Understanding the 5-Digit Number Problem

The question asks us to find the number of 5-digit numbers PQRST where P, Q, R, S, and T are all distinct digits. We are given several conditions relating these digits:

• P, Q, R, S, T are distinct digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).
• T ≠ 0.
• P is thrice T: \( P = 3T \).
• S is greater than Q by 4: \( S = Q + 4 \).
• Q is greater than R by 3: \( Q = R + 3 \).
• The number is PQRST, which is a 5-digit number, meaning P ≠ 0.

Analyzing the Relationships and Constraints

Let's break down the given conditions to find possible values for the digits.

Relationship between P and T

We know \( P = 3T \). Since P is the first digit of a 5-digit number, \( P \ne 0 \). Also, T ≠ 0. Both P and T must be single digits (0-9).

Possible pairs for (P, T) that satisfy \( P = 3T \) and \( P \ne 0, T \ne 0 \) are:

• If \( T = 1 \), then \( P = 3 \times 1 = 3 \). (P=3, T=1) - Valid pair.
• If \( T = 2 \), then \( P = 3 \times 2 = 6 \). (P=6, T=2) - Valid pair.
• If \( T = 3 \), then \( P = 3 \times 3 = 9 \). (P=9, T=3) - Valid pair.
• If \( T = 4 \), then \( P = 3 \times 4 = 12 \). 12 is not a single digit. No further possibilities for T.

So, the possible (P, T) pairs are (3, 1), (6, 2), and (9, 3).

Relationships between Q, R, and S

We are given \( Q = R + 3 \) and \( S = Q + 4 \).

Substitute the first equation into the second:

\( S = (R + 3) + 4 \)

\( S = R + 7 \)

So, the digits Q, R, and S must be in the form R, R+3, and R+7. All these must be distinct digits between 0 and 9.

Finding the Possible 5-Digit Numbers

Now we combine the conditions. For each possible (P, T) pair, we find valid digits for R, Q, and S such that all five digits P, Q, R, S, T are distinct and within the range 0-9.

Case 1: (P, T) = (3, 1)

Digits used so far are 3 and 1. The remaining digits Q, R, S must be R, R+3, R+7 and must be distinct from 3 and 1, and also distinct from each other (which R, R+3, R+7 naturally are if R >= 0).

Let's check possible values for R (R must be a digit 0-9):

• If \( R = 0 \): \( Q = 0 + 3 = 3 \), \( S = 0 + 7 = 7 \). Digits are {P=3, T=1, Q=3, R=0, S=7}. Q=3 is not distinct from P=3. Invalid.
• If \( R = 1 \): \( Q = 1 + 3 = 4 \), \( S = 1 + 7 = 8 \). Digits are {P=3, T=1, Q=4, R=1, S=8}. R=1 is not distinct from T=1. Invalid.
• If \( R = 2 \): \( Q = 2 + 3 = 5 \), \( S = 2 + 7 = 9 \). Digits are {P=3, T=1, Q=5, R=2, S=9}. These are {3, 1, 5, 2, 9}, all distinct digits. Valid. The number is 35291.
• If \( R = 3 \): \( Q = 3 + 3 = 6 \), \( S = 3 + 7 = 10 \). S=10 is not a single digit. Invalid.

Any value of R greater than 2 will result in S being greater than 9. So, for (P, T) = (3, 1), there is 1 valid number.

Case 2: (P, T) = (6, 2)

Digits used so far are 6 and 2. The remaining digits Q, R, S must be R, R+3, R+7 and must be distinct from 6 and 2.

Let's check possible values for R:

• If \( R = 0 \): \( Q = 0 + 3 = 3 \), \( S = 0 + 7 = 7 \). Digits are {P=6, T=2, Q=3, R=0, S=7}. These are {6, 2, 3, 0, 7}, all distinct digits. Valid. The number is 63072.
• If \( R = 1 \): \( Q = 1 + 3 = 4 \), \( S = 1 + 7 = 8 \). Digits are {P=6, T=2, Q=4, R=1, S=8}. These are {6, 2, 4, 1, 8}, all distinct digits. Valid. The number is 64182.
• If \( R = 2 \): \( Q = 2 + 3 = 5 \), \( S = 2 + 7 = 9 \). Digits are {P=6, T=2, Q=5, R=2, S=9}. R=2 is not distinct from T=2. Invalid.
• If \( R = 3 \): \( Q = 3 + 3 = 6 \), \( S = 3 + 7 = 10 \). Q=6 is not distinct from P=6. Invalid.

Any value of R greater than 2 will result in Q or S overlapping with P, T or exceeding 9. So, for (P, T) = (6, 2), there are 2 valid numbers.

Case 3: (P, T) = (9, 3)

Digits used so far are 9 and 3. The remaining digits Q, R, S must be R, R+3, R+7 and must be distinct from 9 and 3.

Let's check possible values for R:

• If \( R = 0 \): \( Q = 0 + 3 = 3 \), \( S = 0 + 7 = 7 \). Digits are {P=9, T=3, Q=3, R=0, S=7}. Q=3 is not distinct from T=3. Invalid.
• If \( R = 1 \): \( Q = 1 + 3 = 4 \), \( S = 1 + 7 = 8 \). Digits are {P=9, T=3, Q=4, R=1, S=8}. These are {9, 3, 4, 1, 8}, all distinct digits. Valid. The number is 94183.
• If \( R = 2 \): \( Q = 2 + 3 = 5 \), \( S = 2 + 7 = 9 \). Digits are {P=9, T=3, Q=5, R=2, S=9}. S=9 is not distinct from P=9. Invalid.

Any value of R greater than 1 will result in Q or S overlapping with P, T or exceeding 9. So, for (P, T) = (9, 3), there is 1 valid number.

Total Number of Such 5-Digit Numbers

Summing the valid numbers from each case:

• Case 1: 1 number
• Case 2: 2 numbers
• Case 3: 1 number

Total number of possible 5-digit numbers = 1 + 2 + 1 = 4.

The possible numbers are 35291, 63072, 64182, and 94183.

Revision Table: Key Conditions and Derivations

Additional Information: Solving Digit Puzzles

Digit puzzles like this involve combining multiple constraints on single digits to narrow down possibilities. A systematic approach is crucial.

Steps often involve:

1. Writing down all given conditions mathematically.
2. Identifying constraints on individual digits (e.g., \(P \ne 0\), digits must be 0-9).
3. Deriving new relationships between digits from the given ones (like \(S = R + 7\) from \(Q=R+3\) and \(S=Q+4\)).
4. Starting with the most constrained relationships (like \(P=3T\) with \(P \ne 0, T \ne 0\)) to list initial possibilities.
5. For each initial possibility, checking if the remaining digits can be found satisfying their conditions and the distinctness requirement for all digits.
6. Using a table can help keep track of possibilities and check for distinctness efficiently.

This method ensures all conditions are met simultaneously and no valid combinations are missed.