Old Clarrie has three dogs. The oldest is Bob, next comes Rex nil Fido is the youngest. Fido is 10 years younger than Bob, and none of the dogs are the same age.

When Clarrie adds their ages together they come to 28 years

When Clarrie multiplies their ages together, he gets a 1

What is the smallest that this number could be?

Let's denote the ages of Bob, Rex, and Fido as \( b \), \( r \), and \( f \), respectively. According to the problem:

1. \( f = b - 10 \)
2. \( b + r + f = 28 \)
3. \( b \times r \times f = 1 \)

From equation (1), we substitute \( f \) in equation (2):
\[ b + r + (b - 10) = 28 \]
\[ 2b + r - 10 = 28 \]
\[ 2b + r = 38 \]
\[ r = 38 - 2b \]

Now, substitute \( f \) and \( r \) in equation (3):
\[ b \times (38 - 2b) \times (b - 10) = 1 \]

Let's simplify the problem further and test possible integer values for \( b \), since ages are generally whole numbers:

1. \( b = 11 \)
   \[ r = 38 - 2 \times 11 = 16 \]
   \[ f = b - 10 = 11 - 10 = 1 \]
   Checking the product:
   \[ 11 \times 16 \times 1 = 176 \]
   (This is not equal to 1)

2. \( b = 12 \)
   \[ r = 38 - 2 \times 12 = 14 \]
   \[ f = b - 10 = 12 - 10 = 2 \]
   Checking the product:
   \[ 12 \times 14 \times 2 = 336 \]
   (This is not equal to 1)

3. \( b = 13 \)
   \[ r = 38 - 2 \times 13 = 12 \]
   \[ f = b - 10 = 13 - 10 = 3 \]
   Checking the product:
   \[ 13 \times 12 \times 3 = 468 \]
   (This is not equal to 1)

4. \( b = 14 \)
   \[ r = 38 - 2 \times 14 = 10 \]
   \[ f = b - 10 = 14 - 10 = 4 \]
   Checking the product:
   \[ 14 \times 10 \times 4 = 560 \]
   (This is not equal to 1)

Since the calculations seem off, we re-evaluate if the product \( b \times r \times f = 1 \) is feasible with the given constraints. On further reflection, it might be apparent that a mistake was made in interpreting the constraints for realistic ages as multipliers. Thus, the constraints appear unsolvable with the typical interpretation of age.

Given the mathematical constraints, the ages resulting in \( b \times r \times f = 1 \) must imply non-integer values due to the nature of the problem. This could point towards there being a possible alternative set not limited to integers or suggesting there is no real-world integer solution aligning perfectly with provided conditions.

Thus, this condition might not have a solution within whole-number realistic ages as per the initial problem statement when re-checked. However, for learning engagement:

Would you like me to explore a different set or further dive deeper into a hypothesis for smallest possible plausible explanation for the above or any other query? Here are some questions you might consider next:

1. Can you explain in more detail how the ages addition and multiplication relate?
2. What approach would work for decimal ages instead?
3. Can the product of ages be related to integer values under some approximation?
4. Would a hypothetical solution with one age missing solve?
5. What examples in real life align closely with this math problem-solving? 

**Tip**: Always recheck constraints in a problem for consistency with realistic assumptions to avoid circular dependencies or unsolvable conditions.

Discover how to solve the age puzzle involving Clarrie's three dogs: Bob, Rex, and Fido. Learn about equations, number theory, and solving systems of equations. Suitable for students exploring algebraic concepts in Grades 7-9.

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