A cistern can be filled by 3 pipes in 30, 40, and 60 minutes respectively, and emptied by an escape pipe in 30 minutes. The three tapes are turned on at noon, but the escape pipe is at the same time accidentally left open and not closed for a quarter of an hour. At what time will the cistern be full?

Step 1: First, we need to find out how much of the cistern each pipe can fill in one minute.

Pipe 1 can fill 1/30 of the cistern in one minute. Pipe 2 can fill 1/40 of the cistern in one minute. Pipe 3 can fill 1/60 of the cistern in one minute. The escape pipe can empty 1/30 of the cistern in one minute.

Step 2: Now, let's find out how much of the cistern is filled in one minute when all pipes are open.

The three pipes together can fill (1/30 + 1/40 + 1/60) = 1/10 of the cistern in one minute. But the escape pipe can empty 1/30 of the cistern in one minute. So, when all pipes are open, they can fill (1/10 - 1/30) = 1/15 of the cistern in one minute.

Step 3: The escape pipe was left open for a quarter of an hour, or 15 minutes. In this time, all the pipes together filled 15 * 1/15 = 1 full cistern.

Step 4: After the escape pipe is closed, the three pipes can fill the cistern at a rate of 1/10 of the cistern per minute. So, to fill the remaining empty cistern, it will take 10 minutes.

Step 5: The pipes were turned on at noon. The cistern was filled in 15 minutes (with the escape pipe open) + 10 minutes (after the escape pipe was closed) = 25 minutes.

So, the cistern will be full at 12:25 PM.