To comply with the write-up rule of root-me.org, in this write-up, I just talk about some hints related this challenge. Here is no solution and correct answer.
Check the provided file the "ch4". It executes files regularly with cron service and makes directory "._cron". It is driven with the permission to get the flag.
And this is some operations of bash script if statement :
1. -f : Exist & Regular file
2. -a : And
3. -x : Exist & Executable file
Using the above conditions, you can do anything with the given permission. But the directory creation strangely couldn't do.