Halloween day
One year there were exactly four Tuesdays and four Fridays in October. On what day of the week was Halloween.
Age
11 to 14
Challenge level
Problem
One 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?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
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 exactly four Tuesdays and exactly 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.