1. When writing love letters people sometimes end with a sequence of O’s and X’s signifying hugs and kisses. Suppose you don’t like giving two hugs in a row (i.e. OO) so that writing XOXXOXXX would be fine, but you wouldn’t want to write XOOXXOXO because of the OO in it.With this constraint in mind, how many ways can you write out a sequence of eight X and O symbols to end your love letters?

2. (One of Juliana’s favorites) **Treasure Island** Two pirates Alice and Bob are marooned on a dessert* island in the south pacific. Alice discovers a secret hut and Bob finds the entrance to a hidden cave. Inside each discovers half of a treasure map. When the two return to the beach they combine the two halves and start searching for the treasure. Here is the description given on the map: *The two largest palm trees on the island serve as the markers for the treasure. You must first start at the secret location. Count your paces to the western most palm tree. Once you arrive at the tree take a ninety-degree turn to the right and walk the same number of paces. Mark this spot. Return to the secret location and walk to the eastern most palm tree, again counting your paces. Take a ninety-degree turn to the left and walk the same number of paces and the mark this spot. The treasure is located directly in the middle of the two spots that you have marked.*

Bob thinks that the secret location is the cave while Alice thinks the secret location is the hidden hut. Sketch a possible map of the island with the palm trees, the secret hut, and the hidden cave. Follow the directions given by the map and mark where Alice and Bob each think the treasure is located. What happens if you choose a different point as the “secret location”? Where do you think the treasure is?

*Yes, this is normally spelled “desert”, but we thought a dessert island is somewhere we’d all want to be.

3. Here’s a quickie! Find two distinct postive integers (greater than 1!) that are factors of 999,991 (big hint – think difference of squares!)

4. (Dan’s all-time favorite) A math professor (P) is walking on the campus with one of her students (S). Three people (X, Y, Z) known to P but not to S cross their path. P turns to S and says, “What are the ages of X, Y and Z, if the product of their ages is 2450, and the sum is twice yours?” S, after calculating and reflecting for several minutes, is puzzled and disturbed. She turns to P and says, “I’m sorry, but I can’t answer your question. You have not given me enough information!” P is shocked, sits down herself goes through some calculations. After a few moments, and with some embarrassment, she turns to S and says, “I’m sorry. You’re absolutely right. The additional information you need is that I am older than the oldest of those three people.” S then quickly replies with the correct answer. What are the ages of X, Y, Z, P and S?