Find Maximum of Three Numbers

⚠️ About the Header Files

The code includes <cmath>, but the max() function is actually defined in <algorithm>. The code may still work on some compilers because <cmath> sometimes transitively includes <algorithm>, but this is compiler-dependent and not guaranteed by the C++ standard. For portable code, always include <algorithm> when using min() or max().

🔑 Key Concepts

• Comparison Operators: Using greater than (>) to compare values
• Logical AND (&&): Both conditions must be true for a number to be maximum
• If-Else-If Chain: Testing conditions sequentially until one is true
• Else Clause: Handles remaining case when neither a nor b is maximum
• Inverse Logic: Maximum finding is the opposite of minimum finding
• Long Long Data Type: Handling very large values (up to 10^15)
• Alternative Methods: Built-in max() function vs manual comparison

💡 Understanding the Logic Flow

Decision Tree:

Is a > b AND a > c?
├── YES → maximum = a ✓ (DONE)
└── NO → Continue...
    
    Is b > a AND b > c?
    ├── YES → maximum = b ✓ (DONE)
    └── NO → maximum = c ✓ (DONE)
    
Logic: If neither a nor b is largest, then c must be!
                    

Why the Else Clause Works:

• If a is not greater than both others → it's not maximum
• If b is not greater than both others → it's not maximum
• Therefore, c must be the maximum (or equal to maximum)
• This process of elimination works even when values are equal

Min vs Max Comparison:

🚀 Using the Built-in max() Function

C++ provides the max() function from the <algorithm> header:

#include <iostream>
#include <algorithm>  // Required for max() function

using namespace std;

int main() {
    long long a, b, c;
    cin >> a >> b >> c;
    
    // Nested max() - find max of (a,b), then compare with c
    long long maximum = max(max(a, b), c);
    
    cout << "Maximum is " << maximum;
    
    return 0;
}

How it works:

• max(a, b) → returns the larger of a and b
• max(result, c) → compares that result with c
• Final result is the maximum of all three

C++11 and later: You can also use initializer lists:

long long maximum = max({a, b, c});  // Requires C++11 or later

⚠️ Important Note:

The code includes <cmath> instead of <algorithm>, but it still works on many compilers because <cmath> sometimes implicitly includes <algorithm>. However, this is non-standard behavior! Always use <algorithm> when you need min() or max() functions for portability.

⚠️ Common Mistakes to Avoid

• Incomplete Comparisons: Only checking one condition

  // ❌ WRONG: Only checks if a > b, not if a > c
  if (a > b)
      maximum = a;  // What if c is larger than a?

• Using OR instead of AND: Would give wrong results

  // ❌ WRONG: a must be larger than BOTH b and c
  if (a > b || a > c)  // Using OR is incorrect!
      maximum = a;  // This would choose a even if c is largest

• Mixing up operators: Using < when you need >
• Copy-paste errors: Copying minimum code and forgetting to change operators
• Not handling equal values: Logic should work even when numbers are equal
• Data type overflow: Using int instead of long long for large values

🎯 Alternative Approaches

Approach 1: Simplified If-Else (Assuming a is maximum first)

long long maximum = a;  // Assume a is maximum

if (b > maximum)
    maximum = b;  // Update if b is larger

if (c > maximum)
    maximum = c;  // Update if c is larger

cout << "Maximum is " << maximum;

Advantage: Cleaner, more intuitive logic. Easy to extend to more numbers.

Approach 2: Ternary Operator (Nested)

long long maximum = (a > b) ? ((a > c) ? a : c) : ((b > c) ? b : c);

cout << "Maximum is " << maximum;

Note: Compact but harder to read. Not recommended for beginners.

Approach 3: Using Array and Loop (Scalable)

long long arr[3] = {a, b, c};
long long maximum = arr[0];

for (int i = 1; i < 3; i++) {
    if (arr[i] > maximum)
        maximum = arr[i];
}

cout << "Maximum is " << maximum;

Advantage: Easily scalable to any number of values.

🌍 Real-world Applications

• E-commerce: Finding the highest bidder in auctions
• Gaming: Determining highest score, strongest character stats
• Finance: Identifying peak stock prices, maximum profit
• Data Analysis: Finding maximum values in datasets
• Sports: Determining winning scores and records
• Resource Management: Finding server with maximum capacity
• Weather Systems: Tracking maximum temperatures, peak rainfall
• Performance Monitoring: Identifying maximum CPU/memory usage
• Quality Control: Finding maximum acceptable thresholds
• Route Planning: Selecting longest/furthest route options

🎓 Combining Min and Max

You can find both minimum and maximum in a single program:

#include <iostream>
#include <algorithm>

using namespace std;

int main() {
    long long a, b, c;
    cin >> a >> b >> c;
    
    // Method 1: Using built-in functions
    long long minimum = min({a, b, c});
    long long maximum = max({a, b, c});
    
    cout << "Minimum is " << minimum << endl;
    cout << "Maximum is " << maximum << endl;
    cout << "Range: " << (maximum - minimum) << endl;
    
    return 0;
}

Useful Derived Values:

• Range: maximum - minimum (spread of values)
• Average: (a + b + c) / 3
• Middle Value: a + b + c - minimum - maximum
• Sum of Extremes: minimum + maximum

🎓 Extension Challenges

• Level 1: Find maximum of 4 or 5 numbers
• Level 2: Find both minimum AND maximum in one program
• Level 3: Find the second largest number among three
• Level 4: Find maximum of N numbers (user inputs N)
• Level 5: Find maximum and its position (which input was it?)
• Level 6: Calculate the range (difference between max and min)
• Level 7: Sort three numbers in descending order
• Bonus: Find max, min, and middle value simultaneously

🎯 What You'll Learn

• Understanding comparison logic with greater-than operator
• Using logical AND (&&) to combine multiple conditions
• Building if-else-if chains for sequential checks
• Recognizing inverse relationships (min vs max logic)
• Working with built-in library functions (max)
• Choosing optimal approaches for different scenarios
• Handling edge cases (equal values, negative numbers)
• Understanding process of elimination in logic

💪 Practice Tips

• Test All Positions: Try cases where maximum is first, second, third position
• Test Equal Values: (50, 50, 30), (50, 30, 50), (30, 50, 50), (50, 50, 50)
• Test Negative Numbers: (-5, -10, -3), (-1, -2, -3), (-100, -200, -50)
• Test Zero: (0, 5, -5), (0, 0, 0), (10, 0, -10)
• Test Large Values: Use values close to 10^15 and -10^15
• Compare with Minimum: Write both programs side-by-side to see patterns
• Trace Logic: Manually walk through conditions with different inputs
• Try All Methods: Implement manual logic, built-in function, and simplified approach

🔗 Connection to Previous Problem

This problem is the complementary pair to Question #23 (Find Minimum):

Key Insight: Understanding one makes the other trivial - just flip the comparison operator!