SQL script that automatically detects free appointments in MySQL database

Question

In my application, users are able to check if a specific appointment is free. So they choose a specific date and time and specify the length of the appointment.

An SQL script checks if there is already an appointment at the specific time. What I want to do now is to offer further appointment options to the user.

Example: User selects 2013-08-24 18:00:00 as start date and 2013-08-24 18:30:00 as end date. The appointment is already taken but options have to be offered:

• offered appointment can be at the same day
• offered appointment can be some hours before or after the prefered appointment
• offered appointment can also be the same date in one week

This is my table (at least the important part of it):

CREATE TABLE `ios_appointments` (
  `appointmentId` int(11) NOT NULL AUTO_INCREMENT,
  `start` timestamp NOT NULL DEFAULT '0000-00-00 00:00:00',
  `end` timestamp NOT NULL DEFAULT '0000-00-00 00:00:00',
  ...
)

Example data:

| appointmentId  |  start              |  end
=========================================================
|  1             | 2013-08-24 18:00:00 | 2013-08-24 18:30:00
|  2             | 2013-08-24 15:00:00 | 2013-08-24 16:00:00
|  3             | 2013-08-24 09:00:00 | 2013-08-24 12:00:00

If the specified length of the appointment is 2 hours, the proposed appointments should be something like:

|  start              |  end
=========================================================
| 2013-08-24 12:00:00 | 2013-08-24 14:00:00
| 2013-08-24 13:00:00 | 2013-08-24 15:00:00

How can I accomplish some kind of "smart" sql script?