Problem Description

You need to make burgers using given ingredients. There are two types of burgers you can make:

• Jumbo Burger: requires 4 tomato slices and 1 cheese slice
• Small Burger: requires 2 tomato slices and 1 cheese slice

Given tomatoSlices (total number of tomato slices) and cheeseSlices (total number of cheese slices), you need to determine how many of each type of burger to make so that:

• All tomato slices are used (remaining = 0)
• All cheese slices are used (remaining = 0)

Return an array [total_jumbo, total_small] where:

• total_jumbo is the number of Jumbo Burgers to make
• total_small is the number of Small Burgers to make

If it's impossible to use all ingredients exactly (no ingredients left over), return an empty array [].

This is essentially a system of linear equations problem where you need to find non-negative integer solutions for:

• 4 * jumbo + 2 * small = tomatoSlices
• jumbo + small = cheeseSlices

Intuition

We have two unknowns (number of jumbo burgers and number of small burgers) and two constraints (total tomato slices and total cheese slices). This naturally forms a system of two linear equations with two unknowns, which can be solved algebraically.

Let's denote:

• x = number of Jumbo Burgers
• y = number of Small Burgers

From the problem constraints, we can write:

• 4x + 2y = tomatoSlices (each jumbo uses 4 tomatoes, each small uses 2)
• x + y = cheeseSlices (each burger uses exactly 1 cheese slice)

To solve this system, we can use substitution or elimination. From the second equation, we get x = cheeseSlices - y. Substituting this into the first equation:

4(cheeseSlices - y) + 2y = tomatoSlices 4 * cheeseSlices - 4y + 2y = tomatoSlices 4 * cheeseSlices - 2y = tomatoSlices 2y = 4 * cheeseSlices - tomatoSlices y = (4 * cheeseSlices - tomatoSlices) / 2

Once we have y, we can find x = cheeseSlices - y.

For a valid solution to exist:

1. (4 * cheeseSlices - tomatoSlices) must be even (divisible by 2) for y to be an integer
2. Both x and y must be non-negative (can't make negative burgers)

If any of these conditions fail, there's no valid solution and we return an empty array.

Learn more about Math patterns.

Solution Approach

Based on our mathematical analysis, we implement the solution by directly calculating the values of x (jumbo burgers) and y (small burgers).

The implementation follows these steps:

1. Calculate k = 4 * cheeseSlices - tomatoSlices: This represents the numerator in our formula for y. We store it in a variable k for clarity.

2. Calculate the number of small burgers: y = k // 2. We use integer division since we need a whole number of burgers.

3. Calculate the number of jumbo burgers: x = cheeseSlices - y. This comes from our second equation x + y = cheeseSlices.

4. Validate the solution:

   • Check if k % 2 != 0: If k is odd, then y cannot be an integer, so no valid solution exists
   • Check if y < 0: The number of small burgers must be non-negative
   • Check if x < 0: The number of jumbo burgers must be non-negative

5. Return the result:

   • If any validation fails, return an empty list []
   • Otherwise, return [x, y] representing the number of jumbo and small burgers respectively

The solution has a time complexity of O(1) since we're only performing constant-time arithmetic operations. The space complexity is also O(1) as we only use a few variables to store intermediate results.

This approach is optimal because it directly computes the answer using the closed-form solution derived from the system of equations, avoiding any iteration or search.

Example Walkthrough

Let's walk through an example with tomatoSlices = 16 and cheeseSlices = 7.

