project_euler_57.rb

# Square root convergents
# Problem 57
# It is possible to show that the square root of two can be expressed as an infinite continued fraction.
#
# √ 2 = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213...
#
# By expanding this for the first four iterations, we get:
#
# 1 + 1/2 = 3/2 = 1.5
# 1 + 1/(2 + 1/2) = 7/5 = 1.4
# 1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666...
# 1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...
#
# The next three expansions are 99/70, 239/169, and 577/408, but the eighth expansion, 1393/985, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.
#
# In the first one-thousand expansions, how many fractions contain a numerator with more digits than denominator?

def expand(i)
  i > 2 ? 2 + Rational(1, expand(i - 1)) : 2 + Rational(1, 2)
end

puts (2..1000).inject(0) { |memo, i|
  e = 1 + Rational(1, expand(i))
  e.numerator.to_s.size > e.denominator.to_s.size ? memo + 1: memo
}