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.

ANSWERS

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.