Social Distancing

Given a 10x10 room to be filled with people, people standing at least 1 m from every other person. How many people can you fit in the room?

I was trying to find the source of a similar puzzle, that was planting trees in a square but i could not find it, I fit 115 people, but i do not know if that is the answer

#Geometry

Easy Math Editor

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Comments

I believe the maximum number of people this room could hold would be $121$ . This would assume we are measuring from the center of each person, as if they were points.

If we take the bottom edge of the room first, we see that we could fit at most $11$ people along this wall. Remember, $11$ people can fit, not just $10$ ( $2$ in the corners, $9$ in between, with $1$ meter between each person).

Then, we can copy this row of $11$ throughout the whole room $11$ times (using the same procedure we used for the bottom wall). This gives us $11 \cdot 11 = \boxed{121}$ .

David Stiff - 1 year, 1 month ago

I'm sure we can do better than this, using circle packing. We essentially represent each person as a circle with radius $0.5$ metres. I haven't crunched the numbers yet, but I bet we can probably get to at least 150.

Elijah L - 1 year, 1 month ago

I think this is actually equivalent to circle packing. I would add a picture here, but I don't think I have the ability to do so in a reply. But here's how you would draw it:

Draw a square $10$ x $10$ .
Draw $11$ circles with radius $0.5$ , all tangent to each other along the top edge. Draw the first circle centered on a corner, then continue along until you reach the adjacent corner. You will have drawn $11$ circles.
Then (if you're really up to it!) draw $10$ more rows of $11$ circles, each tangent to the preceding one. You'll reach the bottom edge on the last row.

You would now have $11$ rows of $11$ circles, giving us a total of $\boxed{121}$ circles.

David Stiff - 1 year, 1 month ago

Alt text , If the image doest not load here it is(https://imgur.com/cCGxv8T)

I think i went about this way, and fit less than 121.(equilateral) i feel that some other shape triangle would work, buut i have no idea how

Bean Ender - 1 year, 1 month ago

Actually Bean Ender's problem generated a few more comments, and I see now what you mean about circle packing. I'm using a square lattice, but other's have suggested an equilateral triangle lattice. Not familiar with this, so can't really comment.

David Stiff - 1 year, 1 month ago

I went ahead and created a problem thats exactly thr same, but i dont actually have a solution , so feel free

https://brilliant.org/problems/social-distancing-edited/

Bean Ender - 1 year, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$