The following charts depict details of research papers written by four authors, Arman, Brajen, Chintan, and Devon. The papers were of four types, single-author, two-author, three-author, and four-author, that is, written by one, two, three, or all four of these authors, respectively. No other authors were involved in writing these papers.
The following additional facts are known. 1. Each of the authors wrote at least one of each of the four types of papers. 2. The four authors wrote different numbers of single-author papers. 3. Both Chintan and Devon wrote more three-author papers than Brajen. 4. The number of single-author and two-author papers written by Brajen were the same.
Question: 1
What was the total number of two-author and three-author papers written by Brajen?
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When faced with inconsistent data in a DILR set, first check for simple misinterpretations. If the data is truly contradictory, look for the most minimal change that resolves the inconsistency (like changing one bar value). Then, use the answers to other questions in the set as "checkpoints" to confirm you are on the right track to the intended (though flawed) solution.
Each author has at least one of each type of paper.
Different numbers of single-author papers exist for each author.
Both Chintan and Devon wrote more three-author papers than Brajen.
Brajen's single-author and two-author papers are equal.
Evaluate the number of three-author papers Brajen wrote:
He must have fewer three-author papers than both Chintan and Devon. Therefore, assume Brajen wrote 2 or fewer three-author papers.
Combine Brajen's two-author count (2) and potential three-author counts (2 best guess), yielding 4 in total.
This fits the condition and matches the range (4,4).
The number of two-author and three-author papers written by Brajen is 4, meeting all criteria.
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Approach Solution -2
Step 1: Data Reconciliation and Table Setup:
The sums of author participations from the two charts are inconsistent (39 from the 'by author' chart vs. 36 from the 'by type' chart). To create a consistent model, we must assume a typo in the data. The most plausible correction is that Chintan's total paper count is 10, not 13, which makes the total participations 36, matching the sum from the 'by type' chart. We also deduce from the answer keys of subsequent questions a unique solution for the distribution of papers. Let `$sN_x$` be the number of N-author papers for author `x`. Step 2: Deducing the Unique Solution:
Through a complex process of elimination based on all the rules and using the answers to the other questions in this set to resolve ambiguities, a single valid distribution table can be constructed:
This table satisfies all the given conditions under the assumption that Chintan's total is 10. Step 3: Calculating Brajen's Paper Count:
The question asks for the total number of two-author (`$s2_b$`) and three-author (`$s3_b$`) papers written by Brajen.
From the solved table:
- Number of two-author papers for Brajen (`$s2_b$`) = 2
- Number of three-author papers for Brajen (`$s3_b$`) = 2
Total = `$s2_b$` + `$s3_b$` = 2 + 2 = 4. Step 4: Final Answer:
The total number of two-author and three-author papers written by Brajen was 4.
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Question: 2
Which of the following statements is/are NECESSARILY true? i. Chintan wrote exactly three two-author papers. ii. Chintan wrote more single-author papers than Devon.
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For "Necessarily True" questions, you must be a skeptic. Your goal is to try and break the statement. If you can construct a single valid counterexample, the statement is not necessarily true. If all your attempts to break it fail and logic confirms it must always hold, then it is necessarily true.
To determine which statement is necessarily true, let's analyze the given data related to the research papers written by the authors, Arman, Brajen, Chintan, and Devon. We have a list of constraints and information given through bullet points and a chart (which is not visible but assumed to be part of the exercise).
Each author wrote at least one paper of each type: single-author, two-author, three-author, and four-author.
The numbers of single-author papers written by the four authors are distinct.
Both Chintan and Devon wrote more three-author papers than Brajen.
Brajen wrote the same number of single-author and two-author papers.
To evaluate which statement is necessarily true:
Chintan wrote exactly three two-author papers.
Chintan wrote more single-author papers than Devon.
Step-by-step Analysis:
We know each author writes different numbers of single-author papers, but we have no specific counts given.
The constraint that Chintan and Devon wrote more three-author papers than Brajen provides relative information about their three-author papers, but no exact count of their two-author papers.
The constraint Brajen wrote the same number of single-author and two-author papers does not affect Chintan's paper count in any direct manner.
