Just thinking about it logically I can see that this will happen between 1:35 and 1:40 So any calc I do can be checked against this.
The hour hand starts 30 degrees past the hour and the minute hand starts on the hour.
After 1 minute the hour hand travels 0.5 degrees and the minute hand travels 6 degrees
so
After x minute the hour hand travels 0.5x degrees and the minute hand travels 6x degrees
the hour hand starts on the 1 so it has that 30 degrees 'head start'
The angle between for the first half hour of so will be 6x-(30+0.5x) and we want that equal to 180
Solve the equation and we get x= \(38\frac{2}{11}\;\;minutes\)
So at \(38\frac{2}{11}\;\;minutes\) past 1 the hands will be exactly opposite each other.
(just as our guest already determined. - thanks guest)