We are playing the Guessing Game. The game will work as follows:<ol>
<li>I pick a number between <code>1</code> and <code>n</code>.</li>
<li>You guess a number.</li>
<li>If you guess the right number, <strong>you win the game</strong>.</li>
<li>If you guess the wrong number, then I will tell you whether the number I picked is <strong>higher or lower</strong>, and you will continue guessing.</li>
<li>Every time you guess a wrong number <code>x</code>, you will pay <code>x</code> dollars. If you run out of money, <strong>you lose the game</strong>.</li>
</ol>Given a particular <code>n</code>, return <em>the minimum amount of money you need to <strong>guarantee a win regardless of what number I pick</strong></em>.
Input: n = 10Output: 16Explanation: The winning strategy is as follows:- The range is [1,10]. Guess 7. - If this is my number, your total is $0. Otherwise, you pay $7. - If my number is higher, the range is [8,10]. Guess 9. - If this is my number, your total is $7. Otherwise, you pay $9. - If my number is higher, it must be 10. Guess 10. Your total is $7 + $9 = $16. - If my number is lower, it must be 8. Guess 8. Your total is $7 + $9 = $16. - If my number is lower, the range is [1,6]. Guess 3. - If this is my number, your total is $7. Otherwise, you pay $3. - If my number is higher, the range is [4,6]. Guess 5. - If this is my number, your total is $7 + $3 = $10. Otherwise, you pay $5. - If my number is higher, it must be 6. Guess 6. Your total is $7 + $3 + $5 = $15. - If my number is lower, it must be 4. Guess 4. Your total is $7 + $3 + $5 = $15. - If my number is lower, the range is [1,2]. Guess 1. - If this is my number, your total is $7 + $3 = $10. Otherwise, you pay $1. - If my number is higher, it must be 2. Guess 2. Your total is $7 + $3 + $1 = $11.The worst case in all these scenarios is that you pay $16. Hence, you only need $16 to guarantee a win.
Input: n = 1Output: 0Explanation: There is only one possible number, so you can guess 1 and not have to pay anything.
Input: n = 2Output: 1Explanation: There are two possible numbers, 1 and 2.- Guess 1. - If this is my number, your total is $0. Otherwise, you pay $1. - If my number is higher, it must be 2. Guess 2. Your total is $1.The worst case is that you pay $1.
class Solution: # Time: O(n^3) # Space: O(n^2) def get_money_amount(self, n: int) -> int: # dp[lo][hi] = min worst-case cost to guarantee a win within [lo, hi]. dp = [[0] * (n + 2) for _ in range(n + 2)] for length in range(2, n + 1): for lo in range(1, n - length + 2): hi = lo + length - 1 dp[lo][hi] = min(x + max(dp[lo][x - 1], dp[x + 1][hi]) for x in range(lo, hi)) return dp[1][n]
