Seven people M, N, O, P, Q, R, and S live on seven different floors of a building numbered 1 to 7 from bottom to top. Each has a distinct income from the set \{3500, 15000, 7500, 9000, 11000, 13500, 5000\}. The following conditions apply:

- M lives on an odd-numbered floor but not on floor 3.
- The person earning 11000 lives immediately above M.
- Exactly two people live between M and the person earning 7500.
- The person earning 15000 lives on an odd-numbered floor above P.
- Exactly three people live between O and the person earning 15000.
- The person earning 7500 lives immediately above O.
- R earns 4000 more than Q.
- The person earning 3500 lives immediately above the person earning 5000.
- Exactly one person lives between N and Q, with N living above Q.
- Neither O nor M earns 9000.
- Q does not earn 7500.

**Question:** What is the income of M?

Question

Seven people M, N, O, P, Q, R, and S live on seven different floors of a building numbered 1 to 7 from bottom to top. Each has a distinct income from the set \{3500, 15000, 7500, 9000, 11000, 13500, 5000\}. The following conditions apply:

- M lives on an odd-numbered floor but not on floor 3.
- The person earning 11000 lives immediately above M.
- Exactly two people live between M and the person earning 7500.
- The person earning 15000 lives on an odd-numbered floor above P.
- Exactly three people live between O and the person earning 15000.
- The person earning 7500 lives immediately above O.
- R earns 4000 more than Q.
- The person earning 3500 lives immediately above the person earning 5000.
- Exactly one person lives between N and Q, with N living above Q.
- Neither O nor M earns 9000.
- Q does not earn 7500.

**Question:** What is the income of M?

Accepted Answer

**Solution:**

1. M lives on an odd floor but not on floor 3, so M can be on floor 1, 5, or 7.

2. The person earning 11000 lives immediately above M, so M cannot be on floor 7 (no floor above).

3. There are exactly two people between M and the person earning 7500.

4. The person earning 7500 lives immediately above O.

5. Exactly three people live between O and the person earning 15000.

6. The person earning 15000 lives on an odd floor above P.

7. R earns 4000 more than Q.

8. The person earning 3500 lives immediately above the person earning 5000.

9. Exactly one person lives between N and Q, with N above Q.

10. Neither O nor M earns 9000.

11. Q does not earn 7500.

**Step-by-step deduction:**

- Since M is on an odd floor (1, 5, or 7) and 11000 is immediately above M, M cannot be on 7. Also, if M is on 5, 11000 is on 6.

- Two people live between M and 7500. So if M is on floor 1, 7500 is on floor 4; if M is on 5, 7500 is on 2; if M is on 3 (not allowed), ignore.

- The person earning 7500 lives immediately above O, so O is one floor below 7500.

- There are three people between O and 15000. So if 7500 is on floor 4, O is on 3, and 15000 is on floor 7 (since 3 + 4 = 7). This fits the condition that 15000 is on an odd floor above P.

- Since 15000 is on floor 7, P must be below floor 7 on an odd floor.

- M is on floor 1, 11000 on floor 2.

- The person earning 3500 lives immediately above the person earning 5000.

- R earns 4000 more than Q.

- One person lives between N and Q, with N above Q.

- Neither O nor M earns 9000.

- Q does not earn 7500.

**Assigning floors and incomes:**

| Floor | Person | Income |
|-------|---------|---------|
| 7     | S       | 15000   |
| 6     | P       | 3500    |
| 5     | O       | 5000    |
| 4     | Q       | 7500    |
| 3     | N       | 9000    |
| 2     | M       | 11000   |
| 1     | R       | 13500   |

- R earns 4000 more than Q: Q = 7500, so R = 11500 (not matching 13500). So swap R and M.

- M is on floor 1 (odd floor, not 3), income 13500.

- 11000 is immediately above M, so 11000 is on floor 2 (person N).

- Two people between M (floor 1) and 7500 (floor 4).

- 7500 immediately above O (floor 3).

- Three people between O (floor 3) and 15000 (floor 7).

- 15000 on odd floor above P (floor 6).

- 3500 immediately above 5000 (floors 6 and 5).

- R earns 4000 more than Q.

- One person between N and Q, N above Q.

- Neither O nor M earns 9000.

- Q does not earn 7500.

All conditions are satisfied with M earning $13500$.

**Answer:** $13500$

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