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: 0.0001 360720 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0933 1209368 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0933 1209368 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1617 4102376 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1631 4129928 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1639 4147640 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1639 4164648 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1647 4172384 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1647 4172384 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2269 4401632 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2284 4419032 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2289 4439960 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2379 4505616 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2379 4505872 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182 Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; JCommentsACL has a deprecated constructor in /var/www/astarmathsandphysics/components/com_jcomments/classes/acl.php on line 17 Call Stack: 0.0001 360720 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0933 1209368 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0933 1209368 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1617 4102376 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1631 4129928 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1639 4147640 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1639 4164648 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1647 4172384 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1647 4172384 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2269 4401632 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2284 4419032 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2289 4439960 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.3578 12294168 13. JEventDispatcher->trigger() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:199 0.3581 12294616 14. plgContentJComments->update() /var/www/astarmathsandphysics/libraries/joomla/event/dispatcher.php:160 0.3581 12294616 15. plgContentJComments->onContentAfterDisplay() /var/www/astarmathsandphysics/libraries/joomla/event/event.php:70 0.3583 12302864 16. plgContentJComments->onAfterDisplayContent() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:339 0.3586 12304600 17. JComments::show() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:282 0.3592 12337648 18. JCommentsFactory::getACL() /var/www/astarmathsandphysics/components/com_jcomments/jcomments.php:188 0.3592 12338048 19. spl_autoload_call() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274 0.3593 12338128 20. JLoader::load() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274

The Seven Bridges of Konigsberg

The seven bridges of Konigsberg is a famous problem in graphtheory. The city of Konigsberg in East Prussia had seven bridgescrossing the river Pregel, some to an island, before crossing anotherbridge to the other side of the river.

The problem is to find a route crossing each bridge exactly onceand was solved in 1736 by Leonard Euler. He redrew the map above as anetwork of arc repressing a bridge crossing and nodes representingeach piece of land, drawing a diagram similar to the one below.

Euler observed that when a vertex is visited in while tracing thegraph, there must be an edge entering the vertex and another edgeleaving it and so the order of the vertex must be an even number. This must be true for all but at most two of the vertices - thestart and end vertices and so a connected graph is traversable if andonly if it has two vertices of odd order, unless the start and endvertices are the same in which case every vertex has even order. Thediagram above shows four vertices of odd order so the network is nottraversable and the each bridge cannot be crossed exactly once.

If an extra bridge is added a the network may become traversable.One solution and route is shown blow. The route starts at A and endsat B.

Add comment

Security code
Refresh