Social distancing in a 5x5 room

Question

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.

https://puzzling.stackexchange.com/questions/98462/a-covid-19-puzzle-how-large-a-class-do-you-need-to-fit-30-pupils here is very close Q&A — Nick, Nov 04 '21 at 07:57

score 35 · Answer 1 · answered Nov 04 '21 at 09:36

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.

score 29 · Accepted Answer · answered Nov 04 '21 at 03:45

29

I'll get things started with:

18

answered Nov 04 '21 at 03:45

RobPratt

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1

Nice work! This will be hard to beat. – Dmitry Kamenetsky Nov 04 '21 at 05:55
7

Man, I'd hate to be one of the people forced to stand right in a corner! – Stiv Nov 04 '21 at 11:28
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@Stiv those people are the safest because they only have to keep track of 90 degrees of potential covid infection vectors – Luke Griffiths Nov 04 '21 at 19:26

Social distancing in a 5x5 room

2 Answers2