Problem 1494. Hungry Snake

Created by Yaroslav

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It's quite hard to finish this simple game; nonetheless</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://en.wikipedia.org/wiki/Chuck_Norris_facts\"><w:r><w:t>Chuck Norris</w:t></w:r></w:hyperlink><w:r><w:t> has accomplished the task in the most difficult level, naturally. Now, since the game was</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>too easy</w:t></w:r><w:r><w:t> for him, he did it with the highest number of turns possible. We shall now inspect Chuck Norris' solution.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Suppose you have a 2ª×2ª matrix</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>M</w:t></w:r><w:r><w:t> (with an integer</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>a</w:t></w:r><w:r><w:t> ≥ 0). The path of the snake is denoted with consecutive numbers 1÷4ª. The matrix</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>M</w:t></w:r><w:r><w:t> must obey the following conditions:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr></w:pPr><w:r><w:t>All the numbers between 1 to 4ª exist</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>once</w:t></w:r><w:r><w:t> in a 2ª×2ª matrix.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr></w:pPr><w:r><w:t>These numbers form a snake; i.e., each number</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>n</w:t></w:r><w:r><w:t> must be adjacent to both</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>n</w:t></w:r><w:r><w:t> -1 and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>n</w:t></w:r><w:r><w:t> +1 (with the obvious exception of 1 and 4ª).</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr></w:pPr><w:r><w:t>There</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>cannot</w:t></w:r><w:r><w:t> be more than 4 consecutive numbers in a row or a column.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Hints</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>Since Chuck Norris can draw an infinite fractal, you may want to check the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://en.wikipedia.org/wiki/Hilbert_curve\"><w:r><w:t>Hilbert Curve</w:t></w:r></w:hyperlink><w:r><w:t> or other</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://en.wikipedia.org/wiki/Space-filling_curves\"><w:r><w:t>Space-filling curves</w:t></w:r></w:hyperlink><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Examples</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ hungry_snake(0)\n ans = \n 1\n\n hungry_snake(1)\n ans = \n 1 2\n 4 3\n\n hungry_snake(2)\n ans = \n 1 4 5 6\n 2 3 8 7\n 15 14 9 10\n 16 13 12 11]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Bad Solutions</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a=1; eye(2^a)</w:t></w:r><w:r><w:t> — doesn't have all the numbers 1 to 4.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a=2; reshape(1:4^a,2^a,2^a)</w:t></w:r><w:r><w:t> — 4 and 5 aren't adjacent.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a=3; spiral(2^a)</w:t></w:r><w:r><w:t> — has 8 consecutive numbers in a row.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The usual cheats</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>are not</w:t></w:r><w:r><w:t> allowed!</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

Who hasn't played <a href = "http://en.wikipedia.org/wiki/Snake_%28video_game%29">the Snake game</a> when they were little? It's quite hard to finish this simple game; nonetheless <a href = "http://en.wikipedia.org/wiki/Chuck_Norris_facts">Chuck Norris</a> has accomplished the task in the most difficult level, naturally. Now, since the game was too easy for him, he did it with the highest number of turns possible. We shall now inspect Chuck Norris' solution.Suppose you have a 2&ordf;&times;2&ordf; matrix M (with an integer a &ge; 0). The path of the snake is denoted with consecutive numbers 1&divide;4&ordf;. The matrix M must obey the following conditions:<ol><li>All the numbers between 1 to 4&ordf; exist once in a 2&ordf;&times;2&ordf; matrix.</li><li>These numbers form a snake; i.e., each number n must be adjacent to both n -1 and n +1 (with the obvious exception of 1 and 4&ordf;).</li><li>There cannot be more than 4 consecutive numbers in a row or a column.</li></ol>Hints<ul><li>Since Chuck Norris can draw an infinite fractal, you may want to check the <a href = "http://en.wikipedia.org/wiki/Hilbert_curve">Hilbert Curve</a> or other <a href = "http://en.wikipedia.org/wiki/Space-filling_curves">Space-filling curves</a>.</li></ul>Examples<pre> hungry_snake(0)
 ans = 
 1</pre><pre> hungry_snake(1)
 ans = 
 1 2
 4 3</pre><pre> hungry_snake(2)
 ans = 
 1 4 5 6
 2 3 8 7
 15 14 9 10
 16 13 12 11</pre>Bad Solutions<ul><li><tt>a=1; eye(2^a)</tt> &mdash; doesn't have all the numbers 1 to 4.</li><li><tt>a=2; reshape(1:4^a,2^a,2^a)</tt> &mdash; 4 and 5 aren't adjacent.</li><li><tt>a=3; spiral(2^a)</tt> &mdash; has 8 consecutive numbers in a row.</li></ul>The usual cheats are not allowed!

Who hasn't played <http://en.wikipedia.org/wiki/Snake_%28video_game%29 the Snake game> when they were little? It's quite hard to finish this simple game; nonetheless <http://en.wikipedia.org/wiki/Chuck_Norris_facts Chuck Norris> has accomplished the task in the most difficult level, naturally. Now, since the game was _too easy_ for him, he did it with the highest number of turns possible. We shall now inspect Chuck Norris' solution.

Suppose you have a 2&ordf;&times;2&ordf; matrix _M_ (with an integer _a_ &ge; 0).  The path of the snake is denoted with consecutive numbers 1&divide;4&ordf;.  The matrix _M_ must obey the following conditions:

# All the numbers between 1 to 4&ordf; exist _once_ in a 2&ordf;&times;2&ordf; matrix.
# These numbers form a snake; i.e., each number _n_ must be adjacent to both _n_ -1 and _n_ +1 (with the obvious exception of 1 and 4&ordf;).
# There _cannot_ be more than 4 consecutive numbers in a row or a column.

*Hints*

* Since Chuck Norris can draw an infinite fractal, you may want to check the <http://en.wikipedia.org/wiki/Hilbert_curve Hilbert Curve> or other <http://en.wikipedia.org/wiki/Space-filling_curves Space-filling curves>.

*Examples*

hungry_snake(0)
 ans = 
     1

hungry_snake(1)
 ans = 
     1     2
     4     3

hungry_snake(2)
 ans = 
     1     4     5     6
     2     3     8     7
    15    14     9    10
    16    13    12    11

*Bad Solutions*

The usual cheats *are not* allowed!

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Problem 1494. Hungry Snake

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