euler - bestof
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@ -52,6 +52,23 @@ $problems = Euler::listAll();
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I recommend using <a href="/programs/view/BefunUtils">BefunExec</a>. I specially made that interpreter for this project. It can run befunge code with around 6.5 MHz <i>(on my machine)</i>
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</p>
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<h1>My favorites:</h1>
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<ul>
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<li><a href="/blog/1/Project_Euler_with_Befunge/problem-101">Problem 101 - Optimum polynomial</a></li>
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<li><a href="/blog/1/Project_Euler_with_Befunge/problem-098">Problem 098 - Anagramic squares</a></li>
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<li><a href="/blog/1/Project_Euler_with_Befunge/problem-091">Problem 091 - Right triangles with integer coordinates</a></li>
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<li><a href="/blog/1/Project_Euler_with_Befunge/problem-079">Problem 079 - Passcode derivation</a></li>
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<li><a href="/blog/1/Project_Euler_with_Befunge/problem-059">Problem 059 - XOR decryption</a></li>
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<li><a href="/blog/1/Project_Euler_with_Befunge/problem-053">Problem 053 - Combinatoric selections</a></li>
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<li><a href="/blog/1/Project_Euler_with_Befunge/problem-043">Problem 043 - Sub-string divisibility</a></li>
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<li><a href="/blog/1/Project_Euler_with_Befunge/problem-017">Problem 017 - Number letter counts</a></li>
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<li><a href="/blog/1/Project_Euler_with_Befunge/problem-015">Problem 015 - Lattice paths</a></li>
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</ul>
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<h1>All solved problems</h1>
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<div class="mdtable_container">
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<table id="PEB_tableProblems" class="mdtable">
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<thead>
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@ -53,7 +53,7 @@ $max = ceil($max / 20) * 20;
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'code' => file_get_contents($problem['file_code']),
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'url' => $problem['url_raw'],
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'interactive' => !$problem['abbreviated'],
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'speed' => $problem['steps'] < 500000 ? 2 : 3,
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'speed' => $problem['steps'] < 15000 ? 1 : ($problem['steps'] < 500000 ? 2 : 3),
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'editable' => false,
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];
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echo require (__DIR__ . '/../fragments/widget_befunge93.php');
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@ -185,7 +185,6 @@ try {
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//TODO better gh widget
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//TODO remove db table prefixes
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//TODO euler insert+show 32bit | 64bit mode
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//TODO euler best of on top of list (?)
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//TODO optimize image sizes for display/download (? - auto?)
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//TODO send cache header (?)
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//TODO global uncached error logger -> send by mail
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@ -12,31 +12,29 @@ $post = Blog::getBlogpost($id);
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if ($post === NULL) httpError(404, 'Blogpost not found');
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$isSubEuler = ($post['type'] === 'euler' && $subview !== '');
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$eulerproblem = null;
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if ($isSubEuler)
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{
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require_once(__DIR__ . '/../internals/euler.php');
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$eulerproblem = Euler::getEulerProblemFromStrIdent($subview);
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}
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if ($eulerproblem === null) $isSubEuler = false;
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?>
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<head>
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<meta charset="utf-8">
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<title>Mikescher.com - Blog</title>
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<title>Mikescher.com - <?php echo ($isSubEuler ? $eulerproblem['title'] : $post['title']); ?></title>
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<meta name="google-site-verification" content="pZOhmjeJcQbRMNa8xRLam4dwJ2oYwMwISY1lRKreSSs"/>
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<link rel="icon" type="image/png" href="/data/images/favicon.png"/>
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<?php printCSS(); ?>
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<?php
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if ($post['type'] === 'euler' && $subview !== '')
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{
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require_once(__DIR__ . '/../internals/euler.php');
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$problem = Euler::getEulerProblemFromStrIdent($subview);
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if ($problem !== NULL) echo '<link rel="canonical" href="' . $problem['canonical'] . '"/>';
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}
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else
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{
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echo '<link rel="canonical" href="' . $post['canonical'] . '"/>';
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}
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?>
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<?php echo '<link rel="canonical" href="' . ($isSubEuler ? $eulerproblem['canonical'] : $post['canonical']) . '"/>'; ?>
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</head>
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<body>
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<div id="mastercontainer">
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<?php $HEADER_ACTIVE = ($post['type']==='euler' && $subview==='') ? 'euler' : 'none'; include (__DIR__ . '/../fragments/header.php'); ?>
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<?php $HEADER_ACTIVE = ($isSubEuler ? 'euler' : 'none'); include (__DIR__ . '/../fragments/header.php'); ?>
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<div id="content" class="content-responsive">
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@ -8,3 +8,117 @@ The code here was a tight fit into the Befunge-93 80x25 grid, not because of som
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Here we generate all combinations from the last digit to the first, we significantly limit the amount of possible combinations by checking the constraints for every number.
