Question

If O sits at the same table as K, which of the following must sit at the same table as each other?

• A. J and T
• B. L and R
• C. N and K
• D. N and T
• E. O and N
• A. J
• B. K
• C. N
• D. O
• J. T
• A. J sits at a table with only one other person.
• B. L sits at a table with only one other person.
• C. K sits at the same table as O.
• D. J sits at the same table as N.
• O. L sits at the same table as S.
• A. J and O
• B. K and S
• C. L and R
• D. N and S
• T. O and T
• A. J, K, and N
• B. K, L, and T
• C. K, N, and T
• D. K, R, and T
• Y. L, N, and R

Answer 1

The Correct Option is D

Step 1: Place O and K.
We are told O sits with K. Since O must also sit with S (rule: O and S together), we get the group:
\[ \{O, K, S\} \] This table is now full (3 people). ✅

Step 2: Place L.
K and L cannot sit together, so L cannot join this table. L must go to a different table. ✅

Step 3: Place children.
Each of the 3 tables must have exactly one child. S is already at Table 1 with O and K.
Thus, R and T must each go to separate remaining tables (Table 2 and Table 3). ✅

Step 4: Place N.
N cannot sit with R. So if R is at Table 2, N must go to Table 3 with T. ✅

Step 5: Check groups.
- Table 1: {O, K, S} (full).
- Table 2: {R, L, ?}.
- Table 3: {N, T, J}. (J has to sit somewhere, and only Table 3 works). ✅

Step 6: Conclusion.
From the arrangement, N and T always end up together. This is forced by the rules (N cannot go with R, so must go with T). ✅

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Final Answer:

\[ \boxed{\text{(D) N and T}} \]

Answer 2

Step 1: Understanding the Concept:
This is a "could be true" question from a logic game. We need to test if a third person can legally join a table that already contains L and R.
Initial Rules Recap: 1. Max table size is 3. 2. Each table must have a child (R, S, or T). 3. O and S are together. 4. K and L are not together. 5. N and R are not together.
Step 2: Detailed Explanation:
1. Setup: We are creating a table with L and R. Since R is a child, Rule 2 is satisfied for this table. Let's call this Table 1: {L, R, ?}.
2. Apply Rules to this table:
- Rule 5 (N \(\neq\) R): Since R is at this table, N cannot be the third person. This eliminates option (C).
- Rule 4 (K \(\neq\) L): Since L is at this table, K cannot be the third person. This eliminates option (B).
- We know O must be with S (Rule 3), and S is a child. So O is at the table with S, not the table with R. This eliminates option (D).
- The children R, S, and T must be at different tables. Since R is at this table, T cannot be the third person. This eliminates option (E).
3. Conclusion by Elimination: The only person left who can sit at this table is J.
4. Verification: Let's see if we can build a full valid arrangement with the table {L, R, J}.
- Table 1: {L, R, J} (full)
- This leaves 5 people to be placed at two other tables: K (woman), N, O (men), S, T (children).
- We know the other two tables must be anchored by S and T.
- Table 2: {S, ...}
- Table 3: {T, ...}
- Rule 3 (O and S together): O must go to Table 2. So Table 2 is {S, O, ...}.
- The remaining people are K and N. The remaining spots are one at Table 2 and one at Table 3 (to make a total of 8 people, the table sizes must be 3, 3, 2).
- We must place K and N.
- We can place K at Table 3: {T, K}.
- We can place N at Table 2: {S, O, N}.
- Let's check this full arrangement: T1:{L,R,J}, T2:{S,O,N}, T3:{T,K}.
- Max 3 per table: OK (3,3,2).
- Each table has a child: OK (R,S,T).
- OS together: OK.
- K not with L: OK (K is T3, L is T1).
- N not with R: OK (N is T2, R is T1).
- This is a valid arrangement.
Step 3: Final Answer:
J is the only person who can sit at the table with L and R.

Answer 3

Step 1: Understanding the Concept:
This is a conditional question. We add the new condition (N sits with S) and then test which of the options is possible by trying to construct a valid scenario.
Step 2: Initial Deductions from the Condition:
1. New Condition: N sits with S.
2. Rule 3 (O with S): O also sits with S. So, one table is {N, S, O}. This table is full. Let's call it Table 2.
3. Rule 2 (Children at separate tables): The other two tables are anchored by R and T. - Table 1: {R, ...} - Table 3: {T, ...}
4. Rule 5 (N not with R): This is satisfied since N is at Table 2 and R is at Table 1.
5. Remaining People: The women J, K, L are left to be placed at Table 1 and Table 3. There are 3 women for 4 available spots (max 2 at T1, max 2 at T3). Since we have only 3 people for 2 tables, one table will have 2 people and the other will have 3. - Total table sizes must be 3, 3, 2. We already have one table of 3 ({N,S,O}). So the other two tables must have sizes 3 and 2.
6. Placing J, K, L: - Rule 4 (K \(\neq\) L): K and L must be at different tables. So one goes to Table 1, the other to Table 3. - The third woman, J, must join one of them. - Scenario A: Table 1 has 3 people, Table 3 has 2. T1={R, K, J}, T3={T, L}. (Here K/L can be swapped). - Scenario B: Table 1 has 2 people, Table 3 has 3. T1={R, K}, T3={T, L, J}. (Here K/L can be swapped).
Step 3: Test the Options:


