Quanto vale o retângulo?

# http://fb.com/todaamatematica/photos/a.602147333194729.1073741828.601759066566889/1265019473574175/
#
# Se o retângulo representar um único dígito, então
# (10 + x)x = 90 + x ⇒ x = 6
#
# Mas e se ele representar um inteiro positivo qualquer?
# Então, se x tiver n dígitos,
#
# (10ⁿ + x)x = 9·10ⁿ + x ⇒ x = ( √( (10ⁿ - 1)² + 4·9·10ⁿ ) - (10ⁿ - 1) ) / 2
#
# Não aparece x>0 inteiro quando n>1, mas vemos uma bela regularidade se formando na parte decimal de x :)

for n in {0..102}
{
  printf "# n = %3i ⇒ x = " $n
  BC_LINE_LENGTH=0 bc <<< "scale=100; ( sqrt( (10^$n-1)^2 + 4*9*10^$n ) - (10^$n-1) ) / 2"
}

# n =   0 ⇒ x = 3.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
# n =   1 ⇒ x = 6.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
# n =   2 ⇒ x = 8.3813441447242135484526932614234208108559778868739304467335831809700903469635381166639671948923313049
# n =   3 ⇒ x = 8.9291986107800241115851121381226310374274560733984679960213230626675317693154668283585643878826693011
# n =   4 ⇒ x = 8.9928122140696549285747664999105648294395128036445309363489475637128332803588646799706594499553145081
# n =   5 ⇒ x = 8.9992801223740141796255138557875637742297392147555092964182377691450820591391464058016284780989584557
# n =   6 ⇒ x = 8.9999280012239740086181042513563601814010918765397754670293416098999285432662643267051987375161556757
# n =   7 ⇒ x = 8.9999928000122399740080618118425097803706237105478841244253454198918645303123343446201391886875662975
# n =   8 ⇒ x = 8.9999992800001223999740080061811984250940203716679707591669266057603016782332972162701658087227923816
# n =   9 ⇒ x = 8.9999999280000012239999740080006181199842509364203717723970729604693514193577972966977710507256619832
# n =  10 ⇒ x = 8.9999999928000000122399999740080000618119998425093604203717828397072664846935995553575468584705215185
# n =  11 ⇒ x = 8.9999999992800000001223999999740080000061811999984250936004203717838839707263528469360809153575218146
# n =  12 ⇒ x = 8.9999999999280000000012239999999740080000006181199999842509360004203717839883970726323284693608945153
# n =  13 ⇒ x = 8.9999999999928000000000122399999999740080000000618119999998425093600004203717839988397072632032846936
# n =  14 ⇒ x = 8.9999999999992800000000001223999999999740080000000061811999999984250936000004203717839998839707263200
# n =  15 ⇒ x = 8.9999999999999280000000000012239999999999740080000000006181199999999842509360000004203717839999883970
# n =  16 ⇒ x = 8.9999999999999928000000000000122399999999999740080000000000618119999999998425093600000004203717839999
# n =  17 ⇒ x = 8.9999999999999992800000000000001223999999999999740080000000000061811999999999984250936000000004203717
# n =  18 ⇒ x = 8.9999999999999999280000000000000012239999999999999740080000000000006181199999999999842509360000000004
# n =  19 ⇒ x = 8.9999999999999999928000000000000000122399999999999999740080000000000000618119999999999998425093600000
# n =  20 ⇒ x = 8.9999999999999999992800000000000000001223999999999999999740080000000000000061811999999999999984250936
# n =  21 ⇒ x = 8.9999999999999999999280000000000000000012239999999999999999740080000000000000006181199999999999999842
# n =  22 ⇒ x = 8.9999999999999999999928000000000000000000122399999999999999999740080000000000000000618119999999999999
# n =  23 ⇒ x = 8.9999999999999999999992800000000000000000001223999999999999999999740080000000000000000061811999999999
# n =  24 ⇒ x = 8.9999999999999999999999280000000000000000000012239999999999999999999740080000000000000000006181199999
# n =  25 ⇒ x = 8.9999999999999999999999928000000000000000000000122399999999999999999999740080000000000000000000618119
# n =  26 ⇒ x = 8.9999999999999999999999992800000000000000000000001223999999999999999999999740080000000000000000000061
# n =  27 ⇒ x = 8.9999999999999999999999999280000000000000000000000012239999999999999999999999740080000000000000000000
# n =  28 ⇒ x = 8.9999999999999999999999999928000000000000000000000000122399999999999999999999999740080000000000000000
# n =  29 ⇒ x = 8.9999999999999999999999999992800000000000000000000000001223999999999999999999999999740080000000000000
# n =  30 ⇒ x = 8.9999999999999999999999999999280000000000000000000000000012239999999999999999999999999740080000000000
# n =  31 ⇒ x = 8.9999999999999999999999999999928000000000000000000000000000122399999999999999999999999999740080000000
# n =  32 ⇒ x = 8.9999999999999999999999999999992800000000000000000000000000001223999999999999999999999999999740080000
# n =  33 ⇒ x = 8.9999999999999999999999999999999280000000000000000000000000000012239999999999999999999999999999740080
# n =  34 ⇒ x = 8.9999999999999999999999999999999928000000000000000000000000000000122399999999999999999999999999999740
# n =  35 ⇒ x = 8.9999999999999999999999999999999992800000000000000000000000000000001223999999999999999999999999999999
# n =  36 ⇒ x = 8.9999999999999999999999999999999999280000000000000000000000000000000012239999999999999999999999999999
# n =  37 ⇒ x = 8.9999999999999999999999999999999999928000000000000000000000000000000000122399999999999999999999999999
# n =  38 ⇒ x = 8.9999999999999999999999999999999999992800000000000000000000000000000000001223999999999999999999999999
# n =  39 ⇒ x = 8.9999999999999999999999999999999999999280000000000000000000000000000000000012239999999999999999999999
# n =  40 ⇒ x = 8.9999999999999999999999999999999999999928000000000000000000000000000000000000122399999999999999999999
# n =  41 ⇒ x = 8.9999999999999999999999999999999999999992800000000000000000000000000000000000001223999999999999999999
# n =  42 ⇒ x = 8.9999999999999999999999999999999999999999280000000000000000000000000000000000000012239999999999999999
# n =  43 ⇒ x = 8.9999999999999999999999999999999999999999928000000000000000000000000000000000000000122399999999999999
