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
Line: 315
Function: require_once

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
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 22
Function: driver

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Cacher.php
Line: 17
Function: initiateCache

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 22
Function: driver

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Cacher.php
Line: 17
Function: initiateCache

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 22
Function: driver

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Cacher.php
Line: 17
Function: initiateCache

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 64
Function: _error_handler

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 24
Function: initContentsStatusInfo

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 71
Function: _error_handler

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 24
Function: initContentsStatusInfo

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 15
Function: database

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 440
Function: model

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 29
Function: initMenu

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 15
Function: database

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 440
Function: model

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 29
Function: initMenu

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 17
Function: _error_handler

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 440
Function: model

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 29
Function: initMenu

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 18
Function: _error_handler

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 440
Function: model

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 29
Function: initMenu

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 20
Function: model

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 440
Function: model

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 29
Function: initMenu

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 440
Function: model

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 29
Function: initMenu

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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
Line: 561
Function: _error_handler

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 29
Function: initMenu

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: 8192

Message: Creation of dynamic property Contents::$common is deprecated

Filename: core/Loader.php

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 14
Function: __construct

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 14
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Line: 315
Function: require_once

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Line: 799
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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Function: view

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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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Line: 387
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
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File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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Function: require_once

수학동아 - 폴리매스
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주니어 폴리매스 문제
수학 문제를 푸는 매력에 빠져 보세요!
[방승진 교수의 수학 퍼즐] 방1. 역삼각형 숫자 퍼즐
수학동아 2018.08.01 01:10 조회 3486
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페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

주니어 폴리매스를 시작하면서 아주대 수학과 방승진 교수님은 다음과 같은 조언을 해주셨습니다. 앞으로 교수님의 의견에 따라 문제를 풀어 보세요!

요즘 수학 문제는 일회용 물수건처럼 쓰고 버리는 것이라는 인상을 지울 수 없습니다. 학생들은 셀 수 없을 정도로 많은 수학 문제를 풀고 또 풀지만 수학 실력은 늘어나지 않습니다. 왜 일까요? 머리가 나빠서 일까요? 연구에 따르면 학생들은 누구나 잠재력이 있다고 합니다. 수학도 마찬가지 아닐까요?

제 경험에 의하면 수학 문제를 마치 껌 씹듯이 생각에 생각을 거듭하면 다양한 생각을 하게 되고 어느 덧 그 수학 문제는 영원히 잊지 않을 정도로 친숙하게 됩니다. 좀 더 정확하게 "어떤 문제를 푸는가도 중요하지만 문제를 어떻게 푸는가?" 가 더 중요합니다. 결국 문제를 풀고 난 뒤에 '어떤 창의적인 산출물을 만들었는가?' 가 중요합니다.

앞으로 게재하는 문제들은 주로 다양하게 생각하도록 유도하는 문제 즉, 개방형 문제(open-ended problem)로서 여러분들이 좀 더 창의적인 태도를 가질 수 있도록 안내할 겁니다.

모든 문제는 주어진 질문에 국한하지 말고 될 수 있는 한 일반화 시키고, 남들과는 다르게 생각하려고 노력해야 좋습니다. 수학 논문을 쓰면 더욱 좋습니다.

 

문제 <그림 2>에서 원이 의 모양으로 놓인 곳은 어디나 A와 B의 차가 C가 된다. 또 6개의 ○안에는 1에서 6까지의 정수가 모두 한 번씩 들어가 있다. 이 규칙에 따라 <그림2>처럼 맨 아래 ○안에 숫자가 3일 때, 10개의 ○안에 1에서 10까지의 정수를 한 번씩만 사용해서 채워라.

이런 경우를 4가지 찾고, 3 대신 다른 수를 써도 되는지도 조사하여라. .

문제 찜하기 목록보기
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  •  
    쓰다만연습장 Lv.5 2018.08.01 10:06

    1-1은 많이 도전을 해보았으나 안되서 일단 패스....

    1-2를 보면 일단 10은 1~9 중에서 그 어느 수의 차로도 나타낼 수 없으니 불가능하고,

    9는 10 - 1의 경우만 가능한데, 이 역시 10을 그 어느 수의 차로 나타낼 수 없으니 불가능하고,

    8은 10 - 2와 9 - 1의 경우만 가능합니다. 그러나 10 - 2는 위의 이유와 같은 이유로 불가능하고, 9 - 1의 경우는 9가 10 - 1의 경우로만 가능한데 1을 이미 써버렸기 때문에 불가능하고,

    7은 10 - 3, 9 - 2, 8 - 1의 경우만 가능합니다. 그러나 10 - 3은 불가능하고, 9 - 2는 위의 이유과 같은 이유로 불가능하고.........