Given: Three tables anchored by $R,S,T$. $O$ sits with $S$ and (by condition) $K$ sits with $O$, so \[ T_2=\{S,O,K\} \quad(\text{full}). \]
Remaining people: $J,L,N$ to be placed at $T_1$ (with $R$) and $T_3$ (with $T$). Rule $N\neq R$ $\Rightarrow$ $N\notin T_1$, so $N\in T_3$.
Since $T_2$ has 3 people, the remaining seats sum to 5; thus $(|T_1|,|T_3|)=(3,2)$ or $(2,3)$.
The only valid assignments (up to ordering within a table) are: \[ \begin{array}{c|l} \# & \text{Assignment}
\hline 1 & T_1=\{R,J,L\},\; T_2=\{S,O,K\},\; T_3=\{T,N\}
2 & T_1=\{R,J\},\; T_2=\{S,O,K\},\; T_3=\{T,N,L\}
3 & T_1=\{R,L\},\; T_2=\{S,O,K\},\; T_3=\{T,N,J\} \end{array} \]
Consequences for the options:

[(A)] ``J sits at a table with only one other person'': occurs in assignment \#2, but not in \#1 or \#3. Possible, not guaranteed.
[(B)] ``L sits at a table with only one other person'': occurs in assignment \#3, but not in \#1 or \#2. Possible, not guaranteed.
[(C)] ``K sits at the same table as O'': true in every assignment (given). Must be true.
[(D)] ``J sits at the same table as N'': occurs in assignment \#3 only. Possible, not guaranteed.
[(E)] ``L sits at the same table as S'': never true (S is in $T_2$, L is in $T_1$ or $T_3$). Impossible.

\fbox{% \parbox{\dimexpr\linewidth-2\fboxsep-2\fboxrule}{ Final summary: Only (C) is necessarily true. Options (A), (B), and (D) can occur in some valid arrangements; (E) cannot occur. }}

Answer 4

Step 1: Understanding the Concept:
This is an EXCEPT question, which means we need to find the one pair that can NEVER sit at the same table. The other four pairs must be possible.
Step 2: Detailed Explanation:
Let's analyze the core rules and constraints. The three tables are anchored by the children R, S, and T. Table S must also contain O. So we have one table that is {S, O, ...}. Table R cannot contain N. Table K's cannot contain L. We need to test each pair. - (A) J and O: Can J sit with O? O is at the table with S. Let's try to make the table {J, O, S}. This is a full table of 3. - Remaining people: K, L, N, R, T. - Remaining tables: T1={R, ...}, T2={T, ...}. - We must place K, L, N. K and L must be at separate tables. N cannot be with R. - T1={R, L}. T2={T, K, N}. This is a valid arrangement ({J,O,S}, {R,L}, {T,K,N}). So, J and O can sit together. - (B) K and S: Can K sit with S? S is at the table with O. Let's try to make the table {K, S, O}. This is a full table of 3. - This is exactly the condition we analyzed in question 10. We found multiple valid arrangements starting with this table. So, K and S can sit together. - (C) L and R: Can L sit with R? Let's try to make a table {L, R, ?}. We found a valid arrangement for this in question 11: {L, R, J}. So, L and R can sit together. - (D) N and S: Can N sit with S? S is at the table with O. Let's try to make the table {N, S, O}. This is a full table of 3. - This was the condition for question 12, and we found multiple valid arrangements. So, N and S can sit together. - (E) O and T: Can O sit with T? - O must sit with the child S (Rule 3). - T is a child, and must anchor a different table from S (Rule 2). - Therefore, O and T must always be at different tables. They can never sit together.
Step 3: Final Answer:
O and T can never sit at the same table because O must be with child S, and T must be at a separate table.

Answer 5

Step 1: Understanding the Concept:
This is a conditional question. We are given the exact composition of one table and asked to find a possible composition for one of the other two tables.
Step 2: Initial Deductions from the Condition:
1. New Condition: One table consists of exactly {O, S}. This is Table 1. Size = 2. 2. Remaining People: J, K, L (women), N (man), R, T (children). Total of 6 people left. 3. Setup of Other Tables: These 6 people must be seated at the other two tables (Table 2 and Table 3). Since the maximum size is 3, the only way to seat 6 people at two tables is to have 3 people at each. 4. Rule 2 (Children at separate tables): The children R and T must be at different tables. So, R is at Table 2 and T is at Table 3. - Table 2: {R, ...} (Size 3) - Table 3: {T, ...} (Size 3)
5. Remaining people to place: J, K, L, N. We have 4 people to fill 4 spots (2 at each table).
Step 3: Placing the Remaining People and Evaluating Options:
- Rule 5 (N \(\neq\) R): N cannot be at Table 2 with R. Therefore, N must be at Table 3 with T. - Table 3 is now {T, N, ?}. - Rule 4 (K \(\neq\) L): K and L must be at different tables. One must go to Table 2, the other to Table 3. - Scenario A: K goes to Table 2, L goes to Table 3. - Table 2 becomes {R, K, ?} - Table 3 becomes {T, N, L} (full). - The last person, J, must go in the last spot at Table 2. - Final tables: {O,S}, {R, K, J}, {T, N, L}. This is a valid arrangement. - Scenario B: L goes to Table 2, K goes to Table 3. - Table 2 becomes {R, L, ?} - Table 3 becomes {T, N, K} (full). - The last person, J, must go to Table 2. - Final tables: {O,S}, {R, L, J}, {T, N, K}. This is also a valid arrangement.
Now we look at the options and see which group composition is possible for one of the tables. - (A) J, K, and N: This is impossible. N is always with T. K can be with R or T. J can be with R or T. They are never all together. - (B) K, L, and T: Impossible, K and L must be at separate tables. - (C) K, N, and T: This matches the composition of Table 3 in Scenario B. So this can be a group. - (D) K, R, and T: Impossible, R and T must be at separate tables. - (E) L, N, and R: Impossible, N and R must be at separate tables.
Step 4: Final Answer:
The group {K, N, and T} is a possible grouping for one of the tables.