In a special operation denoted by "x", the result is defined by the difference between the sum of the digits of the first number and the sum of the digits of the second number. It is given that 56 x 11 = 9, 37 x 13 = 6 and 42 x 12 = 3. Using the same rule, what is the value of 87 x 77 ?

Introduction / Context:
This question defines a custom operation "x" between two numbers based on their digit sums. You are told the outcomes for three pairs and asked to compute the outcome for a fourth pair. These problems test pattern recognition with a focus on simple properties of digits rather than full numeric operations like multiplication.

Given Data / Assumptions:

• 56 x 11 = 9.
• 37 x 13 = 6.
• 42 x 12 = 3.
• The same rule is used to compute 87 x 77.
• The definition suggested in the question description is that the result equals the sum of the digits of the first number minus the sum of the digits of the second number.

Concept / Approach:
We first verify that the supposed rule, digit sum difference, fits all three given examples. To do this, we compute the sum of digits for each of the two numbers in an example pair, subtract the second sum from the first sum and check if the result matches the given output. If the rule holds for all examples, we can safely apply it to 87 and 77. This approach is simpler and more consistent than trying to interpret x as normal multiplication.

Step-by-Step Solution:
Step 1: For 56 x 11, find the digit sums. Sum of digits of 56: 5 + 6 = 11. Sum of digits of 11: 1 + 1 = 2. Difference: 11 - 2 = 9, which matches the given value. Step 2: For 37 x 13, compute the digit sums. Sum of digits of 37: 3 + 7 = 10. Sum of digits of 13: 1 + 3 = 4. Difference: 10 - 4 = 6, matching the given value. Step 3: For 42 x 12, compute the digit sums. Sum of digits of 42: 4 + 2 = 6. Sum of digits of 12: 1 + 2 = 3. Difference: 6 - 3 = 3, which again matches the given value. Step 4: The rule is confirmed: a x b = (sum of digits of a) - (sum of digits of b). Step 5: Now apply this rule to 87 x 77. Sum of digits of 87: 8 + 7 = 15. Sum of digits of 77: 7 + 7 = 14. Difference: 15 - 14 = 1.

Verification / Alternative check:
Because the same simple digit sum difference rule explains all three given examples exactly, there is no need to look for a more complicated pattern. Any other rule would need to be at least as consistent and would also have to give 1 for 87 and 77, which is unlikely without becoming contrived. Recomputing 8 + 7 and 7 + 7 confirms that the difference really is 1, so the result is secure.

Why Other Options Are Wrong:
Options 2, 3, 4 and 5 would require different differences of digit sums or alternative definitions, but these would fail when tested on 56, 11 and the other given pairs. Because the question insists that all equations are solved on the basis of one system, only the value that arises from that system can be accepted. Thus, a difference of exactly 1 is the only correct outcome for 87 x 77.

Common Pitfalls:
A common pitfall is to read x as normal multiplication and attempt to relate the product to the output, which does not yield a consistent pattern here. Another is to test a rule on just one example and not verify it across all three. Always ensure that any pattern you conjecture matches every given instance before applying it to the unknown case.

Final Answer:
Using the digit sum difference rule, the value of 87 x 77 is 1.