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
멘토링&인터뷰
폴리매스 활약한 회원들을 만나보세요!
[폴리매스 멘토링] (멘토링) 사상 최초! 멘토링에서 폴리매스 문제 탄생!
수학동아 2018.12.29 10:28
조회 2562
참 오랜만이죠~?
폴리매스 멘토링 현장 스케치
2018년 11월, 국가수리과학연구소에서 여섯 번째 폴리매스 멘토링이 열렸어요.(무려 1년만!) ‘해결’ 가뭄에 허덕이던 폴리매스 마을에 단비를 내려준 문제는 국가수리과학연구소 19, 20, 21번 문제입니다. 9~12월 불과 4개월 사이 문제가 잇달아 풀리면서 해가 넘어가기 전에 멘토링을 할 수 있었는데요. 문제를 푼 친구가 전부 참석하지 못했지만, 수돌이 친구(19, 21번)와 수학자 친구(21번), 멘토링 기회를 묵혀뒀던 shine 친구(9.5번) 그리고 예전에 문제를 풀었는데 멘토링에 참가하지 못했던 디듀우 친구(8번)가 멘토링에 참여했어요.
디듀우 친구는 첫 멘토링이고 수학자 친구는 두 번째, shine 친구와 수돌이 친구는 대한수학회 멘토링까지 포함해 무려 세 번째 멘토링이었는데요! 베테랑(?)답게 김 기자보다 앞장서서 국가수리과학연구소로 향했습니다. 백진언 연구원과 커다란 칠판 앞에 모여앉아 자기가 푼 문제와 풀이, 아이디어를 발표할 준비를 마친 그들. 어떤 이야기들이 오갔는지 한번 살펴볼까요?
<1> 밝혀진 8번 문제의 정체! 첫 번째 문제는 2017년 7월 출제한 8번 ‘핑순이의 단어 놀이’(www.polymath.co.kr/math/62)입니다. 친구들이 풀이를 설명하기 전 백진언 연구원은 문제를 낸 이유에 관해 설명해줬는데요. 여기서 재밌는 사실이 밝혀졌습니다. 사실 8번 문제는 알파벳을 몇 가지 규칙에 따라 조합해 단어를 만드는 문제가 아니라 평면에 원과 곡선이 서로 만날 때 생기는 교점을 수학적으로 표현하려고 낸 문제였다는 것! 백진언 연구원이 복잡한 도형 문제를 재밌게 바꾼 것이었습니다.(배려 뿜뿜) 그렇다면 원과 곡선의 교점과 단어 만들기 사이에 어떤 관계가 있을까요? 아래 백진언 연구원의 설명을 보고 도형 문제로 바꿔 풀어보세요~.
각 알파벳은 원 안에 놓인 '곡선 1개'를 나타냅니다.
원주 위를 한 방향으로 돌 때 만나는 교점이 어떤 곡선 위의 점인지 읽으면 단어가 만들어져요!
'단어 합치기'는 두 원을 하나로 붙여 단어를 만드는 것,
'중복인 두 글자 없애기'는 단어를 합쳤을 때 중복되는 곡선은 하나만 남기는 거예요!
"문제를 낼 때 몇몇 친구들이 금방 해결하지 않도록 문제를 내려고 해.
그러다 보면 늘 새로운 문제를 만들어낼 순 없고 기존에 알려진 문제를 변형하거나 참고하는데
8번 문제는 100% 새로 만든 문제야~!"
(백)
"중학생이라 아직 배우지 않았지만,
책에서 경우의 수를 계산하는 방법(조합)을 이용해 풀었어요"
(디듀우)
"저녁 7시부터 8시 40분까지 풀었던 거로 기억해요.
저도 문제를 풀 때 단어의 처음과 끝을 이어 원으로 배치해서 풀었는데,
각 알파벳이 원 위의 교점들이었네요!"
(수돌이)
"1시간 40분 만에? 후덜덜"
(일동)
<2> 9.5번 문제와 수돌이의 팩트 폭행(?) 9.5번 ‘제설기의 마법’(www.polymath.co.kr/math/73)는 shine 친구가 7번 문제의 핵심 아이디어였던 ‘불변량’을 이용해 가뿐히 해결했습니다. shine 친구는 직접 ‘구멍’이라는 불변량을 정의해 문제를 해결했다고 해요. shine 친구는 양쪽에 눈이 있고 그 자리에는 아무것도 없는 상태를 ‘구멍’이라고 정의한 뒤 구멍이 두 칸 연속 있을 수 없다는 사실을 증명해 문제를 해결했답니다.
"종종 문제가 빨리 풀리면 추가 문제를 내는데,
9.5번 문제는 9번 문제를 수돌이 친구가 금방 풀어버렸다고 해서 추가로 낸 문제란다"
(백)
"제가요?"
(수돌이둥절)
"이 문제는 ‘이렇게 하면 풀릴 것 같네’라고 감은 오는데
설명하기는 무~척 귀찮은, 그런 문제였어요"
(수학자)
"집은 무한하고 눈은 유한하니까 눈을 양쪽으로 퍼뜨리다 보면
균형을 이루는 단계가 오지 않을까요?"
(디듀우)
"예스! 그걸 수학적으로 설명한 게 바로
‘불’ to the ‘변’ to the ‘량’!"
(shine)
여기 펄.펙.트.한 설명 받으시죠! 수학이 얼만큼 재밌냐면 말이지~.
펜을 든 수돌이 친구! 갑자기 멋짐이 폭발한다!
