# Problem of the Month (May 2014)

In the April 2000 Math Magic problem, we asked the following question: If you pack n squares of one size and m squares of another (possibly equal) size inside a unit square, what is the largest area they can cover? This month we ask the same question for circles in a square. These pictures of unit circles in a square may prove helpful.

Solutions were sent by Maurizio Morandi and Joe DeVincentis.

Configurations with 1 Circle
 A1,1 = (9/2 – 3√2)π = .808+ A1,2 = (35/4 – 6√2)π = .831+ A1,3 = (13 – 9√2)π = .854+ A1,4 = (69/4 – 12√2)π = .877+ A1,5 = .845+ A1,6 = .857+ A1,7 = .868+ A1,8 = .880+

Configurations with 2 Circles
 A2,2 = .817+ A2,3 = .733+ (MM) A2,4 = 33π/128 = .809+ A2,5 = .742+ (MM) A2,6 = .783+ A2,7 = .781+ (JD) A2,8 = .815+ (JD) A2,9=(15-4√2)π/36=.815+ (JD)

Configurations with 3 Circles
 A3,3 .758+ A3,4 = 67π/256 = .822+ A3,5 = .780+ (MM) A3,6 = .815+ (MM) A3,7 = .787+ (JD) A3,8 = .802+ (JD) A3,9 = (3-√2)π/6 = .830+ A3,10 = .802+ (JD)

Configurations with 4 Circles
 A4,4 = 17π/64 = .834+ A4,5 = 69π/256 = .846+ A4,6 = 35π/128 = .859+ A4,7 = .850+ A4,8 = .859+ A4,9 = (21-8√2)π/36 = .845+ A4,10 = 3(29-20√2)π/8 = .843+ A4,11 = .848+

Configurations with 5 Circles
 A5,5 = .763+ (JD) A5,6 = .781+ (JD) A5,7 = .799+ (JD) A5,8 = .817+ (JD) A5,9 = 149π/576 = .812+ (JD) A5,10 = .800+ (JD) A5,11 = .813+ (JD) A5,12 = .825+ (JD)

Configurations with 6 Circles
 A6,6 = .771+ (JD) A6,7 = .783+ (JD) A6,8 = .803+ (JD) A6,9 = 75π/288 = .818+ (JD) A6,10 = .805+ (JD) A6,11 = .812+ (JD) A6,12 = .832+ (JD) A6,13 = .821+ (JD)

Configurations with 7 Circles
 A7,7 = .790+ (JD) A7,8 = .815+ (JD) A7,9 = 151π/576 = .823+ (JD) A7,10 = .799+ (JD) A7,11 = .794+ (JD) A7,12 = .805+ (JD) A7,13 = .812+ (JD) A7,14 = .819+ (JD)

Configurations with 8 Circles
 A8,8 = .827+ (JD) A8,9 = .839+ (JD) A8,10 = .815+ (JD) A8,11 = .815+ (JD) A8,12 = .816+ (JD) A8,13 = .823+ (JD) A8,14 = .830+ (JD) A8,15 = .837+ (JD)

Configurations with 9 Circles
 A9,9 = 153π/576 = .834+ (JD) A9,10 = ? A9,11 = ? A9,12 = ?

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 5/23/14.