Packing Circles 

   1.A 10x10 square can obviously hold 100 unit circles (diameter=1) when arranged in rows and columns. How many additional circles
     can it hold if the circles are packed closer, in a hexagonal arrangement? 
   2.What are the dimensions of the smallest area rectangle which can hold an additional 10 unit circles when the arrangement changes
     from rows and columns to hexagonal? 
   3.What are the dimensions of the smallest area rectangle which can hold 100 unit circles in a hexagonal arrangement? 

Source: Original. 

Solutions were received from Henry Bottomley, Doug Dickson, Philippe Fondanaiche, and Glenn Simon. Philippe pointed out there are
several other web sites which investigate packing circles more closely. One such is Erich Friedman's, which has links to more. 

   1.106 circles can fit in a 10x10 square. 8 rows of row/column circles (which take up 8 units of space) can be replaced by 9 hexagonally
     packed rows (which take up 1+8*sqrt(3)/2, or 7.928 units of space.) This leaves 2 more units of space in the square which can be
     filled with two complete rows of 10 circles. Altogether, there are 4 rows with 9 circles and 7 rows with 10 circles. [Most people,
     myself included, simply packed all the rows as close together as possible and found the answer of 105 circles. Thanks to Philippe for
     finding an even better answer!] 
   2.
          Henry Bottmley sent:
          A 10.866x1.866 rectangle =(10+sqrt(3)/2)x(1+sqrt(3)/2) =20.276 has space for one row of ten circles in rectangular formation,
          but two rows of ten in hexagonal formation. 
          Philippe found the smallest rectangle to hold 10 circles above the original 100:
          The smallest area rectangle which can pack 110 unit circles, is equal to 22.5 * [ 1 + 2*sqrt(3) ] = 100.4423... This configuration
          is obtained by packing in a hexagonal arrangement 22 rows of 5 unit circles each. The length of the rectangle is equal to 22*1 +
          0.5(radius of a circle representing the shift due to the hexagonal arrangement) = 22.5 while its width is defined by 0.5 +
          4*0.5*sqrt(3) + 0.5 = 1 + 2*sqrt(3). This result corresponds to the integer n=5 such as n minimizes the quantity
          [INT(110/n)+0.5]*[1+0.5*(n-1)*sqrt(3)] where INT is the rounded up integer of 110/n. 
   3.
          Henry Bottmley's solution:
          A 7.062x13 rectangle =(1+7*sqrt(3)/2)x13 =91.808 has space for 8 rows of alternately 13 and 12 circles making 100 circles in
          total. 
          Philippe's slightly better solution:
          With the same approach, the smallest area rectangle containing 100 unit circles is equal to 20.5*[ 1 + 2*sqrt(3) ] = 91.51408..
          that is to say 20 rows of 5 unit circles each. 

http://www.ecst.csuchico.edu/~kend/potw/ 2000-05-12