IMC 2008 Q8
8. In a certain year, there were exactly four Tuesdays and exactly four Fridays in October. On what day of the week did Halloween, October 31st, fall that year?
- A Monday
- B Wednesday
- C Thursday
- D Saturday
- E Sunday
Intermediate Mathematical Challenge 2008, Question 8. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A Monday
October has 31 days so, in any year, in October there are three days of the week which occur five times and four days which occur four times. As there were four Tuesdays and four Fridays, there could not have been five Wednesdays or five Thursdays, so the days which occurred five times were Saturday, Sunday and Monday. Hence October 1st fell on a Saturday, which means that October 31st was a Monday.
Solution © UK Mathematics Trust, from the IMC 2008 solutions.