make_change.rb

# dynamic programming version
def make_change(amount, coins)
  result    = Array.new(amount + 1, 0)
  result[0] = 1

  coins.each do |coin|
    (coin..amount).each do |higher_amount|
      remainder              = higher_amount - coin
      result[higher_amount] += result[remainder]
    end
  end

  result[amount]
end

# recursive version:
def make_change2(target, coins = [25, 10, 5, 1])
  return [] if target == 0
  return nil if coins.none? { |coin| coin <= target }

  coins = coins.sort.reverse

  best_change = nil
  coins.each_with_index do |coin, index|
    # can't use this coin, it's too big
    next if coin > target

    # use this coin
    remainder = target - coin

    # Find the best way to make change with the remainder (recursive
    # call). Why `coins.drop(index)`? This is an optimization. Because
    # we want to avoid double counting; imagine two ways to make
    # change for 6 cents:
    #   (1) first use a nickel, then a penny
    #   (2) first use a penny, then a nickel
    # To avoid double counting, we should require that we use *larger
    # coins first*. This is what `coins.drop(index)` enforces; if we
    # use a smaller coin, we can never go back to using larger coins
    # later.
    best_remainder = make_change2(remainder, coins.drop(index))

    # We may not be able to make the remaining amount of change (e.g.,
    # if coins doesn't have a 1cent piece), in which case we shouldn't
    # use this coin.
    next if best_remainder.nil?

    # Otherwise, the best way to make the change **using this coin**,
    # is the best way to make the remainder, plus this one coin.
    this_change = [coin] + best_remainder

    # Is this better than anything we've seen so far?
    best_change = this_changeif best_change.nil? || this_change.count < best_change.count
  end

  best_change
end