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1. What does this program print? ``

python
def max_profit_cooldown(prices):
    hold = float('-inf')
    sold = float('-inf')
    rest = 0
    for p in prices:
        new_hold = max(hold, rest - p)
        new_sold = hold + p
        new_rest = max(rest, sold)
        hold, sold, rest = new_hold, new_sold, new_rest
    return max(sold, rest)


print(max_profit_cooldown([3, 1, 2, 5]))

``

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2. What does this program print? ``

python
def max_profit_k(k, prices):
    buy = [float('-inf')] * (k + 1)
    sell = [0] * (k + 1)
    for p in prices:
        for j in range(1, k + 1):
            buy[j] = max(buy[j], sell[j - 1] - p)
            sell[j] = max(sell[j], buy[j] + p)
    return max(sell)


print(max_profit_k(2, [2, 4, 1, 6]))

``

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3. In the cooldown-only method, why does the buy line use rest - p and not max(rest, sold) - p?

A sale yesterday starts a cooldown, so today's buy must come from a state that had no sale yesterdayrest is always bigger than sold, so the max is never neededBuying on the same day as a sale costs less than buying on a later dayhold already includes yesterday's sale, so no extra term is needed

4. Why is the final answer max(sold, rest) and never hold?

hold is always smaller than sold, so it can never be the best valuehold stores the last price, not moneyhold is only used for the cooldown rule and has no effect on the answerhold means a share is still in hand, and the plan must finish with no share

5. This version is meant to block a buy on the day after a sale. What does it print for the prices [1, 2, 3, 0, 2]? ``

python
def max_profit_cooldown(prices):
    hold = float('-inf')
    sold = float('-inf')
    rest = 0
    for p in prices:
        new_hold = max(hold, max(rest, sold) - p)
        new_sold = hold + p
        new_rest = max(rest, sold)
        hold, sold, rest = new_hold, new_sold, new_rest
    return max(sold, rest)


print(max_profit_cooldown([1, 2, 3, 0, 2]))

``

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State machine for stocks