Six ducks in a pond

Question

This puzzle has got me stumped.

$6$ ducks are swimming on a pond radius $5$. Show that at any moment there are two ducks a distance at most $5$ apart.

@Samuel, two of the 6 ducks must be inside the same disk by the pigeonhole principle. — vadim123, May 27 '13 at 15:04
@vadim: "Can you cover a circle of radius 5 with five circles, each of diameter 5?" No, you can't. — TonyK, May 27 '13 at 19:34

score 6 · Accepted Answer · answered May 27 '13 at 18:44

If one duck sits at the center of the pond we are done. Otherwise each duck has a well defined argument (polar angle), and we can number the docks cyclically according to increasing arguments. In this way we have $$\arg(D_i)-\arg(D_{i-1})\geq0,\quad \sum_{i=1}^6 \bigl(\arg(D_i)-\arg(D_{i-1})\bigr)=2\pi\ .$$ It follows that $\arg(D_i)-\arg(D_{i-1})\leq{\pi\over3}$ for at least one $i\in[6]$, and since a circular sector of radius $5$ and central angle ${\pi\over3}$ has diameter $5$ it follows that $|D_i-D_{i-1}|\leq5$ for such $i$.

That's a lovely solution. Thank you. – marshall May 27 '13 at 20:55 — marshall, May 27 '13 at 20:55

score 0 · Answer 2 · answered May 27 '13 at 15:47

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Let's see if this answer makes sense. Let's say the ducks spread out as far as possible. It would make sense that to do this, they spread evenly around the circumference. Draw a circle of radius 5 and place 6 points along the perimeter to represent the ducks. Line segments connecting each point to its closest 2 neighbors creates a regular hexagon. Connecting these points to the center of the circle creates 6 equilateral triangles with a side length equal to the radius of 5.

answered May 27 '13 at 15:47

Mike

13,318

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The fact that this arrangement is optimal is 'obvious' but not obvious to prove. – jwg May 27 '13 at 16:35
@jwg Sadly, that's true. If one of the ducks is moved inside the hexagon, I believe I can prove it's moving closer to another duck. However, what seems trivial but harder to actually prove is if one duck moves towards another duck along the circumference, it's getting closer to that duck. – Mike May 27 '13 at 17:06
@jwg Actually, I think I can complete the proof proving if I move any duck, I'm moving it closer to another duck. I'll edit that in later. – Mike May 27 '13 at 17:11
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@Stefan: "The minimum distance between ducks is not decreased." No, this is only true for the first duck. After two such moves, the first and second duck may be closer together than when they started. If they started on the same radius, for instance, their distance is reduced to zero. – TonyK May 27 '13 at 19:38
@TonyK : you're right. I'll delete my comment. Thanks. The accepted answer renders my comment irrelevent anyway. – Stefan Smith May 28 '13 at 11:43

Six ducks in a pond

2 Answers2