<3> 뛰어봤자 벼루...ㄱ...ㅐ구리! 19번 ‘도망자 개구리’(www.polymath.co.kr/math/162)는 백진언 연구원이 페이스북에 올라온 소문제 1번을 보고 만든 문제예요. 개구리의 상태(처음 위치, 이동 거리)를 순서쌍으로 나타낸 뒤, 손을 뻗는 위치와 각 상태 사이의 일대일대응을 찾아야 합니다. 손을 뻗을 때마다 개구리가 계속 이동하니까 개구리가 어떤 상태이든 간에 시간이 지나면 잡을 수 있도록 손 뻗는 위치를 찾는 게 핵심이었지요. 사실 소문제 3번은 백진언 연구원도 답을 구해보지 않고 낸 문제인데, 수돌이 친구가 간단해 보이지만, 오묘한 부분을 수학적으로 잘 풀었다고 해요. 풀이에 아주 만족!
"중학생인 디듀우 친구도 제 풀이를 이해하도록 설명해보도록 하죠.
2번은 가산 집합과 불가산 집합을 알아야 해서 좀 어려울 수도 있을 것 같거든요.
그러니까 이 문제는 가갸거겨고교우유…"
(수돌이)
"저, 가산집합이 뭔지 알아요"
(영재교육원 다니는 디듀우)
"음!?"
(일동)
"소문제 2번에서 처음 위치와 이동 거리가 정수가 아닌 ‘실수’면 잡는 게 불가능하다고 했는데,
손을 뻗을 때 점이 아닌 구간 안에 있는 개구리를 잡을 수 있다면 어떨까요?"
(디듀우)
"호오오오. 괜찮은 문제가 될 것 같은데요."
(일동)
그냥 하면 재미없으니 n번째 손을 뻗을 때마다 구간의 길이가 1/n로 줄면 어떨까?
(백)
"앗! 아아…. "
(일동 질색)
<4> 백진언 연구원의 큰 그림 21번 ‘내 맘대로 숫자 레시피’(www.polymath.co.kr/math/181)는 주어진 숫자 사이에 수학 기호를 넣어 등식을 만드는 문제에서 착안해 만든 문제예요. 문제 자체로도 의미가 있지만, 사실 더 깊은 뜻이 숨어있죠. 백진언 연구원이 프로그래밍으로 수학 문제를 풀 때 ‘교차 검증’이 필요하다는 사실을 알려주려고 만든 문제거든요. 그래서 답이 나온 뒤에도 ‘해결’ 딱지를 바로 붙이지 않고 다른 친구들이 다른 방식으로 프로그래밍했을 때도 같은 답이 나오는지 확인한 뒤 해결 딱지를 붙였답니다. 단순 프로그래밍 문제인 줄 알았는데 백진언 연구원이 큰 그림을 그리고 있었네요! 화가 뺨침!
"명령어 순서 하나만 바꿔도 오류가 나는 게 바로 프로그래밍이야.
프로그래밍으로 푼 수학 난제 ‘4색 문제’도 모든 경우를 다 고려했음에도
검증 단계에서 오류가 발견됐었지"
(백)
"맞아요. 문제를 프로그래밍으로 풀 때 가장 중요한 건 ‘모든 경우를 다루고 있는지’ 확인하는 거예요.
비슷하게 프로그래밍 문제를 만들 때도 마찬가지고요!
(수학자)
비록 답이 맞았어도 다른 친구도 풀어서 교차 검증을 하길 바라는 마음에서 바로 ‘해결’ 딱지를 붙이지 않았어.
문제를 푼 수돌이, 수학자 친구 심지어 나도 틀렸을지 모르잖아?"
(백)
(일동 강한 긍정의 끄덕임)
쉬라니깐! 2.5번 문제에 관해 토론하는 멘티들. 국가수리과학연구소 너무나 신기방기!
오랜만에 열린 멘토링 현장, 재밌었나요? 열정을 다해 폴리매스 문제, 수학에 관해 대화를 나눈 4명의 멘티는 저녁 식사 자리에서도 대화를 그칠 줄 몰랐답니다. 과학고 및 대학 입시에 관한 것부터 수학 공부 방법, 최근 폴리매스 문제 관한 얘기 등 재밌는 이야기가 많이 오갔는데요. 특히 교내 프로그래밍 경시대회에서 문제 출제진으로 참여했다는 수학자 친구는 폴리매스배 프로그래밍 대회를 열자는 제안을 하기도 했어요.
이번 멘토링이 열리고 난 직후 대한수학회 2번과 21번 문제, 국가수리과학연구소 24번 문제가 풀리면서 곧 다음 멘토링이 열릴 예정이랍니다. 정답을 찾지 않아도 괜찮아요. 엉뚱하고 기발한 아이디어만으로도 정답자가 될 수 있으니 열심히 도전해보세요!
< 멘토링 후기! >
과연 형들은 정말 대단했다. 열심히 공부해서 형들처럼 되고 싶다.
앞으로 더 많은 사람들과 멘토링을 했으면 좋겠다~.
-디듀우
‘모두가 힘을 합쳐 문제를 푼다’는 건 항상 흥미롭다.
문제를 푸는 것 자체도 즐겁지만, 다른 사람의 의견을 들어볼 수 있다는 건 더 즐겁다.
-shine
폴리매스 문제를 풀 때 앞서 올라온 댓글을 읽고 시작한다.
친구들의 아이디어를 보고 쉽게 푼 문제도 많고 반대로 내 아이디어로 다른 사람이 풀이를 완성하기도 했다.
함께 연구한다는 것, 여러 사람의 힘을 합친다는 건 정말 대단한 것 같다.
문제도 참신해서 연구에 활용하기 좋다.
-수돌이
중·고등학생 친구를 보니 ‘늙었다’는 생각도 들었다.
수학 문제를 열심히 푸는 친구들과 함께 수학 이야기를 하면서
앞으로 더 많은 사람이 폴리매스 프로젝트에 참여했으면 좋겠다는 생각이 들었다.
- 수학자
-끝-
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