We are given a 6-digit number and several conditions about the relationships between the digits. Let's define the digits as follows:

• Let the 6-digit number be $abcdef$, where:
  • $a$ is the hundred thousands digit,
  • $b$ is the ten thousands digit,
  • $c$ is the thousands digit,
  • $d$ is the hundreds digit,
  • $e$ is the tens digit,
  • $f$ is the ones digit.

The problem provides the following relationships:

1. Ones place (f) and ten thousands place (b): $f = b + 7$

2. Hundreds place (d) and thousands place (c): $d = c - 2$

3. Tens place (e) and hundred thousands place (a): $e = a + 4$

4. Hundred thousands place (a) and hundreds place (d): $a = d - 3$

5. Thousands place (c) and ones place (f): $c = f - 1$

6. The ones place digit (f) is 8: $f = 8$

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Step-by-Step Solution:

Using the information that $f = 8$, let's solve the equations step by step:

Step 1: Solve for $b$

From the first condition $f = b + 7$, we know: $8 = b + 7 \implies b = 1$

Step 2: Solve for $c$

From the fifth condition $c = f - 1$, we know: $c = 8 - 1 = 7$

Step 3: Solve for $d$

From the second condition $d = c - 2$, we know: $d = 7 - 2 = 5$

Step 4: Solve for $a$

From the fourth condition $a = d - 3$, we know: $a = 5 - 3 = 2$

Step 5: Solve for $e$

From the third condition $e = a + 4$, we know: $e = 2 + 4 = 6$

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Final Answer:

The digits are $a = 2$, $b = 1$, $c = 7$, $d = 5$, $e = 6$, and $f = 8$.

Therefore, the 6-digit number is: $\boxed{217568}$

Would you like more details or have any questions?

Here are 5 related questions to extend the problem:

1. Can you create a similar problem with a different digit relationship?
2. How would the solution change if the ones digit was different?
3. Can we form a 7-digit number using similar constraints?
4. How does the order of solving these equations affect the outcome?
5. Can this method be applied to finding relationships in larger numbers?

Tip: When solving word problems with multiple conditions, write down the relationships clearly and solve them step by step, substituting known values as soon as possible.