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For example we are at `???????410`, we then don't need to evaluate the first 7 digits because constraint 7 is violated (divisibility of d_789 by 17).
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This way instead of evaluating `3,628,800` possibilities we only have to look at `1702`.
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Pen-and-paper solution
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======================
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We can rule out a lot of combinations by simply looking at our rules. I divided my thought processes into steps:
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**Step 1:**
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- d_1 != d_2 != d_3 != d_4 != d_5 != d_6 != d_7 != d_8 != d_9 *(palindromic)*
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- d_1 is not `0`
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- d_234 is divisible by 2 => d_4 is even
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- d_345 is divisible by 3 => `(d_3 + d_4 + d_5) % 3 = 0`
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- d_456 is divisible by 5 => d_6 is `0` or `5`
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- d_567 is divisible by 7
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- d_678 is divisible by 11
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- d_789 is divisible by 13
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- d_89A is divisible by 17
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**Step 2:**
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if d_6 is `0` then d_678 starts with a `0`, and because d_678 is divisible by 11: `d_7 == d_8`. This conflicts with our palindrome rule.
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Therefore d_6 = `5`.
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**Step 3:**
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d_6 is `5` and d_678 is divisible by 11. There are only 8 possibilities for d_678:
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~~~
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506, 517, 528, 539, 561, 572, 583, 594
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~~~
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**Step 4:**
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d_789 is divisible by 13, together with the possibilities for d_78 there are only 4 possibilities left for d_6789:
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~~~
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5286, 5390, 5728, 5832
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~~~
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**Step 5:**
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d_89A is divisible by 17, then we can again - together with the previous result - look at the possibilities for d_6789A:
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~~~
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52867, 53901, 57289
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~~~
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**Step 6:**
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d_567 is divisible by 7, d_6 is `5` and d_7 is `2`, `3` or `7`. There are now only 2 possible values for d_56789A left:
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~~~
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357289, 952867
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~~~
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**Step 7:**
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Both values contain `2`, `5`, `7`, `8`, `9`, so d_1 - d_4 can't be any of those
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**Step 8:**
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d_345 is divisible by 3. d_5 is `3 or 9` and d_4 is even and the possible digits are `0`, `1`, `2`, `3`, `4`, `6` and `9` (`9` only for d_5).
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The possible values for d_345 are therefore:
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~~~
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063, 069, 309, 603, 609
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~~~
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And so the possible values for d_3456789A:
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~~~
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06357289, 06952867, 30952867, 60357289, 60952867
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~~~
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number 2 and 5 are not palindromic, so the numbers left are:
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~~~
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06357289, 30952867, 60357289
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~~~
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**Step 9:**
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The only numbers left are now `1` and `4` and there are no rules left to consider.
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We can now list the resulting numbers by using the 3 numbers from before and every combination of `1` and `4`:
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- 14 06357289
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- 41 06357289
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----
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- 14 30952867
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- 41 30952867
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----
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- 14 60357289
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- 41 60357289
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**Step 10:**
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The final step is now to calculate the sum of these numbers:
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~~~
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1406357289
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+ 4106357289
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+ 1430952867
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+ 4130952867
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+ 1460357289
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+ 4160357289
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= 16695334890
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~~~
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