In a coded arithmetic pattern, the expressions 37 * 14 = 17, 69 * 33 = 34 and 91 * 125 = 72 follow the same rule. Using this rule, what should be the value of 28 * 56 in the same code?

Introduction / Context:
This question presents a set of unusual equations where the symbol star (*) does not stand for ordinary multiplication. Instead, each pair of numbers is transformed into a different result according to a hidden numerical pattern. The challenge is to identify this pattern from the given examples and then apply it to a new pair of numbers.

Given Data / Assumptions:

• 37 * 14 results in 17.
• 69 * 33 results in 34.
• 91 * 125 results in 72.
• The same rule must be used to find the value of 28 * 56.
• The symbol * indicates a special operation defined by the pattern, not standard multiplication.

Concept / Approach:
When working with coded equations, we look for simple relationships involving addition, subtraction, averages or ratios. Here, each equation involves two numbers on the left and a smaller number on the right. A natural idea is to test whether the result might be related to the sum of the two numbers, possibly scaled or divided by a constant factor. We check the first example, and if it works, we verify against the remaining examples to establish the rule.

Step-by-Step Solution:
Step 1: For 37 * 14, compute the sum of the two numbers: 37 + 14 = 51.Step 2: Compare 51 with the coded result 17. Noting that 51 ÷ 3 = 17 suggests that the rule might be to divide the sum by 3.Step 3: Test this rule with the second equation. For 69 * 33, sum the numbers: 69 + 33 = 102. Now 102 ÷ 3 = 34, which matches the coded result.Step 4: Test the rule with the third equation. For 91 * 125, sum the numbers: 91 + 125 = 216. Then 216 ÷ 3 = 72, which again matches the coded result.Step 5: Since all three examples fit the rule result = (first number + second number) ÷ 3, we accept this as the governing pattern.Step 6: Apply the rule to 28 * 56. Sum the numbers: 28 + 56 = 84.Step 7: Now divide by 3: 84 ÷ 3 = 28.Step 8: Therefore, 28 * 56 should be coded as 28.

Verification / Alternative check:
To ensure there is no alternative rule, you can test whether any simple product or difference based pattern could match all three given equations. None of these alternatives give 17 from 37 and 14, and also give 34 from 69 and 33, and 72 from 91 and 125 simultaneously. The sum divided by 3 rule works consistently in every case, and recomputing 84 ÷ 3 once more confirms that the answer 28 is correct.

Why Other Options Are Wrong:
The option 26 might come from subtracting two or misapplying division. The option 34 or 42 could arise if someone mistakenly uses one of the intermediate sums from the examples instead of applying the rule properly. The option 30 might be guessed by rounding or by using an incorrect divisor. Since only 28 matches the pattern confirmed by all three given equations, the other values must be rejected.

Common Pitfalls:
Students may try to invent very complicated patterns involving multiplication, subtraction or digit manipulation and overlook a simple sum and division rule. Another common error is to identify a rule that fits only the first equation but not the others. Always verify the pattern against every given example before applying it to the target expression to avoid unreliable shortcuts.

Final Answer:
According to the established pattern, the coded value of 28 * 56 is 28.