Step 1: Set up our equations

• We need to find x (jumbo burgers) and y (small burgers)
• Equation 1: 4x + 2y = 16 (tomato constraint)
• Equation 2: x + y = 7 (cheese constraint)

Step 2: Calculate k

• k = 4 * cheeseSlices - tomatoSlices
• k = 4 * 7 - 16 = 28 - 16 = 12

Step 3: Calculate small burgers (y)

• y = k // 2 = 12 // 2 = 6

Step 4: Calculate jumbo burgers (x)

• x = cheeseSlices - y = 7 - 6 = 1

Step 5: Validate the solution

• Is k % 2 == 0? Yes, 12 % 2 = 0 ✓
• Is y >= 0? Yes, y = 6 ✓
• Is x >= 0? Yes, x = 1 ✓

Step 6: Verify our answer

• Jumbo burgers: 1 burger × 4 tomatoes = 4 tomato slices
• Small burgers: 6 burgers × 2 tomatoes = 12 tomato slices
• Total tomatoes used: 4 + 12 = 16 ✓
• Total cheese used: 1 + 6 = 7 ✓

Result: [1, 6]

Example with no solution: tomatoSlices = 17 and cheeseSlices = 4

Step 1-2: Calculate k

• k = 4 * 4 - 17 = 16 - 17 = -1

Step 3: Check if k is even

• k % 2 = -1 % 2 = -1 ≠ 0
• Since k is odd, we cannot get an integer value for y

Result: [] (no valid solution exists)

Solution Implementation

Time and Space Complexity

The time complexity is O(1) because the algorithm performs a fixed number of arithmetic operations regardless of the input size. The operations include:

• One multiplication: 4 * cheeseSlices
• One subtraction: 4 * cheeseSlices - tomatoSlices
• One integer division: k // 2
• One subtraction: cheeseSlices - y
• One modulo operation: k % 2
• Two comparison operations: y < 0 and x < 0

All these operations execute in constant time.

The space complexity is O(1) because the algorithm uses only a fixed amount of extra space to store the variables k, y, and x. The space usage does not depend on the input values and remains constant. The returned list, when non-empty, always contains exactly 2 elements, which is also constant space.

Learn more about how to find time and space complexity quickly.

Common Pitfalls

1. Integer Overflow with Large Input Values

When calculating difference = 4 * cheeseSlices - tomatoSlices, if cheeseSlices is very large (close to the maximum integer value), the multiplication by 4 could cause integer overflow in languages with fixed integer sizes. While Python handles arbitrary precision integers automatically, this could be problematic when translating the solution to other languages.

Solution: Add an early bounds check before the multiplication:

# Early check to prevent potential overflow issues
if cheeseSlices > tomatoSlices // 2:
    return []  # Impossible case since minimum tomatoes needed is 2 per cheese

2. Incorrect Order of Validation Checks

Performing division before checking if the difference is odd can lead to incorrect results. The code difference // 2 performs integer division, which truncates any decimal part. If we check for negative values before checking divisibility, we might miss cases where the difference is odd but positive.

Solution: Always check divisibility by 2 first:

# Check divisibility BEFORE using the division result
if difference % 2 != 0:
    return []
  
num_small_burgers = difference // 2

3. Missing Edge Case: Zero Ingredients

The code doesn't explicitly handle the edge case where both tomatoSlices and cheeseSlices are 0. While the current implementation handles this correctly (returning [0, 0]), it's not immediately obvious and could be confusing during debugging.

Solution: Add explicit handling for clarity:

# Handle the trivial case explicitly
if tomatoSlices == 0 and cheeseSlices == 0:
    return [0, 0]
if tomatoSlices == 0 or cheeseSlices == 0:
    return []  # One is zero but not both - impossible

4. Forgetting to Validate the Mathematical Relationship

A common mistake is assuming that any non-negative solution is valid without checking if the ingredients actually match the burger requirements. For instance, not verifying that the total tomatoes used equals the input.

Solution: Add a verification step (though mathematically unnecessary if the formula is correct, it helps catch implementation errors):

# Verify the solution (optional but helps catch bugs)
if (4 * num_jumbo_burgers + 2 * num_small_burgers == tomatoSlices and 
    num_jumbo_burgers + num_small_burgers == cheeseSlices):
    return [num_jumbo_burgers, num_small_burgers]
return []