To determine the greatest number of platters Evan can prepare such that no cake is left over, we need to find the greatest common divisor (GCD) of the number of pieces of chocolate cake (6) and cheesecake (18).

Step-by-Step Solution:

1. List the factors of each number:

   • Factors of 6: $1, 2, 3, 6$
   • Factors of 18: $1, 2, 3, 6, 9, 18$

2. Identify the common factors:

   • Common factors of 6 and 18: $1, 2, 3, 6$

3. Find the greatest common factor:

   • The greatest common factor is $6$.

Conclusion:

The greatest number of identical platters Evan can prepare is 6.

Each platter will have:

• $\frac{6}{6} = 1$ piece of chocolate cake,
• $\frac{18}{6} = 3$ pieces of cheesecake.

Would you like a detailed explanation of GCD or any questions related to this? Here are 5 further exploration points:

1. How is the GCD of two numbers calculated using other methods (e.g., Euclidean algorithm)?
2. Could Evan use a different number of platters if he allowed leftover cake?
3. How many cakes and cheesecakes does Evan need to serve 12 guests with equal portions?
4. Can this approach be applied to more than two desserts? How?
5. What are the real-life implications of understanding GCD in planning and logistics?

Tip: When planning identical groupings, always check for the GCD to simplify your arrangements!