     

    요롷게 해보면 4에서 가능한 거 같은데.......(위에서부터 8, 10, 3, 9, 2, 7, 6, 5, 1, 4)

    저도 헷갈리니깐 제가 미쳐 못 봤거나 잘못된 부분은 지적해주세요!!

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    •  
      수학짱 Lv.1 2018.08.01 20:33

      이어서 계속 해보면 6에서는 7  1, 8  2, 9  3이 가능한데(10 은 어차피 안돼니까 빼겠습니다.) 7은 또 8  1. 9 2 인데 1은 이미 써서 안돼고 9 2에서 9는 10  1이지만 10을 써서 안됍니다.

      그리고 5인 경우는 6  1,7 2,8  3,9  4입니다.                              6은 7 1, 8  2, 9  3인데 1은 써서 안돼고,8도 9  1인데 불가능하고 9는 10  1이므로 불가능합니다.      

      4인 경우는 코딩의 달인님께서 말씀해주신 것처럼 되고,3인 경우도 있습니다.

      밑이 2인 경우에는 9  7, 8  6, 7  5,6  4,5  3,3  1이  있는데 9,8은 위 경우처럼 불가능합니다.더 계산을 해야겠네요!                            

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  •  
    21세기오일러 Lv.11 2018.08.01 18:58

    주니어폴리매스는 나이제한이 있나요?

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    •  
      조가현 편집장 Lv.15 2018.08.01 19:31

      나이 제한은 없습니다. 다만 실력이 너무 뛰어난 분이 와서 단번에 답을 내면 서로 의견을 나누며 문제를 풀 수가 없기 때문에 답을 바로 아실 경우 비밀댓글로 달아주세요. 그리고 다른 참여자분들에게 코멘트를 해주시면 좋을 것 같습니다. ^^* 앞으로 많은 관심 부탁드립니다.

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  •  
    구머 Lv.6 2018.08.01 19:03

    이거 이번 imo문제 이닌가 ㅋㅋ

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    •  
      21세기오일러 Lv.11 2018.08.01 19:42

      역파스칼삼각형 뒤집으면 답나오네요.

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  •  
    21세기오일러 Lv.11 2018.08.01 19:46

    이런경우를 4가지 찾으란 것은 밑이 3일때 4가지 찾는 것인가요?

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  •  
    수학짱 Lv.1 2018.08.01 20:02

    일단 1-1에서 두가지 만 찾으면 뒤집어서 네가지로 만들 수 있을것 같네요

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    •  
      수학짱 Lv.1 2018.08.01 20:05

      그리고 3이 아래 꼭짓점인 경우는 위에서부터 

      6  10  1  8  4  9  7  5  2  3,    8  1  10  6  7  9  4  2  5  3(첫번째 뒤집은 것)이 있네요

       

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    •  
      수학짱 Lv.1 2018.08.01 20:11

      그리고 마지막 두개는 위에서 부터

      8  10  1  6  2  9  5  7  4  3와  6  1  10  8  5  9  2  4  7  3(첫번째 뒤집은것)이 있네요!

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    •  
      math Lv.2 2018.08.01 20:16

      6 10 9 2

        4 1 7

         5 8

          3

       

      4 10 9 2

        6 1 7

         5 8

          3

      이런 구조들도 있네요.

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    •  
      수학짱 Lv.1 2018.08.01 20:42

      그런데 math님의 댓글에서 7-1=8이라는 것이 잘못되지 않았나요?

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    •  
      math Lv.2 2018.08.01 20:58

      그렇네요 ㅠㅠ

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  •  
    수학짱 Lv.1 2018.08.01 20:19

    뒤집어서 똑같아지는 경우는 1개로 보나요?

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  •  
    21세기오일러 Lv.11 2018.08.01 22:01

    밑이 4이면

    8 3 10 9

      5 7 1

        2 6

          4

    가뇌네요.

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  •  
    쓰다만연습장 Lv.5 2018.08.02 07:42

    그럼 맨 아래가 1인것도 한 번 봅시돱!

    그리고 교수님이 말씀하시길 문제를 일반화 하는 것도 좋은 방법이라고 하시니

    일반화에 대해서도 같이 고민해 봅시다.

    저도 조금(진짜 조금) 고민해 봤는데, 맨 아래가 n이라고 하면,

    그 위에에 올 수 있는 경우의 수는 9 - n(어차피 10이 오면 안되니까 미리 하나 빼놓을께요)인데, 그 위에꺼는 제 머리에 한계가 왔네요..

    그리고 제 생각에는 맨 아랫줄부터 세 번쨰 줄까지 6개로 이루어진 역삼각형 있잖아요 거기에 1, 2, 3이 다 들어가면 삼각형 만들기가 불가능 한 거 같은데..

