Introduction

Petit fours, those delightful, bite-sized pastries, have fascinated chefs, bakers, and math enthusiasts alike. This article explores a unique math problem related to petit fours, delving into their variety and the intriguing world of number theory. We will also discuss the practical application of math in culinary arts.

The Math Problem with Petit Fours

To begin, let's revisit a math problem that Fred presented involving petit fours. The problem states that a pastry chef made 1 kg (1000 grams) of petit fours, each weighing more than 10 grams. The chef has one petite four left over when tried to put them in boxes in rows of 2, 3, 4, 5, or 6.

This problem requires understanding of number theory, specifically the concept of prime numbers and least common multiples (LCM).

Understanding the Constraints: Each petit four must weigh more than 10 grams. If the total weight is 1000 grams, the maximum number of petit fours is 100 (1000 / 10).Mathematical Insight: The chef has one left over in each case, meaning the number of petit fours must be a prime number greater than the LCM of 2, 3, 4, 5, and 6. The LCM of these numbers is 60. Therefore, the number of petits fours must be one more than a multiple of 60, i.e., 60k 1, where k is a non-negative integer.Prime Factorization: We need to find the smallest number greater than 60 that is one more than a multiple of 60, which is also a prime number. The smallest such number is 61, as 61 is a prime number and equals 60 * 1 1.

Thus, the pastry chef made 61 petit fours.

About Petit Fours

Petit fours, whether traditional or modern, add a touch of elegance to any dessert menu. Traditional petit fours consist of multiple layers of Genoese sponge with jam adorned with fondant icing, but they can also be made with marzipan, frangipane, or nuts. These treats come in various shapes and sizes, adding to the complexity of determining the number made by the pastry chef.

The versatility of petit fours makes it challenging to give an exact count. From round to cube shapes, the variations are endless. Nonetheless, if made well, these petite confections often weigh between 10 to 20 grams each. Given the constraints of the problem, the chef could have made a maximum of 100 petit fours, each weighing more than 10 grams.

The Math Problem with Cafeteria Food

Next, let's consider a math problem related to a hungry math major visiting the school cafeteria. The problem involves taking k servings of food, including at least one main course, an even number of side vegetables, an odd number of rolls, and at least two desserts. All food items are distinguishable only in the coarsest way (e.g., a serving is a serving).

The solution to this problem involves combinatorics. Let's break it down step by step:

Main Course: At least one main course. If there are n main courses, there are (2^{n} - 1) ways to choose at least one.Side Vegetables: An even number. Each vegetable can either be included (1) or not (0), with the inclusion of an even number of vegetables. If there are m side vegetables, the number of ways to choose an even number is (2^{(m-1)}).Rolls: An odd number. Similar to side vegetables, if there are r rolls, the number of ways to choose an odd number is (2^{(r-1)}).Desserts: At least two. If there are s desserts, the number of ways to choose at least two is (sum_{k2}^{s} binom{s}{k}).

The total number of ways to take the food is the product of these combinations. This problem showcases the application of combinatorial mathematics in practical scenarios like dining in a cafeteria.

Conclusion

In conclusion, the math problem involving petit fours demonstrates the intricate relationship between number theory and real-world applications. Understanding prime numbers, LCM, and combinatorial mathematics provides valuable insights into problems that might seem complex at first glance. Whether it's in the kitchen or the cafeteria, the beauty of mathematics lies in its ability to offer elegant solutions to practical problems.