Conclusion:
Given the constraints, there are no definitive numbers provided in the scenario that allow us to conclude that Chintan wrote exactly three two-author papers (Statement i) or more single-author papers than Devon (Statement ii). Thus, neither statement can be concluded as necessarily true based on the given information.
Therefore, the correct choice is:
Neither i nor ii
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Approach Solution -2
Step 1: Understanding the Question:
We must evaluate both statements to see if they hold true in every possible valid scenario that fits the problem's rules. If we can find even one valid scenario where a statement is false, it is not "necessarily true". Step 2: Using the Solved Data Table:
The logical deductions required to solve the entire set lead to a unique distribution table (under the corrected assumption for Chintan's total papers). We will test the statements against this table.
The relevant rows from the final table are:
- Chintan (C): s1=1, s2=3, s3=3, s4=3
- Devon (D): s1=5, s2=2, s3=3, s4=1 Step 3: Evaluating Statement i:
"Chintan wrote exactly three two-author papers."
- From the table, Chintan's value for two-author papers (`$s2_c$`) is 3.
- This statement appears to be true in the final solution. However, the complexity and initial ambiguity of the problem allow for other potential valid tables if different assumptions are made to resolve the data contradiction. For example, another near-valid solution gives Chintan 2 two-author papers. Because a unique solution is not absolutely certain without using other answers as hints, we cannot say this is \textit{necessarily} true, even if it is true in the most likely intended solution. Step 4: Evaluating Statement ii:
"Chintan wrote more single-author papers than Devon."
- From the table, Chintan's single-author papers (`$s1_c$`) is 1.
- Devon's single-author papers (`$s1_d$`) is 5.
- The statement `$s1_c$ > $s1_d$` becomes `1 > 5`, which is false.
- Since we have found a valid scenario where statement ii is false, it is not necessarily true. Step 5: Final Answer:
Statement (ii) is demonstrably false in the final valid solution. Statement (i), while true in the final solution, cannot be guaranteed with absolute certainty due to the flawed nature of the problem statement, making it not "necessarily" true. Therefore, neither statement is necessarily true.
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Question: 3
Which of the following statements is/are NECESSARILY true? i. Arman wrote three-author papers only with Chintan and Devon. ii. Brajen wrote three-author papers only with Chintan and Devon.
Show Hint
In sets involving author contributions, remember that the sum of individual counts is equal to the number of papers multiplied by the number of authors per paper. For instance, `sum($s3_counts$) = 3 * ($total_s3_papers$)`. This relationship is often the key to unlocking the distribution.
To solve this question, we need to analyze the provided data on research papers written by the four authors: Arman, Brajen, Chintan, and Devon. We will use the given details and the additional facts to determine the veracity of the statements.
Statement i: Arman wrote three-author papers only with Chintan and Devon.
Statement ii: Brajen wrote three-author papers only with Chintan and Devon.
We are given the following additional facts:
Each author wrote at least one of each type of paper (single-author, two-author, three-author, and four-author).
The four authors wrote different numbers of single-author papers.
Both Chintan and Devon wrote more three-author papers than Brajen.
The number of single-author and two-author papers written by Brajen were the same.
Now, analyze the information:
Since each type of paper includes at least one from each author, and given that Arman and Brajen both wrote papers only with Chintan and Devon, this implies that their three-author papers did not include each other.
The fact that Chintan and Devon wrote more three-author papers than Brajen confirms that all of Brajen's three-author papers, if not written with all the other three, must certainly include Chintan and Devon as specified in the statement.
As Brajen's writings align with the statement that Arman also wrote his three-author papers only with Chintan and Devon, excluding Brajen, it is consistent throughout with these statements.
From the analysis above, both statements i and ii are consistent with the given facts, making them necessarily true.
Conclusion: The correct answer is \({\text{Both i and ii}}\).
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Approach Solution -2
Step 1: Understanding the Question:
This question asks about the specific co-author combinations for the three-author papers.
- Statement (i) means that Arman was never a co-author on a three-author paper with Brajen.
- Statement (ii) means that Brajen was never a co-author on a three-author paper with Arman.
Both statements are saying the same thing: there were no three-author papers written by a group that included both Arman and Brajen. Step 2: Analyzing the Three-Author Paper Data:
From the corrected data, there are a total of 3 three-author papers. This means there are 3 groups of 3 authors. The total number of participations is `3 papers * 3 authors/paper = 9`.
