It is between 3 pm and 4 pm, and the distance between the hour hand and the…

On a clock face, both hands move at constant rates measured in minute spaces (60 spaces make a full circle, one per minute mark). The minute hand covers 1 minute space every minute, while the hour hand covers only 1/12 of a minute space every minute. So, relative to the hour hand, the minute hand gains 55 minute spaces every 60 minutes — this net gain rate converts any required gap, in minute spaces, into elapsed time.

1. At 3:00 pm, the hour hand is at the 15-minute mark (3 x 5) and the minute hand is at the 0-minute mark, so the minute hand starts 15 minute spaces behind the hour hand.

2. For the hands to be 18 minute spaces apart after 3:00 pm, the minute hand must first close this 15-space gap and then move 18 spaces ahead of the hour hand -- a total gain of 15 + 18 = 33 minute spaces.

3. Since the minute hand gains 55 minute spaces on the hour hand every 60 minutes, gaining 33 minute spaces takes (60/55) x 33 = 36 minutes.

4. Adding 36 minutes to 3:00 pm gives 3:36 pm.

Independent cross-check by degrees: at 3:36 pm the hour hand has swept 3.6 x 30 degrees = 108 degrees and the minute hand has swept 36 x 6 degrees = 216 degrees. The gap is 216 - 108 = 108 degrees, and since each minute space equals 6 degrees, this is 108 / 6 = 18 minute spaces -- confirming the same separation.

So the clock shows 3:36 pm.