Let's solve this step by step:

Problem:

• The number is greater than 400 and smaller than 420.
• The digits of the number must add up to 14.

Solution:

1. List all numbers between 400 and 420:

   • 401, 402, 403, ..., 419.

2. Calculate the sum of the digits for each number:

   • 401: $4 + 0 + 1 = 5$
   • 402: $4 + 0 + 2 = 6$
   • 403: $4 + 0 + 3 = 7$
   • ...
   • 419: $4 + 1 + 9 = 14$

3. Identify the number where the digits add up to 14:

   • The number is 419.

Final Answer:

Paul is thinking of the number 419.

Do you have any further questions or need clarification? Here are 5 related questions to explore further:

1. What strategies can be used to solve similar problems with digit-sum constraints?
2. How would the answer change if the range were 500–520?
3. Can you generalize a rule to determine a digit-sum number for any range?
4. What applications might this type of problem-solving have in other contexts (e.g., cryptography)?
5. How could this be extended to a three-digit or multi-step problem?

Tip: When solving digit-sum problems, systematically test each candidate within the specified range to avoid missing possibilities.