Directory with read and write for a user is permission denied, but fine with execute permissions
Solution 1:
What you are seeing is normal and expected behaviour. The x bit is required on a directory to enter it, search through it, or execute a program in it. As you are trying to cd into the directory then without the x bit being set you can't. When you set the permissions to 744, your user world who is the owner of the directory then has full control and as you observed an do what they want.