Understanding the Batsman Score Comparison

The problem asks us to compare the scores of seven batsmen, B1, B2, B3, B4, B5, B6, and B7, based on several given conditions and determine how many batsmen scored more than B4. This is a type of logical reasoning puzzle that requires careful analysis of each statement to build a relative ranking of scores.

Analyzing the Given Conditions

Let's break down the information provided:

1. The score of B1 is greater than B7 and B3. We can write this as: $Score(B1) > Score(B7)$ and $Score(B1) > Score(B3)$.
2. The score of B6 is neither more nor less than the score of B4. This means their scores are equal: $Score(B6) = Score(B4)$.
3. The score of B6 is more than B1. We write this as: $Score(B6) > Score(B1)$.
4. B5 scored less than only one batsman. This is a crucial piece of information. It means that there is exactly one batsman with a score higher than B5, and B5's score is higher than all other batsmen (except that one). In a sorted list of scores from highest to lowest, B5 has the second-highest score. Let the scores be $S_1 \ge S_2 \ge S_3 \ge S_4 \ge S_5 \ge S_6 \ge S_7$. This condition means $S_1 > S_2$ and $S_2 > S_3$, and B5's score is $S_2$. The highest score ($S_1$) is achieved by exactly one batsman.
5. The score of B7 is not the least score. This means B7 does not have the lowest score among the seven batsmen.

Deducing the Ranking of Batsmen Scores

Let's combine the conditions to figure out the relative scores:

• From condition 2, we know $Score(B6) = Score(B4)$.
• From condition 3, we have $Score(B6) > Score(B1)$. Substituting condition 2, we get $Score(B4) > Score(B1)$.
• From condition 1, we know $Score(B1) > Score(B7)$ and $Score(B1) > Score(B3)$.
• Combining these, we have the partial ranking: $Score(B6) = Score(B4) > Score(B1) > \{Score(B7), Score(B3)\}$. At this point, we don't know the relationship between B7 and B3, or where B2 and B5 fit.

Now, let's use condition 4: B5 scored less than only one batsman. This establishes B5 as the batsman with the second-highest score ($S_2$), and there is exactly one batsman with the highest score ($S_1$). So, the top part of the ranking is: $S_1$ (by one batsman) $> S_2$ (by B5) $> S_3 \ge ...$

We know $Score(B6) = Score(B4) > Score(B1)$. Based on the strict inequality at the top ($S_1 > S_2 > S_3$), B6 and B4 (being equal) cannot be at Rank 1 (single batsman) or Rank 2 (B5). Therefore, B6 and B4 must share the third-highest score level ($S_3$).

So the ranking begins: $S_1$ (Rank 1) $> Score(B5)$ (Rank 2) $> Score(B6) = Score(B4)$ (Rank 3).

Who is the batsman with the highest score ($S_1$, Rank 1)? We have accounted for B4, B5, B6, B1, B3, B7 in the relationships $Score(B6) = Score(B4) > Score(B1) > \{Score(B7), Score(B3)\}$. The only batsman not mentioned in these comparisons is B2. Given that B6 and B4 are already at Rank 3 (below Rank 1 and 2), and B1, B7, B3 are even lower, none of them can be Rank 1. Thus, B2 must be the batsman with the highest score ($S_1$).

So far, the ranking is: $Score(B2)$ (Rank 1) $> Score(B5)$ (Rank 2) $> Score(B6) = Score(B4)$ (Rank 3).

Now let's place the remaining batsmen: B1, B7, B3. We know $Score(B6) = Score(B4) > Score(B1) > \{Score(B7), Score(B3)\}$. This means B1's score is lower than B6/B4's score ($S_3$), and B7 and B3's scores are lower than B1's score. B1 must be at Rank 4 ($S_4$), and B7 and B3 must occupy Rank 5 ($S_5$) and Rank 6 ($S_6$).

The ranking structure is: $Score(B2)$ (Rank 1) $> Score(B5)$ (Rank 2) $> Score(B6) = Score(B4)$ (Rank 3) $> Score(B1)$ (Rank 4) $> \{Score(B7), Score(B3)\}$ (Ranks 5 and 6).

Finally, consider condition 5: B7 is not the least score. The least score in this structure is at Rank 6. Therefore, B7 cannot be at Rank 6. This means B3 must be at Rank 6 (the lowest score), and B7 must be at Rank 5.

The complete ranking from highest score to lowest score is:

$Score(B2) > Score(B5) > Score(B6) = Score(B4) > Score(B1) > Score(B7) > Score(B3)$

Let's verify this ranking against all conditions:

All conditions are satisfied by this ranking.

Answering the Question

The question asks: How many batsman scored more than B4?

Looking at our final ranking:

$Score(B2) > Score(B5) > Score(B6) = Score(B4) > Score(B1) > Score(B7) > Score(B3)$

Batmen who scored more than B4 are those with a strictly higher score. Based on the ranking, the batsmen with scores greater than B4 are:

• B5 (whose score is greater than B4)
• B2 (whose score is greater than B5, and thus greater than B4)

There are two batsmen who scored more than B4: B2 and B5.

Conclusion

By carefully analyzing and combining the given conditions about the batsmen's scores, we determined the relative ranking. Based on this ranking, we found that exactly two batsmen scored more than B4.

The number of batsmen who scored more than B4 is 2.

Revision Table: Key Deductions

Additional Information: Logical Reasoning in Rankings

Ranking and comparison problems like this one are common in logical reasoning sections of competitive exams. They test your ability to process multiple pieces of conditional information, combine them logically, and deduce relationships that are not explicitly stated. Key strategies involve:

• Representing relationships using symbols (e.g., >, <, =).
• Identifying absolute positions or key constraints (like "scored less than only one", which fixes the 2nd position).
• Combining related statements to build chains of inequalities (e.g., if A > B and B > C, then A > C).
• Handling ties carefully, as they occupy a single rank position but include multiple entities.
• Systematically placing elements into a potential ranking structure, testing against all conditions, and refining as needed.
• Using elimination to determine the position of elements not explicitly compared to many others (like B2 in this problem).

Practicing various types of ranking puzzles helps improve analytical skills and speed in solving such questions.