A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property CI_URI::$config is deprecated
Filename: core/URI.php
Line Number: 101
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property CI_Router::$uri is deprecated
Filename: core/Router.php
Line Number: 127
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$benchmark is deprecated
Filename: core/Controller.php
Line Number: 75
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$hooks is deprecated
Filename: core/Controller.php
Line Number: 75
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$config is deprecated
Filename: core/Controller.php
Line Number: 75
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$log is deprecated
Filename: core/Controller.php
Line Number: 75
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$utf8 is deprecated
Filename: core/Controller.php
Line Number: 75
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$uri is deprecated
Filename: core/Controller.php
Line Number: 75
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$exceptions is deprecated
Filename: core/Controller.php
Line Number: 75
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$router is deprecated
Filename: core/Controller.php
Line Number: 75
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$output is deprecated
Filename: core/Controller.php
Line Number: 75
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$security is deprecated
Filename: core/Controller.php
Line Number: 75
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$input is deprecated
Filename: core/Controller.php
Line Number: 75
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$lang is deprecated
Filename: core/Controller.php
Line Number: 75
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$load is deprecated
Filename: core/Controller.php
Line Number: 78
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property CI_Cache::$apc is deprecated
Filename: libraries/Driver.php
Line Number: 188
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Cacher.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Cacher.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property CI_Cache::$file is deprecated
Filename: libraries/Driver.php
Line Number: 188
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Cacher.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Cacher.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$cache is deprecated
Filename: core/Loader.php
Line Number: 1279
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Cacher.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Cacher.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$cacher is deprecated
Filename: core/Loader.php
Line Number: 1279
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$jscript is deprecated
Filename: core/Loader.php
Line Number: 1279
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Return type of CI_Session_files_driver::open($save_path, $name) should either be compatible with SessionHandlerInterface::open(string $path, string $name): bool, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice
Filename: drivers/Session_files_driver.php
Line Number: 132
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Return type of CI_Session_files_driver::close() should either be compatible with SessionHandlerInterface::close(): bool, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice
Filename: drivers/Session_files_driver.php
Line Number: 290
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Return type of CI_Session_files_driver::read($session_id) should either be compatible with SessionHandlerInterface::read(string $id): string|false, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice
Filename: drivers/Session_files_driver.php
Line Number: 164
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Return type of CI_Session_files_driver::write($session_id, $session_data) should either be compatible with SessionHandlerInterface::write(string $id, string $data): bool, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice
Filename: drivers/Session_files_driver.php
Line Number: 233
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Return type of CI_Session_files_driver::destroy($session_id) should either be compatible with SessionHandlerInterface::destroy(string $id): bool, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice
Filename: drivers/Session_files_driver.php
Line Number: 313
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Return type of CI_Session_files_driver::gc($maxlifetime) should either be compatible with SessionHandlerInterface::gc(int $max_lifetime): int|false, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice
Filename: drivers/Session_files_driver.php
Line Number: 354
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 284
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: Warning
Message: session_set_cookie_params(): Session cookie parameters cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 291
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 306
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 316
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 317
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 318
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 319
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 377
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: Warning
Message: ini_set(): Session ini settings cannot be changed after headers have already been sent
Filename: drivers/Session_files_driver.php
Line Number: 108
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: Warning
Message: session_set_save_handler(): Session save handler cannot be changed after headers have already been sent
Filename: Session/Session.php
Line Number: 110
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: Warning
Message: session_start(): Session cannot be started after headers have already been sent
Filename: Session/Session.php
Line Number: 143
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$session is deprecated
Filename: core/Loader.php
Line Number: 1279
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$agent is deprecated
Filename: core/Loader.php
Line Number: 1279
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$contentsStatus is deprecated
Filename: libraries/Common.php
Line Number: 64
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$articleStatus is deprecated
Filename: libraries/Common.php
Line Number: 71
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$db is deprecated
Filename: core/Loader.php
Line Number: 390
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/models/ver3/Contents_model.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property CI_DB_mysqli_driver::$failover is deprecated
Filename: database/DB_driver.php
Line Number: 371
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/models/ver3/Contents_model.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents_model::$best_cnt is deprecated
Filename: ver3/Contents_model.php
Line Number: 17
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/models/ver3/Contents_model.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents_model::$popular_cnt is deprecated
Filename: ver3/Contents_model.php
Line Number: 18
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/models/ver3/Contents_model.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$member_model is deprecated
Filename: core/Loader.php
Line Number: 353
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/models/ver3/Contents_model.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$contents_model is deprecated
Filename: core/Loader.php
Line Number: 353
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$menuList is deprecated
Filename: libraries/Common.php
Line Number: 561
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$common is deprecated
Filename: core/Loader.php
Line Number: 1279
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$dto is deprecated
Filename: core/Loader.php
Line Number: 1279
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property Contents::$together_end_date is deprecated
Filename: ver3/Contents.php
Line Number: 17
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property CI_Cache::$dummy is deprecated
Filename: libraries/Driver.php
Line Number: 188
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/models/ver3/Contents_model.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property CI_Loader::$benchmark is deprecated
Filename: core/Loader.php
Line Number: 925
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property CI_Loader::$hooks is deprecated
Filename: core/Loader.php
Line Number: 925
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property CI_Loader::$config is deprecated
Filename: core/Loader.php
Line Number: 925
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property CI_Loader::$log is deprecated
Filename: core/Loader.php
Line Number: 925
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property CI_Loader::$utf8 is deprecated
Filename: core/Loader.php
Line Number: 925
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
Message: Creation of dynamic property CI_Loader::$uri is deprecated
Filename: core/Loader.php
Line Number: 925
Backtrace:
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
A PHP Error was encountered
Severity: 8192
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수학동아 - 폴리매스
주니어 폴리매스 문제
수학 문제를 푸는 매력에 빠져 보세요!