# n =  44 ⇒ x = 8.9999999999999999999999999999999999999999992800000000000000000000000000000000000000001223999999999999
# n =  45 ⇒ x = 8.9999999999999999999999999999999999999999999280000000000000000000000000000000000000000012239999999999
# n =  46 ⇒ x = 8.9999999999999999999999999999999999999999999928000000000000000000000000000000000000000000122399999999
# n =  47 ⇒ x = 8.9999999999999999999999999999999999999999999992800000000000000000000000000000000000000000001223999999
# n =  48 ⇒ x = 8.9999999999999999999999999999999999999999999999280000000000000000000000000000000000000000000012239999
# n =  49 ⇒ x = 8.9999999999999999999999999999999999999999999999928000000000000000000000000000000000000000000000122399
# n =  50 ⇒ x = 8.9999999999999999999999999999999999999999999999992800000000000000000000000000000000000000000000001223
# n =  51 ⇒ x = 8.9999999999999999999999999999999999999999999999999280000000000000000000000000000000000000000000000012
# n =  52 ⇒ x = 8.9999999999999999999999999999999999999999999999999928000000000000000000000000000000000000000000000000
# n =  53 ⇒ x = 8.9999999999999999999999999999999999999999999999999992800000000000000000000000000000000000000000000000
# n =  54 ⇒ x = 8.9999999999999999999999999999999999999999999999999999280000000000000000000000000000000000000000000000
# n =  55 ⇒ x = 8.9999999999999999999999999999999999999999999999999999928000000000000000000000000000000000000000000000
# n =  56 ⇒ x = 8.9999999999999999999999999999999999999999999999999999992800000000000000000000000000000000000000000000
# n =  57 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999280000000000000000000000000000000000000000000
# n =  58 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999928000000000000000000000000000000000000000000
# n =  59 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999992800000000000000000000000000000000000000000
# n =  60 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999280000000000000000000000000000000000000000
# n =  61 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999928000000000000000000000000000000000000000
# n =  62 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999992800000000000000000000000000000000000000
# n =  63 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999280000000000000000000000000000000000000
# n =  64 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999928000000000000000000000000000000000000
# n =  65 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999992800000000000000000000000000000000000
# n =  66 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999280000000000000000000000000000000000
# n =  67 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999928000000000000000000000000000000000
# n =  68 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999992800000000000000000000000000000000
# n =  69 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999280000000000000000000000000000000
# n =  70 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999928000000000000000000000000000000
# n =  71 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999992800000000000000000000000000000
# n =  72 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999280000000000000000000000000000
# n =  73 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999928000000000000000000000000000
# n =  74 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999992800000000000000000000000000
# n =  75 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999280000000000000000000000000
# n =  76 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999928000000000000000000000000
# n =  77 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999992800000000000000000000000
# n =  78 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999280000000000000000000000
# n =  79 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999928000000000000000000000
# n =  80 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999992800000000000000000000
# n =  81 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999280000000000000000000
# n =  82 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999928000000000000000000
# n =  83 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999992800000000000000000
# n =  84 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999280000000000000000
# n =  85 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999928000000000000000
# n =  86 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999992800000000000000
# n =  87 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999280000000000000
# n =  88 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999928000000000000
# n =  89 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999992800000000000
# n =  90 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999280000000000
# n =  91 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999928000000000
# n =  92 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999992800000000
# n =  93 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999280000000
# n =  94 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999928000000
# n =  95 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999992800000
# n =  96 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999280000
# n =  97 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999928000
# n =  98 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999992800
# n =  99 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999280
# n = 100 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999928
# n = 101 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999992
# n = 102 ⇒ x = 8.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999