Parent ’12 asks a good question:

What are the “rules” for Mountain Day?

For example, I assume not on the Friday of Family Days, but are there other Friday restrictions?

There’s a discussion about it on WSO.

http://wso.williams.edu/discuss/comments.php?DiscussionID=2091

In the WSO thread above, several students report watching the weather forecast obsessively in an attempt to predict Mountain Day.

However, this is all a moot point because, as Prof. Gerrard once demonstrated in class, Mountain Day does not exist.

In the WSO thread, Nathaniel Hewett starts in the right direction but doesn’t go through all the way to the logical conclusion:

I believe that this year, there should be four eligible fridays. This month has five fridays (2,9,16,23,30), and the idea behind not having the last friday be eligible is that if it got to that point, the entire student body would know that mountain day were, in fact, that friday (which raises the question of why it can be the second to last, because we’ll all know it’s then as well). So, there should be four potential fridays on which mountain day could be this year and you all might still have a chance for a day off on the 23rd. I’m not certain about that, but that’s my understanding.

A simple proof, using two basic axioms:

A. We know that Mountain Day will fall on a Friday in October.
B. We know that Mountain Day has to be a surprise when it is announced to the student body in the morning.

This October, there are five Fridays: 10/2, 10/9, 10/16, 10/23, and 10/30.

  1. Now, Mountain Day obviously cannot be 10/30 – if the administration has let the previous four Fridays go by, then it would no longer be a surprise to declare Mountain Day on 10/30. The previous evening, students will plan on missing class the next day because they would know for sure that Mountain Day is coming. This contradicts axiom B. Thus, 10/30 is ruled out.
  2. This means that we are left with four possible days which could be Mountain Day: 10/2, 10/9, 10/16, and 10/23.
  3. However, we discover that Mountain Day cannot possibly be 10/23 either. If the administration has let the first three Fridays go by, and we know from step 1 that Mountain Day cannot be on 10/30, then students will be sure that Mountain Day will be on 10/23, and will be able to slack off accordingly on 10/22. This contradicts the surprise requirement. Thus, Mountain Day cannot be on 10/23.
  4. Thus, we are left with the following possible choices for Mountain Day: 10/2, 10/9, and 10/16.
  5. However, by the same process as in Step 1, we find that Mountain Day cannot possibly be on 10/16, because on the evening before 10/16, students would know that Mountain Day will fall on 10/16: they know that it can’t be on 10/23 or 10/30, as we have proven above, and they know it wasn’t 10/2 or 10/9. Thus, it would no longer be a surprise.
  6. Mountain Day cannot possibly be on 10/9, because on the evening before 10/9, students would know that Mountain Day will fall on 10/9: they know that it can’t be on 10/16 or 10/23 or 10/30, as we have proven above, and they know it wasn’t 10/2. Thus, it would no longer be a surprise.
  7. Mountain Day cannot possibly be on 10/2, because on the evening before 10/2, students would know that Mountain Day will fall on 10/2: they know that it can’t be on 10/9 or 10/16 or 10/23 or 10/30, as we have proven above. Thus, it would no longer be a surprise.

I would give you the formal notation if I remembered it. But you get the general idea. The proof still works if you remove the requirement that Mountain Day be on a Friday.

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