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Number of Burgers with No Waste of Ingredients

Given two integers tomatoSlices and cheeseSlices. The ingredients of different burgers are as follows:

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

Return [total_jumbo, total_small] so that the number of remaining tomatoSlices equal to 0 and the number of remaining cheeseSlices equal to 0. If it is not possible to make the remaining tomatoSlices and cheeseSlices equal to 0 return [].

 

Example 1:

Input: tomatoSlices = 16, cheeseSlices = 7
Output: [1,6]
Explantion: To make one jumbo burger and 6 small burgers we need 4*1 + 2*6 = 16 tomato and 1 + 6 = 7 cheese.
There will be no remaining ingredients.

Example 2:

Input: tomatoSlices = 17, cheeseSlices = 4
Output: []
Explantion: There will be no way to use all ingredients to make small and jumbo burgers.

Example 3:

Input: tomatoSlices = 4, cheeseSlices = 17
Output: []
Explantion: Making 1 jumbo burger there will be 16 cheese remaining and making 2 small burgers there will be 15 cheese remaining.

 

Constraints:

  • 0 <= tomatoSlices, cheeseSlices <= 107

Solution Explanation for Number of Burgers with No Waste of Ingredients

This problem involves finding the number of jumbo and small burgers that can be made using a given number of tomato and cheese slices, without any leftovers. Each jumbo burger requires 4 tomato slices and 1 cheese slice, while each small burger needs 2 tomato slices and 1 cheese slice.

The solution utilizes a mathematical approach to efficiently solve this problem.

Mathematical Formulation:

Let's define:

  • x: the number of jumbo burgers
  • y: the number of small burgers

We can set up two equations based on the ingredient constraints:

  • Tomato slices: 4x + 2y = tomatoSlices
  • Cheese slices: x + y = cheeseSlices

We can solve for x and y using these equations. The most efficient way is to solve for one variable in terms of the other and then check for constraints. One approach is solving the second equation for x (x = cheeseSlices - y), then substituting this into the first equation:

4(cheeseSlices - y) + 2y = tomatoSlices

Simplifying, we get:

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

Once we have y, we can easily find x using: x = cheeseSlices - y

Constraints:

The solutions x and y must satisfy several conditions:

  • Non-negative: Both x and y must be greater than or equal to 0 (you can't make a negative number of burgers).
  • Integer: x and y must be integers (you can't make half a burger).
  • Even: (4 * cheeseSlices - tomatoSlices) must be an even number for y to be an integer.

Algorithm:

  1. Calculate k = 4 * cheeseSlices - tomatoSlices.
  2. Calculate y = k / 2.
  3. Calculate x = cheeseSlices - y.
  4. Check if k is even, and if x and y are both non-negative.
  5. If all conditions are met, return [x, y]; otherwise, return an empty list [].

Time and Space Complexity:

  • Time Complexity: O(1) - The solution involves a fixed number of arithmetic operations regardless of the input size.
  • Space Complexity: O(1) - The solution uses a constant amount of extra space to store variables.

Code Implementation (Python):

class Solution:
    def numOfBurgers(self, tomatoSlices: int, cheeseSlices: int) -> List[int]:
        k = 4 * cheeseSlices - tomatoSlices
        y = k // 2  # Integer division
        x = cheeseSlices - y
        if k % 2 == 0 and x >= 0 and y >= 0:
            return [x, y]
        else:
            return []

The code in other languages (Java, C++, Go, TypeScript, Rust) follows the same algorithm and logic, simply adapting the syntax to each language's specifics. The core operations remain consistent across all implementations.