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.)
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