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Halloween Day

Age 11 to 16 Short Challenge Level:

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.
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.