[KPP의 퍼즐라이프] 정이십면체를 만들어라! 키슬 퍼즐
수학동아 2020.09.26 01:39
조회 1815
KPP와 함께하는
퍼즐라이프
퍼즐러가 사는 법
퍼즐을 좋아하는 사람들이 모인 단체
KPP _Korean Puzzle Party 멤버들이
매달 재밌고 유익한 퍼즐 이야기를 들려드립니다.
KPP와 함께 신기한 퍼즐을 살펴보고
그 속에 숨은 수학도 즐겨보세요!
*사진 출처: 김우현
--------------------------------
코로나19 때문에 외출도 못하고
지루한 하루를 보내고 있나요?
그렇다면 ‘키슬 퍼즐’과 함께
슬기로운 퍼즐 생활을 즐겨봅시다!
1
키슬 퍼즐고 정이십면체
키슬 퍼즐은 체코의 퍼즐 디자이너
바츨라프 옵쉬바츠가 만든 퍼즐로
나무 구슬이 달린 조각 4개를 결합해
정이십면체 모양을 만드는 게 목표예요.
푸는 방법을 분석해보기 전에 먼저
조각의 생김새를 살펴보겠습니다.
각 조각은 모두 커다란 나무 구슬 3개가
얇은 나무 막대로 연결된 ㄱ자 모양을 하고 있습니다.
나무 구슬 3개 중에 2개는 바짝 붙어 있고,
1개는 나무 막대로 연결돼 혼자 떨어져 있죠.
정이십면체는 20개의 정삼각형으로
만들어진 입체도형이라는 사실을 알고 있죠?
한 꼭짓점 당 5개의 삼각형이 모이기 때문에
꼭짓점의 개수는 정삼각형 20개의 총 꼭짓점의 수를
5로 나눈 12(=3×20÷5)개라는 걸 알 수 있어요.
키슬 퍼즐 조각에 달린 나무 구슬도 총 12개이므로
조각을 결합했을 때 나무 구슬의 중심이
정이십면체의 꼭짓점에 위치한다는 것을 알 수 있습니다.
실제로 그렇기도 하고요.
좀더 자세히 이야기해 볼까요?
키슬 퍼즐의 ㄱ자 조각(아래)과 3개의 ㄱ자 조각을 결합한 모습(위).
2
4개의 조각을 어떻게 결합해야 할까?
ㄱ자 조각들로 정이십면체를 만들 수 있다는 건
정이십면체의 어떤 세 꼭짓점을 두 개씩 이어 2개의 변을 그렸을 때
두 변이 이루는 각이 정확히 90°라는 뜻입니다.
정이십면체는 정삼각형으로만 만들어졌는데
어디에서 직각을 찾을 수 있을까요?
비밀은 서로 마주 보는 모서리입니다!
아래 그림①처럼 정이십면체의 중심을 기준으로
서로 마주보는 두 모서리의 꼭짓점들을 이어
사각형을 만들어보세요.
그림①. 정이십면체의 마주보는 두 모서리의 양 끝점을 이으면 직사각형을 만들 수 있다.
이 사각형의 두 대각선은
정이십면체의 내부를 지나는 가장 긴 대각선이기 때문에
서로 길이가 같고, 대각선의 교점이 정이십면체의 중심과 일치합니다.
따라서 이 사각형은 직사각형이죠.
이 직사각형의 세 꼭짓점을 고르면
사잇각이 90°인 ㄱ자 모양을 만들 수 있고
이 모양을 키슬 퍼즐의 조각이라고 생각할 수 있습니다.
그림②는 퍼즐 조각 각각을 정이십면체의 어떤 꼭짓점에
대응시켜야 할지 알 수 있는 힌트이므로
퍼즐을 풀 때 이 그림을 참고해보세요!
그림②. 정이십면체를 위, 아래, 중앙으로 나눠 생각하면
ㄱ자 조각들을 어떤 꼭짓점에 대응시켜야 할지 쉽게 알 수 있다.
앞서 찾은 직사각형의 가로세로 길이 비가
황금비라는 사실도 흥미롭습니다.
그림③처럼 정이십면체의 한 모서리 길이가 1이면
직사각형의 짧은 변의 길이도 1이고, 긴 변은 한 변의 길이가 1인
정오각형의 대각선 길이와 같다는 걸 알 수 있습니다.
그림③. 정이십면체 속에 숨은 정오각형을 이용해
직사각형의 가로세로 길이의 비를 구할 수 있다.
정오각형의 한 변과 대각선의 길이의 비는 황금비를 이루므로,
직사각형의 두 변의 길이도 황금비를 이루죠.
자, 길이의 비까지 알았으니 스티로폼 공과
이쑤시개로 키슬 퍼즐을 만들어 풀어보세요!
--------------------------------
함 께 풀 어 보 세 요
1.
정이십면체에서 세 꼭짓점을 골랐을 때
ㄱ자 모양이 되는 경우의 수는 몇 가지일까요?
2.
정오각형의 한 변과 대각선의 길이의 비가
황금비를 이룬다는 사실을 증명해보세요.
3.
ㄱ자 조각 대신 그림①에 있는 직사각형의 꼭짓점에
나무 구슬을 배치해 만든 사각형 조각 3개로
정이십면체를 만들려고 합니다.
각 조각을 어떻게 배치해야 정이십면체 모양을 만들 수 있을까요?
-끝-
지금 바로
로그인 하면 내가 찜한 문제와 수행 평가를 한곳에서 볼 수 있어요!
