If 29$14 = 6 and 37$25 = 3 under a certain rule, then what is the value of 84$62 under the same rule?

Introduction / Context:
This is a two-number coding puzzle with a custom operator $. Instead of standard arithmetic, the value of a $ b is defined by a pattern using the digits of the numbers. We are given two examples and must use them to decode the rule and then apply it to 84$62.

Given Data / Assumptions:

• 29$14 = 6.
• 37$25 = 3.
• We must find 84$62.
• The rule likely involves the digits of each number rather than the full values directly.

Concept / Approach:
The results 6 and 3 are small numbers, suggesting a relationship involving sums of digits and their difference. We test whether the result equals the difference between the digit sums of the two numbers, which is a common pattern in such puzzles.

Step-by-Step Solution:
Step 1: For 29$14 = 6. Sum of digits of 29 = 2 + 9 = 11. Sum of digits of 14 = 1 + 4 = 5. Difference = 11 − 5 = 6, which matches the given result. Step 2: For 37$25 = 3. Sum of digits of 37 = 3 + 7 = 10. Sum of digits of 25 = 2 + 5 = 7. Difference = 10 − 7 = 3, again matching the given result. Step 3: The rule is now clear: a $ b equals the difference between the sum of digits of a and the sum of digits of b. Step 4: Apply the rule to 84$62. Sum of digits of 84 = 8 + 4 = 12. Sum of digits of 62 = 6 + 2 = 8. Difference = 12 − 8 = 4.

Verification / Alternative check:
We have verified the rule with both given examples and it fits perfectly. No other simple digit-based rule (such as adding or multiplying digit sums) gives 6 and 3 simultaneously for the first two cases, so this rule is robust.

Why Other Options Are Wrong:
The values 8, 13, and 15 do not equal the difference between the digit sums of 84 and 62. Any rule that produced those values would contradict the first two examples, making them inconsistent with the given coding pattern.

Common Pitfalls:
A common error is to treat $ as a normal arithmetic operation like addition or multiplication of the whole numbers. Another is to confuse the sum of the numbers with the sum of their digits. Paying attention to small results often hints that digit-level operations are involved.

Final Answer:
According to this digit-sum difference rule, 84$62 evaluates to 4.