Solving the PQRST Number Puzzle

The problem asks us to find the value of digit P in a five-digit number PQRST. The digits P, Q, R, S, and T are distinct and are chosen from the set {1, 2, 3, 4, 5}. The following conditions must be satisfied:

• $P < Q$
• $S > P > T$
• $R < T$

Deriving Key Inequalities

Let's combine the given inequalities:

• From $R < T$ and $P > T$, we know $T$ is between $R$ and $P$.
• From $P > T$ and $P < Q$, we have $T < P < Q$.
• Combining $R < T$ with $T < P$, we get the sequence $R < T < P$.
• So, the core sequence derived is $R < T < P < Q$.
• We also have the condition $S > P$.

The digits P, Q, R, S, T must be distinct digits from {1, 2, 3, 4, 5}.

Analyzing Possible Values for P

The inequality $R < T < P$ means that P must be at least the third smallest digit used. Since the digits available are {1, 2, 3, 4, 5}, P cannot be 1 or 2. Thus, P must be at least 3.

Case 1: P = 3

If P = 3, the digits available for R and T must be less than 3 and distinct. The available digits smaller than 3 are {1, 2}.

• To satisfy $R < T < P$, we must have T=2 and R=1.
• The digits used so far are R=1, T=2, P=3.
• The remaining digits are {4, 5} for Q and S.
• We need to check the conditions $P < Q$ (3 < Q) and $S > P$ (S > 3).
• If Q=4 and S=5: $3 < 4$ (True) and $5 > 3$ (True). The number PQRST could be 34152. All digits {1, 2, 3, 4, 5} are distinct and used. This is a valid solution.
• If Q=5 and S=4: $3 < 5$ (True) and $4 > 3$ (True). The number PQRST could be 35142. All digits {1, 2, 3, 4, 5} are distinct and used. This is also a valid solution.

Since we found valid assignments for Q and S when P=3, P=3 is a possible value.

Case 2: P = 4

If P = 4, the digits available for R and T must be less than 4 and distinct. The available digits smaller than 4 are {1, 2, 3}.

• To satisfy $R < T < P$, possible pairs (R, T) from {1, 2, 3} are (1, 2), (1, 3), (2, 3).
• The remaining digits for Q and S must be chosen from {1, 2, 3, 5} excluding R and T.
• We need to satisfy $P < Q$ (4 < Q) and $S > P$ (S > 4). This means both Q and S must be greater than 4.
• The only digit available greater than 4 is 5.
• This would require both Q=5 and S=5, which violates the distinct digit condition.

Therefore, P cannot be 4.

Case 3: P = 5

If P = 5, the condition $P < Q$ requires Q to be greater than 5. No such digit exists in the set {1, 2, 3, 4, 5}.

Therefore, P cannot be 5.

Conclusion

The only possible value for P that satisfies all conditions is 3.