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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?

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Here is a similar puzzle for four clocks: Four touching clocks

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    $\begingroup$ I'm VTCing as duplicate because the same strategy gives the optimum for rectangular grid of any size. $\endgroup$
    – Bubbler
    Mar 17, 2021 at 4:08
  • $\begingroup$ I've decided to undelete this puzzle. It is not obvious to me how the previous strategy works here. For one there is an odd number of clocks. Can you show me how it works? $\endgroup$
    – Dmitry Kamenetsky
    Mar 17, 2021 at 4:37
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    $\begingroup$ Color the entire grid in the checkerboard fashion. Let's assume the upper left corner is black. Now assign xx:00 to all black cells and xx:30 to all white. Then all adjacent pairs of clocks have 30 minutes difference, so they will touch every hour. (Alternatively, if you successfully convinced yourself that it works for 2N x 2M, just remove the last row/column to make the dimensions odd. The "touch every hour" property is preserved for all the existing clocks.) $\endgroup$
    – Bubbler
    Mar 17, 2021 at 4:52
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    $\begingroup$ The number of total touches per hour is hard-capped by the number of touching pairs of clocks themselves, which is 12 for 3x3. "x touches every 15 minutes" does not matter. $\endgroup$
    – Bubbler
    Mar 17, 2021 at 5:15
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    $\begingroup$ The crux is that "a pair of adjacent clocks touching once every hour is the optimal configuration between a pair of clocks". The checkboard solution preserves this optimality for every pair of adjacent clocks, regardless of how many clocks there are. So you can even have an irregularly shaped configuration of clocks [in a grid], and this strategy will still give the optimal number of touches. $\endgroup$
    – justhalf
    Mar 17, 2021 at 5:19

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