Let's analyze each of the given statements and select the correct open phrase:

1. Thrice the difference of the integer $N$ and 4

   • "Thrice" means three times, and "the difference of $N$ and 4" means $N - 4$.
   • So, this translates to $3(N - 4)$.
   • Correct Answer: 3(N - 4)

2. Four more than six times the integer $N$

   • "Six times the integer $N$" means $6N$, and "four more than" means we add 4 to this.
   • So, this translates to $6N + 4$.
   • Correct Answer: 6N + 4

3. The cube of the difference of two numbers $N$ and $P$

   • "The difference of $N$ and $P$" means $N - P$, and "the cube of" means raising this difference to the third power.
   • So, this translates to $(N - P)^3$.
   • Correct Answer: (N - P)^3

Summary:

1. Thrice the difference of the integer $N$ and 4: 3(N - 4)
2. Four more than six times the integer $N$: 6N + 4
3. The cube of the difference of two numbers $N$ and $P$: (N - P)^3

Would you like more details on any of these open phrases, or do you have any questions?

Here are 5 related questions:

1. How would you express "twice the sum of $N$ and 7" in algebraic form?
2. What is the difference between a phrase like "thrice" and "three times" in mathematical expressions?
3. How do you represent "five less than four times $N$" algebraically?
4. Can you explain how to translate "the square of a sum" into an expression?
5. What are some common mistakes when translating verbal phrases into algebraic expressions?

Tip: When translating word phrases to algebraic expressions, carefully identify the operations (like addition, subtraction, multiplication) and their order to avoid errors.