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Covering a circle (extended mix) (Posted on 2006-06-20) Difficulty: 3 of 5
In "Covering a circle" you were to cover a unit circle with 3 squares as small as possible.

I would like instead to cover a circle of maximum radius with n non-overlapping unit squares.

The n=3 problem is solved. Try 4, 5, 6, 7, and 8. (Feel free to keep going. I've done up to 36, but that's a bit excessive.)

No Solution Yet Submitted by Jer    
Rating: 4.0000 (1 votes)

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Solution n = 6 (spoiler) | Comment 4 of 5 |
Wow!  Excellent thought, Tristan.  I certainly didn't think of that!
I did run the numbers, though.

See Tristan's picture (preceding comment).
Draw a circle with radius r which touches the left and bottom border and whose edges go through the lower corner of the top square and the left corner of the right-hand square.  (Maybe somebody can draw a picture.  I can't).

The isoceles triangle whose vertices are the bottom of the circle and the lower corners of the top square has height 2.  Its height also equals r + sqrt(r*r - .25).  Solving 2 = r + sqrt(r*r - .25) gives r = 17/16.  Note that the square on the top is not centered above the other two; the center of the circle and of the top square is 17/16 from the left edge.

All credit for this solution goes to Tristan, who had the insight.  Nice work!

Edited on June 21, 2006, 12:13 am
  Posted by Steve Herman on 2006-06-21 00:12:51

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