Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0000 362296 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0580 1211568 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0580 1211568 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1738 4104816 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1752 4132208 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1760 4149920 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1760 4166928 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1768 4174664 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1768 4174664 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2433 4403920 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2449 4421320 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2455 4442248 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2551 4502520 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2551 4502776 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Smith's Magic Multiplication Squares

A magic square is a square of numbers with each row or each column adding up to the same number.
There is another 3x3 square of numbers, with some entries missing, such that the first number in each row or column multiplied by the second equals the third entry in that row or column. It so happens that we can write four numbers in the square in almost any way so that the rest of the square can be completed.
\[ \begin{array}{ccc} 2 & 3 & ? \\ 4 & 6 & ? \\ ? & ? & ? \end{array} \]

The square is completed for the first two rows with
\[2 \times 3=6\]

\[4 \times 6=24\]

We have then  
\[ \begin{array}{ccc} 2 & 3 & 6 \\ 4 & 6 & 24 \\ ? & ? & ? \end{array} \]

Now the columns.
\[ \begin{array}{ccc} 2 & 3 & 6 \\ 4 & 6 & 24 \\ 8 & 18 & 144 \end{array} \]
.
Now notice for the last row  
\[8 \times 18 =144\]
.
This always happens.
If we fill out the square differently, say  
\[ \begin{array}{ccc} 1 & ? & 7 \\ 1 & ? & 8 \\ ? & ? & ? \end{array} \]
.
Completing the columns first gives  
\[ \begin{array}{ccc} 1 & ? & 7 \\ 1 & ? & 8 \\ 1 & ? & 56 \end{array}\]
.
Completing the rows gives  
\[ \begin{array}{ccc} 1 & 7 & 7 \\ 1 & 8 & 8 \\ 1 & 56 & 56 \end{array}\]
.
Again the first enty in a row or column times the second entry in that row or column equals the third entry.