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I have booked a square meeting room that is 5 by 5 meters. Our Covid-19 policy says that each person must be at least 1.5 meters away from any other person. What is the highest number of people that can legally fit in this meeting room?

Assume that people are points that can be placed in two dimensions.

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27
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I'll get things started with:

18

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    $\begingroup$ Nice work! This will be hard to beat. $\endgroup$ Nov 4 '21 at 5:55
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    $\begingroup$ Man, I'd hate to be one of the people forced to stand right in a corner! $\endgroup$
    – Stiv
    Nov 4 '21 at 11:28
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    $\begingroup$ @Stiv those people are the safest because they only have to keep track of 90 degrees of potential covid infection vectors $\endgroup$ Nov 4 '21 at 19:26
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The problem is equivalent to

circle packing in a square, where the square is not 5 by 5 meters but 6.5 by 6.5; if the 1.5m circle fits inside the 6.5 by 6.5 meter square, its midpoint falls inside the 5 by 5 meter square.

Now,

6.5 meter is $8\frac23$ times the radius of these circles (0.75m). We can check the Wikipedia table and verify that highest number of circles for which the square size (second column) is lower than $8\frac23$ is 18, so @RobPratt's solution is optimal.

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