If the month of August starts on a Thursday, what is the date of the last Thursday of October in the same year?

The correct answer is 31.

To find the date of the last Thursday of October, we can trace the days of the week month by month starting from August 1.

Step 1: Analyze the month of August
August has 31 days.
We are given that August 1 is a Thursday.
Adding 7 days repeatedly gives all the Thursdays in August:
• August 1 = Thursday
• August 1 + 7 = August 8 = Thursday
• August 8 + 7 = August 15 = Thursday
• August 15 + 7 = August 22 = Thursday
• August 22 + 7 = August 29 = Thursday
Since August 29 is a Thursday, we can determine the days for the remaining dates of August:
• August 30 = Friday
• August 31 = Saturday

Step 2: Analyze the month of September
Since August ends on Saturday (August 31), September 1 must be a Sunday.
September has 30 days in total.
We can calculate the total number of full weeks and remaining days (odd days) in September:
$30=(4\times 7)+2$
September has 4 full weeks plus 2 extra days.
Since September 1 is Sunday, counting 29 days after September 1 (or 2 extra days past 4 weeks) gives:
• Sunday + 2 extra days = Tuesday (for the 31st shift, so September 30 is a Monday).

Step 3: Analyze the month of October
Since September ends on a Monday (September 30), October starts on a Tuesday (October 1).
Let's find the first Thursday of October:
• October 1 = Tuesday
• October 2 = Wednesday
• October 3 = Thursday

Now, add 7 days sequentially to find all Thursdays in October:
• 1st Thursday = October 3
• 2nd Thursday = October 3 + 7 = October 10
• 3rd Thursday = October 10 + 7 = October 17
• 4th Thursday = October 17 + 7 = October 24
• 5th Thursday = October 24 + 7 = October 31

Since October has 31 days, October 31 is the last day of October and also falls exactly on a Thursday.

Therefore, the date of the last Thursday of October is 31.