Problem Analysis:
Let's break down the information given in the problem:
- A morning rain is always followed by a clear evening.
- An evening rain is always preceded by a clear morning.
- A total of 9 mornings and 12 evenings were clear.

Solution:
To find the number of rainless days, we need to determine the number of days when it neither rained in the morning nor in the evening.

Step 1: Identify the days when it rained in the morning and evening

Since it rained for thirteen days in total, and a morning rain is always followed by a clear evening, we can conclude that there were thirteen clear evenings as well.

Step 2: Identify the number of clear mornings and clear evenings

The problem states that there were nine clear mornings and twelve clear evenings.

Step 3: Find the days when it neither rained in the morning nor in the evening

To find the number of rainless days, we need to subtract the days when it rained in the morning or evening from the total number of days.

Total days = 13
Days when it rained in the morning = 13
Days when it rained in the evening = 13

Days when it neither rained in the morning nor in the evening = Total days - Days when it rained in the morning or evening

Days when it neither rained in the morning nor in the evening = 13 - (Days when it rained in the morning + Days when it rained in the evening)

Days when it neither rained in the morning nor in the evening = 13 - (13 + 13) = 13 - 26 = -13

Step 4: Interpret the result

The result is -13, which doesn't make sense in the context of the problem. It is not possible to have negative rainless days.

Step 5: Consider the given information again

Upon closer inspection of the given information, we can see that there is a contradiction. The problem states that there were nine clear mornings and twelve clear evenings. However, it also states that a morning rain is always followed by a clear evening, and an evening rain is always preceded by a clear morning.

Since there were nine clear mornings, there should have been nine mornings with rain, which contradicts the statement that it rained for thirteen days. Therefore, there is an error in the given information.

Step 6: Determine the correct number of rainless days (assuming the error)

Since we cannot rely on the given information, we cannot determine the exact number of rainless days. However, if we assume that there were thirteen clear mornings and evenings instead of nine, we can solve the problem.

Total days = 13
Days when it rained in the morning = 13
Days when it rained in the evening = 13

Days when it neither rained in the morning nor in the evening = Total days - Days when it rained in the morning or evening

Days when it neither rained in the morning nor in the evening = 13 - (Days when it rained in the morning + Days when it rained in the evening)

Days when it neither rained in the morning nor in the evening = 13 - (13 + 13) =