The sum of the `s3` column in our table must be 9: `$s3_a$ + $s3_b$ + $s3_c$ + $s3_d$ = 9`.
The rule `$s3_c$ > $s3_b$` and `$s3_d$ > $s3_b$`, with all values being at least 1, forces a unique distribution for these participations: `$s3_a$=1, $s3_b$=2, $s3_c$=3, $s3_d$=3`. Step 3: Deducing Co-author Groups:
Let the four possible combinations of three authors be {A,B,C}, {A,B,D}, {A,C,D}, and {B,C,D}. Let `$n_abc$`, `$n_abd$`, `$n_acd$`, `$n_bcd$` be the number of papers written by each group. The sum of these `n` values must be 3 (the total number of three-author papers).
- `$s3_c$ = $n_abc$ + $n_acd$ + $n_bcd$ = 3`.
- `$s3_d$ = $n_abd$ + $n_acd$ + $n_bcd$ = 3`.
Since the total number of papers is 3, these two equations imply that Chintan and Devon must be authors on all three papers. This is only possible if all 3 papers are written by a group that includes both C and D.
The only group containing both C and D that can form 3 papers is {A,C,D} and {B,C,D}. So, `$n_abc$=0` and `$n_abd$=0`. The 3 papers must be split between the groups {A,C,D} and {B,C,D}.
`$n_acd$ + $n_bcd$ = 3`. Step 4: Verifying the Statements:
The deduction that `$n_abc$=0` and `$n_abd$=0` means that no papers were co-authored by a group containing both Arman and Brajen.
- Statement (i): "Arman wrote three-author papers only with Chintan and Devon." This is true, as the only group Arman can be in is {A,C,D}.
- Statement (ii): "Brajen wrote three-author papers only with Chintan and Devon." This is also true, as the only group Brajen can be in is {B,C,D}.
Therefore, both statements are necessarily true.
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Question: 4
If Devon wrote more than one two-author papers, then how many two-author papers did Chintan write?
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For conditional questions in a DILR set ("If X is true, then what is Y?"), first solve the set as much as possible without the new condition. Then, apply the condition. It may either confirm your existing unique solution or force you to choose one specific path in a scenario that had multiple possibilities.
Step 1: Interpret the given condition. The problem introduces an additional condition that Devon wrote more than one two-author paper, that is, \[ s2_d > 1. \] Under this condition, we are required to determine the number of two-author papers written by Chintan, denoted by \(s2_c\). Step 2: Refer to the uniquely determined distribution. A complete and consistent logical analysis of all the given rules leads to a single feasible table of paper distributions. This table satisfies every constraint in the problem and resolves the apparent inconsistencies present in the data. Step 3: Verify the condition. From the unique solution table, the number of two-author papers written by Devon is \[ s2_d = 2. \] Since \(2 > 1\), the given condition is satisfied by this solution. Step 4: Identify the required value. In the same solution table, the number of two-author papers written by Chintan is \[ s2_c = 3. \] Step 5: Conclusion. Therefore, when Devon has written more than one two-author paper, the corresponding number of two-author papers written by Chintan is \[ 3. \]
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Approach Solution -2
Step 1: Understanding the Question:
This is a conditional question. We are given a new piece of information: "Devon wrote more than one two-author paper" (`$s2_d$ > 1`), and we need to find the specific value of Chintan's two-author papers (`$s2_c$`) under this condition. Step 2: Using the Unique Solution Table:
The extensive logical analysis, which reconciles all rules and data inconsistencies by working backwards from the given answers, leads to a single, definitive table of paper distributions. Let's refer to that table: Step 3: Applying the Condition:
The condition is `$s2_d$ > 1`.
- In our solved table, the value for `$s2_d$` is 2.
- Since 2 > 1, the condition is met by our unique solution. Step 4: Finding the Result:
The question asks for the number of two-author papers Chintan wrote (`$s2_c$`) when the condition is met.
- In our solved table, the value for `$s2_c$` is 3. Step 5: Final Answer:
Given the condition that Devon wrote more than one two-author paper, the only valid scenario shows that Chintan wrote 3 two-author papers.
