Seven friends M, N, O, P, Q, R, and S having different heights are standing in ascending order of their heights. The tallest person stands on the right and the shortest on the left. Only three friends are taller than O. R is shorter than only Q. N is taller than both P and M but shorter than S. Who is the third tallest?

Question

Prepp Team · Accepted Answer

Analyzing Height Differences and Ranking

The problem involves determining the relative heights of seven friends: M, N, O, P, Q, R, and S, who are standing in ascending order of their heights from left to right. This means the shortest person is on the left (Position 1) and the tallest is on the right (Position 7).

Breaking Down the Clues

Let's list the information given about the friends' heights:

There are seven friends with different heights.
They are arranged in ascending order of height (Shortest < ... < Tallest).
Position 1: Shortest, Position 7: Tallest.
Only three friends are taller than O.
R is shorter than only Q.
N is taller than both P and M but shorter than S.

Step-by-Step Height Determination

We can deduce the relative positions based on the clues:

"Only three friends are taller than O." Since there are 7 friends total and they are in ascending order, if only three are taller than O, then O must be taller than $7 - 3 - 1 = 3$ friends. This places O at the 4th position from the shortest end (or 4th from the left).
Current potential order (positions 1 to 7): \_\_ \_\_ \_\_ O \_\_ \_\_ \_\_
"R is shorter than only Q." This is a strong clue. If R is shorter than *only* one person, that person must be the absolute tallest, and R must be the second tallest. So, Q is the tallest (Position 7), and R is the second tallest (Position 6).
Current potential order: \_\_ \_\_ \_\_ O \_\_ R Q
The friends placed so far are O (Position 4), R (Position 6), and Q (Position 7). The remaining friends are M, N, P, and S. They must occupy the remaining positions: 1, 2, 3, and 5.
"N is taller than both P and M but shorter than S." This gives us a relative height order among these four: P and M are shorter than N, and N is shorter than S. So, P/M < N < S.
Considering the remaining positions (1, 2, 3, 5), S must be the tallest among M, N, P, S. The highest available position is 5. Since S is shorter than O (at position 4 is impossible as S>N, N>P/M meaning S is taller than at least 2), S must be at position 5 as S is shorter than O, R, Q. If S were at position 3, then N and P/M would be at 1 and 2, which would make S shorter than O at position 4. If S were at position 1 or 2, it would contradict S > N > P/M. Therefore, S is at position 5.
Current potential order: \_\_ \_\_ \_\_ O S R Q
The remaining friends are M, N, P, and they must occupy positions 1, 2, and 3. We know P/M < N. To fit this, N must be at position 3, and M and P will occupy positions 1 and 2 (their specific order relative to each other isn't determined by the clues, but it doesn't affect the other positions).
Final order from shortest (Position 1) to tallest (Position 7):

M/P < P/M < N < O < S < R < Q

Identifying the Third Tallest Person

Based on the final ascending order from left (shortest) to right (tallest):

Position 1 (Shortest): M or P
Position 2: P or M
Position 3: N
Position 4: O
Position 5: S
Position 6 (Second Tallest): R
Position 7 (Tallest): Q

The third tallest person is at Position 5 in the ascending order. Looking at our determined order, the person at Position 5 is S.

Position	Height Rank	Friend
1	Shortest	M/P
2	Second Shortest	P/M
3	Third Shortest	N
4	Fourth Tallest / Shortest	O
5	Third Tallest	S
6	Second Tallest	R
7	Tallest	Q

Thus, the third tallest friend is S.

Revision Table: Height Ranking Clues

Clue	Deduction	Placement
7 friends, ascending height order	Positions 1 (Shortest) to 7 (Tallest)	Framework setup
Only 3 taller than O	O is 4th tallest	Position 4 for O
R shorter than only Q	Q is tallest, R is 2nd tallest	Position 7 for Q, Position 6 for R
N > P, N > M, S > N	P/M < N < S	Relative order for M, N, P, S
Fitting M,N,P,S into remaining slots (1,2,3,5) respecting order	S must be 5th, N must be 3rd	Position 5 for S, Position 3 for N. M/P in 1 & 2.

Additional Information: Logic Puzzles and Ordering

Logic puzzles involving ordering or ranking are common in aptitude tests. They require careful reading of clues and systematic deduction. Key strategies include:

Representing the positions or ranks visually (e.g., slots or a line).
Identifying absolute positions first (like Q and R being 7th and 6th, or O being 4th).
Using relative clues (like N > P/M < S) to place remaining elements into available slots while maintaining the relative order.
Eliminating possibilities as you place individuals.
Checking if all clues are satisfied by the final arrangement.

In this problem, establishing the fixed positions of O, R, and Q early was crucial. Then, using the relative order of M, N, P, and S helped determine their placement in the remaining slots, leading to the final complete ranking.