Electoral considerations aside, what are potential benefits, for the US, of policy changes proposed by the tweet recognizing Golan annexation?
How much character growth crosses the line into breaking the character
Open a doc from terminal, but not by its name
Mixing PEX brands
Why is this estimator biased?
Can disgust be a key component of horror?
Why does AES have exactly 10 rounds for a 128-bit key, 12 for 192 bits and 14 for a 256-bit key size?
Non-trope happy ending?
Why can Carol Danvers change her suit colours in the first place?
How can I cure my players of their suspicion that every object is a mimic?
Why did the EU agree to delay the Brexit deadline?
What is Cash Advance APR?
Keeping a ball lost forever
Four stones on a Go-boardBlock the snake from reaching pointsNumber swapping gameLeast amount of moves is requiredHnefatafl - a lost ArtJonMark Perry's Grid Logic PuzzleGuide the dots to land on the portals at the same timeGuide dots to land on any pair of matching portals at the same time20 cards facing downThe No-Straight Maze
$begingroup$
Suppose you can make a rectangular maze, where each cell (apart from the bottom-right) can contain an arrow in one of the four directions (up, down, left or right) of your choosing, except for those on an edge or corner, which must not point out of the maze.
A ball is then placed into the top-left square and begins to move. It will move in the direction of the arrow in the cell that it is currently in. Then, that arrow will rotate 90 degrees clockwise. If an arrow is pointing out of the maze, it will continue rotating clockwise until it points in a valid direction.
A valid maze is one in which the ball will never reach the bottom-right corner.
Prove or disprove the existence of such a maze. If it exists, find the smallest possible maze (in terms of number of squares).
Here is an example of a maze.
- The ball begins in A1. It moves right to A2, and the A1 arrow rotates to point down.
- The ball moves left to A1, and the A2 arrow rotates to point down (as both up and right point out of the maze).
- The ball moves down to B1, and the A1 arrow rotates to point right (as both left and up point out of the maze).
- The ball moves up to A1, and the B1 arrow rotates to point right.
- The ball moves right to A2, and the A1 arrow rotates to point down.
- The ball moves down to B2, and the A2 arrow rotates to point left.
- Now, the ball is in B2, the bottom-right corner of the maze, so that is the end. It is not a valid maze, but if it were, it would have a score of $4$.
logical-deduction strategy optimization
asked 2 hours ago
ZanyGZanyG
1,086420
$endgroup$
add a comment |
$begingroup$
Suppose you can make a rectangular maze, where each cell (apart from the bottom-right) can contain an arrow in one of the four directions (up, down, left or right) of your choosing, except for those on an edge or corner, which must not point out of the maze.
A ball is then placed into the top-left square and begins to move. It will move in the direction of the arrow in the cell that it is currently in. Then, that arrow will rotate 90 degrees clockwise. If an arrow is pointing out of the maze, it will continue rotating clockwise until it points in a valid direction.
A valid maze is one in which the ball will never reach the bottom-right corner.
Prove or disprove the existence of such a maze. If it exists, find the smallest possible maze (in terms of number of squares).
Here is an example of a maze.
- The ball begins in A1. It moves right to A2, and the A1 arrow rotates to point down.
- The ball moves left to A1, and the A2 arrow rotates to point down (as both up and right point out of the maze).
- The ball moves down to B1, and the A1 arrow rotates to point right (as both left and up point out of the maze).
- The ball moves up to A1, and the B1 arrow rotates to point right.
- The ball moves right to A2, and the A1 arrow rotates to point down.
- The ball moves down to B2, and the A2 arrow rotates to point left.
- Now, the ball is in B2, the bottom-right corner of the maze, so that is the end. It is not a valid maze, but if it were, it would have a score of $4$.
logical-deduction strategy optimization
asked 2 hours ago
ZanyGZanyG
1,086420
$endgroup$
add a comment |
$begingroup$
Suppose you can make a rectangular maze, where each cell (apart from the bottom-right) can contain an arrow in one of the four directions (up, down, left or right) of your choosing, except for those on an edge or corner, which must not point out of the maze.
A ball is then placed into the top-left square and begins to move. It will move in the direction of the arrow in the cell that it is currently in. Then, that arrow will rotate 90 degrees clockwise. If an arrow is pointing out of the maze, it will continue rotating clockwise until it points in a valid direction.
A valid maze is one in which the ball will never reach the bottom-right corner.
Prove or disprove the existence of such a maze. If it exists, find the smallest possible maze (in terms of number of squares).
Here is an example of a maze.
- The ball begins in A1. It moves right to A2, and the A1 arrow rotates to point down.
- The ball moves left to A1, and the A2 arrow rotates to point down (as both up and right point out of the maze).
- The ball moves down to B1, and the A1 arrow rotates to point right (as both left and up point out of the maze).
- The ball moves up to A1, and the B1 arrow rotates to point right.
- The ball moves right to A2, and the A1 arrow rotates to point down.
- The ball moves down to B2, and the A2 arrow rotates to point left.
- Now, the ball is in B2, the bottom-right corner of the maze, so that is the end. It is not a valid maze, but if it were, it would have a score of $4$.
logical-deduction strategy optimization
asked 2 hours ago
ZanyGZanyG
1,086420
$endgroup$
Suppose you can make a rectangular maze, where each cell (apart from the bottom-right) can contain an arrow in one of the four directions (up, down, left or right) of your choosing, except for those on an edge or corner, which must not point out of the maze.
