In a certain code language, 'how many goals scored' is written as '5397'; 'many more matches' is written as '982'; and 'he scored five' is written as '163'. In this language, how is the word 'goals' written?

Introduction / Context:
Here we again see a coded language where each word is represented by a single digit. Three coded sentences are given, and we are asked to find the code for the word 'goals'. This is a classic example where we can find some mappings, but ultimately the information is not enough to uniquely determine the digit for 'goals'. Recognising that the data is insufficient is essential in such problems.

Given Data / Assumptions:

• 'how many goals scored' is written as '5397'.
• 'many more matches' is written as '982'.
• 'he scored five' is written as '163'.
• Each distinct word is represented by exactly one digit.
• The position of digits within a code does not matter for solving; only the set of digits is important.

Concept / Approach:
We use intersection logic on sets of words and digits. The common word between two sentences must correspond to the common digit in their codes. This allows us to identify the digits for 'many' and 'scored'. After assigning these, we see which digits remain for 'how' and 'goals'. If two words share two remaining digits with no extra information, then there is no unique mapping, and the correct logical answer is that the data is not sufficient.

Step-by-Step Solution:
Step 1: Compare 'how many goals scored' (5397) with 'many more matches' (982). The common English word is 'many'. Step 2: The common digit between 5397 and 982 is 9. Thus, 'many' is coded as 9. Step 3: Compare 'how many goals scored' (5397) with 'he scored five' (163). The common English word is 'scored'. Step 4: The common digit between 5397 and 163 is 3. Therefore, 'scored' is coded as 3. Step 5: In the first sentence, the digits are 5, 3, 9 and 7, corresponding to 'how', 'many', 'goals' and 'scored'. We now know that 9 is 'many' and 3 is 'scored'. Step 6: The remaining digits 5 and 7 must correspond to 'how' and 'goals' in some order, but the information given does not tell us which is which.

Verification / Alternative check:
We can attempt both assignments: if 'goals' is 5 and 'how' is 7, all three sentences still translate consistently. If we swap them, letting 'goals' be 7 and 'how' be 5, the translations remain just as valid. Since both possibilities satisfy all given data, the code for 'goals' cannot be uniquely determined from the information provided.

Why Other Options Are Wrong:
Option A: 5 is a possible code for 'goals' but is not uniquely forced. Option B: 7 is also a possible code for 'goals' but again is not uniquely forced. Option C: 3 is already identified as the code for 'scored', not 'goals'. Option E: 9 is already fixed as the code for 'many'.

Common Pitfalls:
Students sometimes assume that when two digits remain, the smaller or larger one must correspond to a particular word, but that is not logically justified. Another error is to overlook that 'how' and 'goals' appear only in one sentence, which makes them indistinguishable from each other based solely on the given data. In such questions, it is crucial to accept 'Data is not sufficient' when multiple mappings work equally well.

Final Answer:
The code for 'goals' cannot be uniquely determined from the given information, so the correct answer is Data is not sufficient.