    혹시 제가 틀린 부분이나 잘못된 부분이 있다면 꼭! 알려주세요!

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  •  
    쓰다만연습장 Lv.5 2018.08.02 07:49

    그럼 한번 지금까지 한 것 좀 정리해 봅시다.

    1 - 1은

     8 10 1 6   6 1 10 8   6 10 1 8   8 1 10 6

       2 9 5       5 9 2       4 9 7       7 9 4

        7 4          4 7         5 2          2 5

         3             3           3             3

    요롷게 해서 된 것 같구요

    1 - 2를 보면

    10, 9, 8, 7, 6, 5는 불가능하다는 것에 증명됬고, 4, 3, 2 는 가능합니다. 그러나 1은 아직까지 증명이 되지 않았습니다!

    우리모두 힘내서 이거 끝내고 정사각형 채우기로 갑시다!!!

     

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    •  
      math Lv.2 2018.08.02 08:42

      1 수작업으로 해보니까 안되는 거 같은데요?

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  •  
    쓰다만연습장 Lv.5 2018.08.02 19:51

    그..럼  다 푼건가요?

     

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    •  
      쓰다만연습장 Lv.5 2018.08.02 19:52

      1 - 1은

       8 10 1 6   6 1 10 8   6 10 1 8   8 1 10 6

         2 9 5       5 9 2       4 9 7       7 9 4

          7 4          4 7         5 2          2 5

           3             3           3             3

      요롷게 해서 된 것 같구요

      1 - 2를 보면

      10, 9, 8, 7, 6, 5, 1는 불가능하다는 것에 증명됬고, 4, 3, 2 는 가능합니다. 요게 답인가요?

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    •  
      math Lv.2 2018.08.02 20:39

      그런데 뭔가 찝찝하네요... 각 숫자들을 문자들로 바꿔서 왜 안되고, 왜 되는지도 증명을 해야하지 않을까요?

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    •  
      21세기오일러 Lv.11 2018.08.02 23:02

      밑이2도 불가능하지않나요?

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    •  
      수학짱 Lv.1 2018.08.02 23:39

      제가 계산한 결과  1,2  는 불가능하다는 결론이 나오네요.

      그럼  9.8.7.6.5.2.1은 불가능하고  3.4는 되는 것  같네요!

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  •  
    21세기오일러 Lv.11 2018.08.02 23:03

    3대신은 4밖에안됩니다.

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  •  
    쓰다만연습장 Lv.5 2018.08.03 20:02

    근데... 왜 정사각형 채우기랑 다르게 주니어 맨토님의 맨토링이 없는거쥐??

     

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    •  
      조가현 편집장 Lv.15 2018.08.03 20:20

      KAIST 문제 주니어 멘토님은 KAIST 과학영재연구원 소속 분이세요. 그래서 그 문제만 의견을 주시는 거랍니다. 방승진 교수님의 문제는 현재 멘토가 따로 있는 건 아니라서 어느 정도 풀이가 모이면 교수님께 의견을 직접 받을 계획입니다.

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  •  
    냐뇨기 Lv.1 2018.08.12 06:12

    매~~~~~~~~~~~~~~~~~~~~~~~~~~우 여럽다!!!!

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  •  
    임소정 Lv.1 2018.08.14 01:45

    저는 초등학교 2학년 입니다.

    처음 이 사이트에 들어와서 수학퍼즐을 해 보았는데 너무 재미있었답니다.

    제가 해 본 정답은

    6 10 1 8

     4 9 7

       5 2

         3

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  •  
    쓰다만연습장 Lv.5 2018.08.15 06:28

    자 정사각형 채우기는 끝났으니깐 우리 이거래도 열심히 해봅시다! (할게 이거밖에 없음..)

    근데 여기 왜 이렇게 썰렁한걸까요?

     

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    •  
      김우현 기자 Lv.5 2018.08.15 16:43

      모두 더위에 지쳐 문제 풀 기운이 없는 걸까요!?crying

      좋아요1
  •  
    Euler Lv.1 2018.08.17 06:18

    잠만 나 아주대학교 영재원하는데 이 교수님이 영재원 원장인데... 이건 무슨 우연이지?

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    •  
      J.O.S Lv.1 2018.08.29 02:13

      저두요 ㅋㅋ

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  •  
    muse Lv.6 2018.08.19 21:59

    1-1. 아래와 같은 4가지 경우입니다.

    8 10 1 6 
2 9 5 
7 4 
3 

    6 1 10 8 
5 9 2 
4 7 
3 

6 10 1 8 
4 9 7 
5 2 
3 

8 1 10 6 
7 9 4 
2 5 
3

     

    1-2. 4를 사용하면 4가지가 가능합니다.

