In a three-digit number, one of the digits is 7 and the difference between any two of the digits is 4 or less.
What is the smallest this number could be?

To determine the smallest three-digit number where one of the digits is 7 and the difference between any two of the digits is 4 or less, we can follow these steps:

1. **Identify the range of digits around 7**:
   Since the difference between any two digits must be 4 or less, the digits can be:
   - \(7 - 4 = 3\) to \(7 + 4 = 11\)
   - However, since the digit range is 0 to 9, this limits us to the digits 3 through 9.

2. **Construct the smallest three-digit number**:
   We need to place the digit 7 in one of the positions (hundreds, tens, or units) and select the smallest possible digits for the remaining positions. Since we want the smallest overall number, we start by placing the smallest digits in the highest place value positions.

3. **Try placing 7 in the hundreds place**:
   - If 7 is the hundreds digit, we consider the smallest digits for the tens and units positions:
     - Possible digits: 3, 4, 5, 6, 8, 9 (since these are all within 4 units of 7).

   - The smallest digits among these are 3 and 4. Therefore, we construct numbers like 734, 743.

4. **Try placing 7 in the tens place**:
   - If 7 is the tens digit, the smallest hundreds digit is 1 (but that’s outside the range of 3 to 9, hence not valid). Therefore, the smallest valid hundreds digit is 3.
   - Units digit can be 3, 4, 5, 6, 8, or 9. Among these, 3 is the smallest.
   - This results in the number 373.

5. **Try placing 7 in the units place**:
   - If 7 is the units digit, the smallest possible hundreds digit is 3.
   - The smallest possible tens digit is also 3.
   - This results in the number 337.

**Comparing all possibilities**:
- 734
- 743
- 373
- 337

Among these numbers, **337** is the smallest three-digit number where one of the digits is 7, and the difference between any two digits is 4 or less.

So, the smallest number that satisfies the conditions is **337**.

Would you like more details or have any questions? Here are 5 questions you might consider asking next:
1. Can you explain how to find the largest number under the same conditions?
2. What if the difference between digits can be up to 5?
3. How does the method change if we use a four-digit number instead?
4. What are the possible numbers if the digit 7 must be in the hundreds place?
5. How would the solution change if the number has to be even?

**Tip**: When solving digit-related problems, always consider the constraints first and systematically test all possible configurations within those constraints to ensure the smallest or largest number is found.

Discover how to find the smallest three-digit number where one of the digits is 7 and the difference between any two digits is 4 or less. This math problem explores number theory and place value concepts. Suitable for students in Grades 4-6.

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