The hundreds digit is half the tens digit.
The sum of the hundreds and tens digit is 12. The ones digit is an odd square number not equal to the thousands digit.
To find the solution, we can break down the problem into smaller steps:
Step 1: Determine the possible values for the thousands digit.
Since the ones digit is an odd square number not equal to the thousands digit, we can list the odd square numbers (1, 9, 25, 49, 81) and remove any number that is equal to the thousands digit if known. In this case, we don't have any information about the thousands digit, so we can keep all the odd square numbers.
Step 2: Determine the possible values for the hundreds digit.
The problem states that the hundreds digit is half the tens digit. We can use this information to narrow down the possible values for the hundreds digit.
Let's consider each odd square number one by one and try to find a valid value for the hundreds digit:
- If the ones digit is 1, then the hundreds digit would be 0. However, the hundreds digit cannot be 0 because it would make the number a two-digit number.
- If the ones digit is 9, then the hundreds digit would be 4. This satisfies the condition that the hundreds digit is half the tens digit (4 is half of 8).
- If the ones digit is 25, then the hundreds digit would be 2. This does not satisfy the condition that the hundreds digit is half the tens digit.
- If the ones digit is 49, then the hundreds digit would be 4. This satisfies the condition that the hundreds digit is half the tens digit (4 is half of 8).
- If the ones digit is 81, then the hundreds digit would be 4. This satisfies the condition that the hundreds digit is half the tens digit (4 is half of 8).
So, the possible values for the hundreds digit are 4, 4, and 2, corresponding to the ones digits being 9, 49, and 81, respectively.
Step 3: Determine the possible values for the tens digit.
The problem states that the sum of the hundreds and tens digit is 12. We can use this information to narrow down the possible values for the tens digit.
Let's consider each value for the hundreds digit one by one and try to find a valid value for the tens digit:
- If the hundreds digit is 4, then the tens digit would be 8 (since 4 + 8 = 12). This satisfies the condition that the sum of the hundreds and tens digit is 12.
- If the hundreds digit is 4, then the tens digit would be 4 (since 4 + 4 = 8). This does not satisfy the condition that the sum of the hundreds and tens digit is 12.
- If the hundreds digit is 2, then the tens digit would be 10 (since 2 + 10 = 12). This does not satisfy the condition that the tens digit is a single digit.
So, the possible values for the tens digit are 8.
Step 4: Determine the possible values for the ones digit.
We already found the possible values for the ones digit in step 1, which are 9, 49, and 81.
Putting it all together, the possible combinations for the thousands, hundreds, tens, and ones digits are:
1. Thousands: unknown, Hundreds: 4, Tens: 8, Ones: 9
2. Thousands: unknown, Hundreds: 4, Tens: 8, Ones: 49
3. Thousands: unknown, Hundreds: 2, Tens: 8, Ones: 81
Therefore, there are three possible 4-digit numbers that satisfy the given conditions.
Let's break down the given information step-by-step:
Step 1: The hundreds digit is half the tens digit.
Let's assume the tens digit is represented by variable T. According to the given information, the hundreds digit (H) will be half of the tens digit. Therefore, we can express this relationship as H = (1/2)T.
Step 2: The sum of the hundreds and tens digit is 12.
We already have H = (1/2)T. According to the given information, H + T = 12. Let's substitute the value of H from Step 1 into this equation. We get (1/2)T + T = 12. Now we need to solve this equation to find the value of T.
Step 3: The ones digit is an odd square number not equal to the thousands digit.
Let's assume the ones digit is represented by variable O, and the thousands digit is represented by variable Th. According to the given information, O is an odd square number. Odd square numbers include 1, 9, 25, 49, etc., but not 0, 4, 16, 36, etc. It is also mentioned that it is not equal to the thousands digit (O ≠ Th).
To summarize:
- The hundreds digit (H) is half the tens digit (T) -> H = (1/2)T.
- The sum of the hundreds and tens digit is 12 -> H + T = 12.
- The ones digit (O) is an odd square number not equal to the thousands digit (O ≠ Th).
Let me know if you would like further assistance!