    8 3 10 9 
5 7 1 
2 6 
4

     

    8 10 3 9 
2 7 6 
5 1 
4 

9 3 10 8 
6 7 2 
1 5 
4 

9 10 3 8 
1 7 5 
6 2 
4

     

    프로그램을 이용해 확인해본 결과 더 이상의 방법은 없었습니다.

     

     

    소스 코드

    #include <bits/stdc++.h>

int board[10][10];
bool chk[20];

int main(){
    for(int i=1; i<=10; i++){
        for(int j=1; j<=10; j++){
            if(i==j) continue;
            for(int k=1; k<=10; k++){
                if(i==k||j==k) continue;
                for(int l=1; l<=10; l++){
                    if(i==l||j==l||k==l) continue;
                    board[1][1] = i;
                    board[1][2] = j;
                    board[1][3] = k;
                    board[1][4] = l;
                    chk[i] = chk[j] = chk[k] = chk[l] = 1;
                    for(int ii=2; ii<=4; ii++){
                        for(int jj=1; jj+ii<=5; jj++){
                            board[ii][jj] = abs(board[ii-1][jj] - board[ii-1][jj+1]);
                            if(chk[board[ii][jj]]) goto next_t;
                            chk[board[ii][jj]] = 1;
                        }
                    }
                    for(int ii=1; ii<=4; ii++){
                        for(int jj=1; jj+ii<=5; jj++){
                            printf("%d ", board[ii][jj]);
                        }
                        puts("");
                    }
                    puts("");
                    next_t:
                    for(int i=1; i<=10; i++) chk[i] = 0;
                }
            }
        }
    }
}
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    •  
      Simon Lv.2 2018.08.25 18:12

      goto문 쓰시네요;;

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    •  
      gunchang Lv.1 2018.08.30 02:05

      그러면 문제 해결인가요!

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  •  
    냐뇨기 Lv.1 2018.08.24 02:55

    crying

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  •  
    냐뇨기 Lv.1 2018.08.24 03:11

    보면 가로세로 합치면 다 9가 되네요!!

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  •  
    냐뇨기 Lv.1 2018.08.25 01:56

    보면 10,9,8,7,6,5는 불가능하고

    4,3,2만 되네여!!!

    1은 모르겟네~?

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    •  
      muse Lv.6 2018.08.25 03:36

      2는 안 되는 것 같습니다.

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    •  
      수학짱 Lv.1 2018.08.29 23:33

      1도 안 돼네요!

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  •  
    루피 Lv.1 2018.08.28 01:24

    5일때 위에서부터 2 10 9 6 8 1 3 7 2 5 이면 가능하지 않나요? 오류 있으면 얘기 해 주세요

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    •  
      상어 Lv.1 2018.09.01 01:31

      2가 두번들어가네요.

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  •  
    상어 Lv.1 2018.09.01 01:29

    방금 왔는데 벌써 거의다 다풀렸네요. (풀게 없다....)

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  •  
    데카르트 Lv.1 2018.09.30 04:48

    뭐가 증명이 안된거죠?

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  •  
    ryan09 Lv.1 2020.05.30 23:46
    확인요청중

    제 답은 반대로 있는게 2개 있어서 진짜로는 2개만 구한 거에요.

    8 10 3 9        9 3 10 8       9 10 3 8         8 3 10 9  

     2 7 6            6 7 2            1 7 5             5 7 1

       5 1              1 5               6 2                2 6

       4                  4                  4                   4

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  •  
    김진선 Lv.1 2020.10.23 07:25
    확인요청중

    전체10개 구멍의 삼각형이지만

    6개 구멍의 삼각형으로 쪼갠후

    그것의 패턴을 찾았습니다

    a3,a4,a5

    a1,a2

    a0

    그러면 a0=|a1-a2|=|a3-a5| 라는 패턴이

    항상 성립할 것 같아요

    이것을 증명한다면?

    더 큰 삼각형에도 일반화 시켜싴

    뭔가 해볼 수 있을거 같아요

     

     

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  •  
    김진선 Lv.1 2020.10.23 07:25
    확인요청중

    전체10개 구멍의 삼각형이지만

    6개 구멍의 삼각형으로 쪼갠후

    그것의 패턴을 찾았습니다

    a3,a4,a5

    a1,a2

    a0

    그러면 a0=|a1-a2|=|a3-a5| 라는 패턴이

    항상 성립할 것 같아요

    이것을 증명한다면?

    더 큰 삼각형에도 일반화 시켜싴

    뭔가 해볼 수 있을거 같아요

     

     

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  •  
    찌니쭈니 Lv.2 2022.04.26 05:21

    일단 손수 계산결과 저는

    8 10 1 6

     2  9  5

       7  4

         3

    이 나왔습니다.

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