In a certain numerical code, the symbol "$" represents a special operation between two numbers. It is given that 23 $ 35 = 13 and 3 $ 5 = 8. Using the same rule, what is the value of 4 $ 13?

Introduction:
This question uses a special operator "$" applied to pairs of numbers. The operator does not carry its usual programming or financial meaning here. Instead, two example equations show how it behaves, and we must infer the underlying rule and then apply it to a new pair, 4 and 13. This is a typical pattern recognition exercise in numerical reasoning.

Given Data / Assumptions:
The relationships provided are:
1) 23 $ 35 = 13
2) 3 $ 5 = 8
We need to find 4 $ 13. We assume:
1) The operation $ is the same for every pair of numbers in the question.
2) The rule is built from simple operations on the digits of the numbers, since the outputs are much smaller than the original values.

Concept / Approach:
Because the results 13 and 8 are small compared to 23 and 35, it is natural to suspect that $ may involve the sum of individual digits. For 23 and 35, both are two digit numbers, so we can test if the sum of all their digits equals the result. If the same pattern also explains the second example, we can confidently apply it to 4 and 13.

Step-by-Step Solution:
Step 1: Consider 23 $ 35 = 13. Break 23 into digits 2 and 3, and 35 into digits 3 and 5.Step 2: Add all digits together: 2 + 3 + 3 + 5 = 13. This matches the given result, so the rule is consistent here.Step 3: Now test the rule on the second example 3 $ 5 = 8. Both numbers are single digits, so add them directly: 3 + 5 = 8, which again matches the provided value.Step 4: Since both examples are explained by "add all digits of both numbers", we adopt this as the rule for the $ operation.Step 5: Apply this rule to 4 $ 13. The digits are 4, 1, and 3.Step 6: Add them: 4 + 1 + 3 = 8.Step 7: Therefore, 4 $ 13 equals 8 under this operation.

Verification / Alternative check:
If we attempted any other simple pattern such as taking differences, products, or averages, they would not fit both examples simultaneously. For instance, 23 + 35 equals 58, not 13, and 23 minus 35 is negative, which is not used. Hence, the digit sum rule is the most straightforward and uniquely consistent explanation.

Why Other Options Are Wrong:
The value 14 would require an extra addition not supported by the examples. The value 6 would correspond to 4 + 2, with no reason to extract only a subset of digits. The value 49 is far from any simple digit based rule, and 10 would require adding an invented constant. None of these align with both given examples.

Common Pitfalls:
Sometimes learners immediately try to apply operations directly to the full numbers, like 23 + 35, and overlook digit wise patterns. Another mistake is to find a rule that fits only one example and then use it without testing on the other. Always verify your proposed rule on all given sample cases before generalising.

Final Answer:
According to the pattern, the value of 4 $ 13 is 8.