factorial-brute-force.rb

# "Proof" of https://twitter.com/tweetsauce/status/965259567322943490
# 1 = 1! 
# 2 = 2! 
# 145 = 1! + 4! + 5! 
# 40,585 = 4! + 0! + 5! + 8! + 5! 
# These are the only four numbers with this property.

# Store the factorials from 0 to 9 for later use.
factorials = (0..9).map {|n| (1..n).inject(:*) || 1 }

# First show that from N=7 onwards the number 999...9 consisting of N 9's is 
# bigger than the sum of its digits' factorials.
(1..10).each do |n| 
  test = ('9' * n).to_i > n * factorials[9]
  puts "#{('9' * n).to_i} #{test ? '>' : '<='} #{(['9!'] * n).join(' + ')}"
  break if test
end

# And now brute force our way up to 7 9's:
(1..9_999_999).each do |n| 
  if n == n.digits.map{ |d| factorials[d] }.sum 
    puts "#{n} = #{n.digits.reverse.map {|d| "#{d}!"}.join(' + ')}" 
  end
end

# Output:
# 9 <= 9!
# 99 <= 9! + 9!
# 999 <= 9! + 9! + 9!
# 9999 <= 9! + 9! + 9! + 9!
# 99999 <= 9! + 9! + 9! + 9! + 9!
# 999999 <= 9! + 9! + 9! + 9! + 9! + 9!
# 9999999 > 9! + 9! + 9! + 9! + 9! + 9! + 9!
# 1 = 1!
# 2 = 2!
# 145 = 1! + 4! + 5!
# 40585 = 4! + 0! + 5! + 8! + 5!