Week 5 · The longest chain

The following iterative sequence is defined for the set of positive integers:

n → n/2 (n is even)
n → 3n + 1 (n is odd)

Using the rule above and starting with 13, we generate the following sequence:

13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.

Which starting number, under one million, produces the longest chain?

NOTE: Once the chain starts the terms are allowed to go above one million.

From the two solutions below, choose the best solution and explain how it can be improved.

Solution A
def next_num(n)
  n % 2 == 0 ? n / 2 : 3 * n + 1
end
sequences = {}
max_sequence_start = 0
max_sequence_length = 0
sequence_length = 0
sequence_number = 0
1.upto(999999) do |num|
  sequence_length = 1
  sequence_number = num
  while sequence_number > 1
    unless sequences[sequence_number]
      sequence_number = next_num(sequence_number)
      sequence_length += 1
    else
      sequence_length += (sequences[sequence_number] - 1)
      break
    end
  end
  
  sequences[num] = sequence_length
  
  if sequence_length > max_sequence_length
    max_sequence_start = num
    max_sequence_length = sequence_length
  end
  
end
puts "Solution: #{max_sequence_start} = #{max_sequence_length}"
 
Solution B
$col={1=>1}
def collatz(i)
  n, c = i, 0
  while $col[n].nil?
    c += 1
    case n % 2
    when 0
      n = n / 2
    when 1
      n = n * 3 + 1
    end
  end
  $col[i] = c + $col[n]
  return $col[i]
end
(1..1_000_000).each {|i| collatz(i)}
$col.sort {|a,b| a[1] <=> b[1]}[-1][0]
 
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