Stone Game III


10 Prerequisite LeetCode Problems
“1406. Stone Game III” is a dynamic programming problem with game theory. Here are 10 problems to build your skills:
“53. Maximum Subarray”: This problem gives a basic introduction to the concept of dynamic programming.
“70. Climbing Stairs”: Another basic dynamic programming problem which helps to understand how subproblems can be used to build up the solution to the overall problem.
“1143. Longest Common Subsequence”: A more complex dynamic programming problem which introduces the concept of a 2D DP array.
“198. House Robber”: A dynamic programming problem where the state transition is not straightforward and requires some thought.
“213. House Robber II”: A variation of the “House Robber” problem that adds a new constraint, increasing the complexity and forcing you to adapt the previous DP approach.
“322. Coin Change”: A classic dynamic programming problem involving finding minimum/maximum values.
“877. Stone Game”: This problem introduces the concept of game theory and how it can be solved using dynamic programming.
“486. Predict the Winner”: This problem is a similar game theory problem to “Stone Game” with a slightly different setup.
“375. Guess Number Higher or Lower II”: This problem combines game theory and dynamic programming in a more complex way.
“416. Partition Equal Subset Sum”: A DP problem with a binary choice similar to what’s required in the “Stone Game III” problem.


Problem Classification
Language Agnostic Coding Drills
This code is solving a problem called Stone Game III. It requires understanding of topics like dynamic programming, loops, list indexing, conditional statements, and ternary expressions.
Here are some of the smaller units of learning, arranged in increasing level of difficulty:
Basic Programming Syntax: Understanding of variables, lists, and loops in programming languages.
Indexing and List manipulation: Understanding of how to access elements from a list and how to use indices to manipulate a list.
Modulo Operator: Understanding of the modulo operator and its use in programming, for example for performing operations in a cyclical manner.
Conditional Statements: Understanding of if, elif, and else statements and their use in controlling program flow.
Functions and Method: Understanding of how to create a function or a method in a class. Understanding of the
self
keyword in class methods.Iteration: Understanding how to iterate through a list in reverse order.
Classes and Objects: Understanding how to create a class in Python and create an object of that class to call a method.
Dynamic Programming: Understanding of dynamic programming, which is an optimization technique used to solve complex problems by breaking them down into simpler subproblems. It involves understanding of concepts like overlapping subproblems, optimal substructure, and the creation and use of a DP table.
Game Theory in Programming Problems: Understanding of how game theory concepts apply to problems like the Stone Game III. This includes understanding strategies and how to determine the best move at any given point.
Ternary Expressions: Understanding how to use ternary expressions to assign values to variables based on a condition.
Optimization Techniques: Understanding how to optimize a program for better performance. In this case, using dynamic programming and a circular DP table (size of 3) to save space.
Problem Analysis and Solution Development: Understanding how to analyze a problem statement, identify the necessary concepts, and develop an efficient solution. This includes mapping realworld problems to known computer science concepts or data structures. For the given problem, this would involve recognizing it as a dynamic programming problem, formulating the DP state transition equations, and implementing the solution.
Targeted Drills in Python
 Basic Programming Syntax
1 2 3
print("Hello, World!") x = 10 print(f"X is {x}")
 Indexing and List manipulation
1 2 3 4
my_list = [1, 2, 3, 4, 5] print(my_list[2]) # prints 3 my_list.append(6) # appends 6 to the list print(my_list) # prints [1, 2, 3, 4, 5, 6]
 Modulo Operator
1
print(10 % 3) # prints 1
 Conditional Statements
1 2 3 4 5 6 7
x = 10 if x > 5: print("X is greater than 5") elif x == 5: print("X is equal to 5") else: print("X is less than 5")
 Functions and Method
1 2 3 4
def greet(name): print(f"Hello, {name}!") greet("Alice") # prints "Hello, Alice!"
 Iteration
1 2
for i in range(10, 0, 1): print(i, end=' ') # prints 10 9 8 7 6 5 4 3 2 1
 Classes and Objects
1 2 3 4 5 6 7 8 9
class Car: def __init__(self, color): self.color = color def show_color(self): print(self.color) my_car = Car("blue") my_car.show_color() # prints "blue"
 Dynamic Programming
1 2 3 4 5 6 7 8
# Fibonacci sequence with DP def fib(n, memo = {}): if n in memo: return memo[n] if n <= 2: return 1 memo[n] = fib(n1, memo) + fib(n2, memo) return memo[n] print(fib(10)) # prints 55
 Game Theory in Programming Problems
1 2 3 4 5 6
# This requires complex problems, such as Nim Game def canWinNim(n): return n % 4 != 0 print(canWinNim(4)) # prints False print(canWinNim(5)) # prints True
 Ternary Expressions
1 2
x = 10 print("Even" if x % 2 == 0 else "Odd") # prints "Even"
 Optimization Techniques
1 2 3 4 5 6 7 8
# Using memoization to optimize a recursive Fibonacci sequence calculation def fib(n, memo={}): if n in memo: return memo[n] if n <= 2: return 1 memo[n] = fib(n1, memo) + fib(n2, memo) return memo[n] print(fib(1000)) # prints the 1000th Fibonacci number
 Problem Analysis and Solution Development
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
# This drill requires a complete problem and solution, such as finding the longest palindrome in a string def longest_palindrome(s): res = "" for i in range(len(s)): odd = find_palindrome(s, i, i) even = find_palindrome(s, i, i+1) res = max(res, odd, even, key=len) return res def find_palindrome(s, l, r): while l >= 0 and r < len(s) and s[l] == s[r]: l = 1 r += 1 return s[l+1:r] print(longest_palindrome("babad")) # prints "bab"
Note that for complex topics such as Game Theory and Problem Analysis and Solution Development, the drills are simplified versions and may not cover the full depth of these topics. These drills are intended as starting points to understand these concepts.