A ball is then placed into the top-left square and begins to move. It will move in the direction of the arrow in the cell that it is currently in. Then, that arrow will rotate 90 degrees clockwise. If an arrow is pointing out of the maze, it will continue rotating clockwise until it points in a valid direction.
A valid maze is one in which the ball will never reach the bottom-right corner.
Prove or disprove the existence of such a maze. If it exists, find the smallest possible maze (in terms of number of squares).
Here is an example of a maze.
- The ball begins in A1. It moves right to A2, and the A1 arrow rotates to point down.
- The ball moves left to A1, and the A2 arrow rotates to point down (as both up and right point out of the maze).
- The ball moves down to B1, and the A1 arrow rotates to point right (as both left and up point out of the maze).
- The ball moves up to A1, and the B1 arrow rotates to point right.
- The ball moves right to A2, and the A1 arrow rotates to point down.
- The ball moves down to B2, and the A2 arrow rotates to point left.
- Now, the ball is in B2, the bottom-right corner of the maze, so that is the end. It is not a valid maze, but if it were, it would have a score of $4$.
logical-deduction strategy optimization
logical-deduction strategy optimization
asked 2 hours ago
ZanyGZanyG
1,086420
asked 2 hours ago
ZanyGZanyG
1,086420
asked 2 hours ago
ZanyGZanyG
1,086420
asked 2 hours ago
ZanyGZanyG
1,086420
asked 2 hours ago
ZanyGZanyG
1,086420
1,086420
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Suppose such a maze exists. Then the balls visits at least one square infinitely many times. Let $S$ be one such square that is closest to the bottom-right. $S$ is not the bottom-right square, so there exists a square $T$ to the right of or below $S$ that is closer to the bottom-right than $S$ is. However, the ball must visit $T$ at least once every $4$ visits to $S$ due to arrow rotation. Therefore, the ball must also visit $T$ infinitely many times, contradicting the minimality of $S$. Then no such maze exists.
answered 2 hours ago
noednenoedne
7,16712057
$endgroup$
$begingroup$
Very succinct; well done. I'll wait a bit before accepting.
$endgroup$
– ZanyG
2 hours ago
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "559"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fpuzzling.stackexchange.com%2fquestions%2f80948%2fkeeping-a-ball-lost-forever%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Suppose such a maze exists. Then the balls visits at least one square infinitely many times. Let $S$ be one such square that is closest to the bottom-right. $S$ is not the bottom-right square, so there exists a square $T$ to the right of or below $S$ that is closer to the bottom-right than $S$ is. However, the ball must visit $T$ at least once every $4$ visits to $S$ due to arrow rotation. Therefore, the ball must also visit $T$ infinitely many times, contradicting the minimality of $S$. Then no such maze exists.
answered 2 hours ago
noednenoedne
7,16712057
$endgroup$
$begingroup$
Very succinct; well done. I'll wait a bit before accepting.
$endgroup$
– ZanyG
2 hours ago
add a comment |
$begingroup$
Suppose such a maze exists. Then the balls visits at least one square infinitely many times. Let $S$ be one such square that is closest to the bottom-right. $S$ is not the bottom-right square, so there exists a square $T$ to the right of or below $S$ that is closer to the bottom-right than $S$ is. However, the ball must visit $T$ at least once every $4$ visits to $S$ due to arrow rotation. Therefore, the ball must also visit $T$ infinitely many times, contradicting the minimality of $S$. Then no such maze exists.
answered 2 hours ago
noednenoedne
7,16712057
$endgroup$
$begingroup$
Very succinct; well done. I'll wait a bit before accepting.
$endgroup$
– ZanyG
2 hours ago
add a comment |
$begingroup$
Suppose such a maze exists. Then the balls visits at least one square infinitely many times. Let $S$ be one such square that is closest to the bottom-right. $S$ is not the bottom-right square, so there exists a square $T$ to the right of or below $S$ that is closer to the bottom-right than $S$ is. However, the ball must visit $T$ at least once every $4$ visits to $S$ due to arrow rotation. Therefore, the ball must also visit $T$ infinitely many times, contradicting the minimality of $S$. Then no such maze exists.
answered 2 hours ago
noednenoedne
7,16712057
$endgroup$
Suppose such a maze exists. Then the balls visits at least one square infinitely many times. Let $S$ be one such square that is closest to the bottom-right. $S$ is not the bottom-right square, so there exists a square $T$ to the right of or below $S$ that is closer to the bottom-right than $S$ is. However, the ball must visit $T$ at least once every $4$ visits to $S$ due to arrow rotation. Therefore, the ball must also visit $T$ infinitely many times, contradicting the minimality of $S$. Then no such maze exists.
answered 2 hours ago
noednenoedne
7,16712057
answered 2 hours ago
noednenoedne
7,16712057
answered 2 hours ago
noednenoedne
7,16712057
answered 2 hours ago
noednenoedne
7,16712057
7,16712057
$begingroup$
Very succinct; well done. I'll wait a bit before accepting.
$endgroup$
– ZanyG
2 hours ago
add a comment |
$begingroup$
Very succinct; well done. I'll wait a bit before accepting.
$endgroup$
– ZanyG
2 hours ago
$begingroup$
Very succinct; well done. I'll wait a bit before accepting.
$endgroup$
– ZanyG
2 hours ago
$begingroup$
Very succinct; well done. I'll wait a bit before accepting.
$endgroup$
– ZanyG
2 hours ago
add a comment |
Thanks for contributing an answer to Puzzling Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fpuzzling.stackexchange.com%2fquestions%2f80948%2fkeeping-a-ball-lost-forever%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
