There are nine 12-hour clocks arranged in a 3x3 grid, as shown in the diagram. The long minute handles of the left two clocks are touching, while the others are not. Two minute handles touch if they are both vertical (0 and 30 minutes) or both horizontal (15 and 45 minutes). You can set the time of each clock separately. How can you set the time of these clocks, so that there are as many touches between pairs of minute handles as possible in a 24 hour period?
Here is a similar puzzle for four clocks: Four touching clocks