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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <title>Unconditional FizzBuzz</title> <style type="text/css"> table { border-collapse: collapse; } table, th, td { border: 1px solid black; } th { background-color: #0FF; } </style> </head> <body> <h1>Unconditional FizzBuzz!</h1> <p>The "secret" to Unconditional FizzBuzz is modular arithmetic. For any number V, there are three possible values of V mod 3: 0,1 and 2. There are five possible values of V mod 5: 0, 1, 2, 3 and 4. Combined, this gives 15 possible combinations of the two. Mathematically, we can create unique values for these 15 combinations by the formula X = (V mod 3) * 5 + (V mod 5). The following table shows this for the numbers 1 through 15:</p> <table width="400"> <tbody> <tr> <th align="right">Value (V)</th> <th align="right">X = V mod 3</th> <th align="right">Y = V mod 5</th> <th align="right">X * 5 + Y</th> <th></th> </tr> <tr> <td align="right">1</td> <td align="right">1</td> <td align="right">1</td> <td align="right">6</td> <td><br> </td> </tr> <tr> <td align="right">2</td> <td align="right">2</td> <td align="right">2</td> <td align="right">12</td> <td><br> </td> </tr> <tr> <td align="right">3</td> <td align="right">0</td> <td align="right">3</td> <td align="right">3</td> <td>Fizz</td> </tr> <tr> <td align="right">4</td> <td align="right">1</td> <td align="right">4</td> <td align="right">9</td> <td><br> </td> </tr> <tr> <td align="right">5</td> <td align="right">2</td> <td align="right">0</td> <td align="right">10</td> <td>Buzz</td> </tr> <tr> <td align="right">6</td> <td align="right">0</td> <td align="right">1</td> <td align="right">1</td> <td>Fizz</td> </tr> <tr> <td align="right">7</td> <td align="right">1</td> <td align="right">2</td> <td align="right">7</td> <td><br> </td> </tr> <tr> <td align="right">8</td> <td align="right">2</td> <td align="right">3</td> <td align="right">13</td> <td><br> </td> </tr> <tr> <td align="right">9</td> <td align="right">0</td> <td align="right">4</td> <td align="right">4</td> <td>Fizz</td> </tr> <tr> <td align="right">10</td> <td align="right">1</td> <td align="right">0</td> <td align="right">5</td> <td>Buzz</td> </tr> <tr> <td align="right">11</td> <td align="right">2</td> <td align="right">1</td> <td align="right">11</td> <td><br> </td> </tr> <tr> <td align="right">12</td> <td align="right">0</td> <td align="right">2</td> <td align="right">2</td> <td>Fizz</td> </tr> <tr> <td align="right">13</td> <td align="right">1</td> <td align="right">3</td> <td align="right">8</td> <td><br> </td> </tr> <tr> <td align="right">14</td> <td align="right">2</td> <td align="right">4</td> <td align="right">14</td> <td><br> </td> </tr> <tr> <td align="right">15</td> <td align="right">0</td> <td align="right">0</td> <td align="right">0</td> <td>FizzBuzz</td> </tr> </tbody> </table> <p>We can sort the table to show the relationships between the mod values a bit more clearly:</p> <table width="400"> <tbody> <tr> <th align="right">Value (V)</th> <th align="right">X = V mod 3</th> <th align="right">Y = V mod 5</th> <th align="right">X * 5 + Y</th> <th></th> </tr> <tr> <td align="right">15</td> <td align="right">0</td> <td align="right">0</td> <td align="right">0</td> <td>FizzBuzz</td> </tr> <tr> <td align="right">6</td> <td align="right">0</td> <td align="right">1</td> <td align="right">1</td> <td>Fizz</td> </tr> <tr> <td align="right">12</td> <td align="right">0</td> <td align="right">2</td> <td align="right">2</td> <td>Fizz</td> </tr> <tr> <td align="right">3</td> <td align="right">0</td> <td align="right">3</td> <td align="right">3</td> <td>Fizz</td> </tr> <tr> <td align="right">9</td> <td align="right">0</td> <td align="right">4</td> <td align="right">4</td> <td>Fizz</td> </tr> <tr> <td align="right">10</td> <td align="right">1</td> <td align="right">0</td> <td align="right">5</td> <td>Buzz</td> </tr> <tr> <td align="right">1</td> <td align="right">1</td> <td align="right">1</td> <td align="right">6</td> <td><br> </td> </tr> <tr> <td align="right">7</td> <td align="right">1</td> <td align="right">2</td> <td align="right">7</td> <td><br> </td> </tr> <tr> <td align="right">13</td> <td align="right">1</td> <td align="right">3</td> <td align="right">8</td> <td><br> </td> </tr> <tr> <td align="right">4</td> <td align="right">1</td> <td align="right">4</td> <td align="right">9</td> <td><br> </td> </tr> <tr> <td align="right">5</td> <td align="right">2</td> <td align="right">0</td> <td align="right">10</td> <td>Buzz</td> </tr> <tr> <td align="right">11</td> <td align="right">2</td> <td align="right">1</td> <td align="right">11</td> <td><br> </td> </tr> <tr> <td align="right">2</td> <td align="right">2</td> <td align="right">2</td> <td align="right">12</td> <td><br> </td> </tr> <tr> <td align="right">8</td> <td align="right">2</td> <td align="right">3</td> <td align="right">13</td> <td><br> </td> </tr> <tr> <td align="right">14</td> <td align="right">2</td> <td align="right">4</td> <td align="right">14</td> <td><br> </td> </tr> </tbody> </table> <p>You can easily see that four of these combinations indicate a number divisible by 3 but <em>not</em> 5; two of these combinations indicate a number divisible by 5 but <em>not</em> 3 and a single combination indicating a number divisible by <em>both</em> 3 and 5.</p> <p>Finally, we can easily see from the "X * 5 + Y" column that the value computed from the two mod results can be used to index into a table of "tags" that can be used to identify the "Fizziness" of a value!</p> </body> </html>
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The "FizzBuzz" application without ANY conditional tests!
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