Problem Solving
Here is a breakdown of the problemsolving approach for the “Stone Game III” problem. This problem belongs to a category of problems that can be solved using Dynamic Programming.
The problem statement is:
Alice and Bob continue their games with piles of stones. There are several stones arranged in a row, and each stone has an associated value which is an integer given in the array stoneValue.
Alice and Bob take turns, with Alice starting first. On each player's turn, that player can take 1, 2 or 3 stones from the first remaining stones in the row.
The score of each player is the sum of values of the stones taken. The score of each player is 0 initially.
The objective of the game is to end with the highest score, and the goal of the game is to maximize the difference in the score of the first player  the score of the second player. If the difference in the score is positive, then Alice wins, if the difference in the score is negative, then Bob wins, if the difference in the score is 0, then the game is a tie.
Here is how you could approach solving this problem:
Understanding the Problem: The first step is to understand the problem. Here, we have Alice and Bob playing a game where they take turns to pick 1, 2, or 3 stones from a row. The goal for each player is to maximize their score. The objective of the problem is to determine who wins the game, or if it is a tie.
Identify the problem type: This problem can be categorized as a dynamic programming problem because we can see that there are overlapping subproblems, which means that we can build the solution to the larger problem using the solutions of smaller subproblems.
Define the approach: Dynamic programming problems are usually solved by defining a state for the problem, and then filling up a table (dp) which gives us the solution to our problem. In this case, our state is defined by the index of the first stone that has not been picked yet. dp[i] represents the maximum score difference the current player can achieve starting from the ith stone.
Pseudocode:
 Initialize dp array with size n+3 to avoid checking if we are out of bounds, and fill with inf.
 Loop from the end of the array to the start, for each stone calculate the maximum score Alice can get for the next 3 possibilities (taking 1, 2, or 3 stones) and select the maximum one.
 Alice’s score will be the sum of stones Alice took minus the best score Bob can achieve from this position.
 Bob plays optimally as well, which means he will maximize his score, which is equivalent to minimizing Alice’s score, so Alice’s best option is to maximize her score considering that Bob will also play optimally.
 Finally, if dp[0](the maximum score difference Alice can achieve starting from the 0th stone) is greater than 0, Alice wins. If dp[0] is less than 0, Bob wins. If dp[0] is 0, the game is a tie.
Translate into code: Now, you can translate the pseudocode into actual code.
Test the code: Finally, test your code with different test cases, including edge cases, to make sure it’s working as expected.
Developing drills that cover the full depth of the “Stone Game III” problemsolving approach and dynamic programming in general would require focusing on each of the key concepts and building up from the basic to the complex. Here’s a proposed progression:
Basic Array Manipulation: Understanding and manipulating arrays is crucial to solving this problem. The learner could practice accessing array elements, slicing arrays, and iterating over arrays.
Python Drill:
1 2 3 4 5 6
# Create an array and practice accessing elements, slicing, and iterating arr = [1, 2, 3, 4, 5] print(arr[2]) # Access an element print(arr[1:4]) # Slice array for i in arr: # Iterate over array print(i)
Understanding and Using Loops: Loops are used extensively in the problem. Drills could involve using both for and while loops to solve various problems.
Python Drill:
1 2 3 4 5 6
# Use a loop to calculate the sum of all numbers in an array arr = [1, 2, 3, 4, 5] sum = 0 for num in arr: sum += num print(sum)
Conditional Statements: Understanding how to use if, elif, and else statements is important in controlling the flow of the game logic.
Python Drill:
1 2 3 4 5 6 7 8
# Write a program that prints whether a number is positive, negative or zero num = 5 # Try with different numbers if num > 0: print("Positive") elif num < 0: print("Negative") else: print("Zero")
Mathematical Operations: The problem involves mathematical operations, including addition, subtraction, multiplication, and finding the maximum.
Python Drill:
1 2 3
# Write a program to find the maximum of three numbers num1, num2, num3 = 5, 10, 3 # Try with different numbers print(max(num1, num2, num3))
Dynamic Programming Basics: The problem is a classic example of dynamic programming. The learner could start with simpler DP problems like Fibonacci sequence, climbing stairs, etc.
Python Drill:
1 2 3 4 5 6 7 8
# Write a program to find the nth Fibonacci number using dynamic programming def fibonacci(n): fib = [0, 1] + [0]*(n1) for i in range(2, n+1): fib[i] = fib[i1] + fib[i2] return fib[n] print(fibonacci(10)) # Try with different numbers
Problem Specific Drills: Now, learner can focus on smaller problems specific to the Stone Game III problem.
Python Drill:
1 2 3 4 5 6 7 8 9 10 11
# Given a list of stone values, write a function to calculate the sum of maximum values a player can get by picking 1, 2, or 3 stones. def max_sum(stones): n = len(stones) dp = [0] * (n + 3) for i in range(n  1, 1, 1): dp[i] = max(stones[i]  dp[i + 1], stones[i] + (stones[i + 1] if i + 1 < n else 0)  dp[i + 2], stones[i] + (stones[i + 1] if i + 1 < n else 0) + (stones[i + 2] if i + 2 < n else 0)  dp[i + 3]) return dp[0] print(max_sum([1,2,3,7]))
Full Problem: Finally, they should try to solve the entire problem now.
Remember that the drills should be done in an interactive way, getting feedback and understanding the learner’s difficulties and addressing them at each stage.
This is the general stepbystep process for problem analysis and solution development in dynamic programming problems.