The following are some sample problems that PACT students tackle:

- Given 8 oranges on the table, 7 of them have exactly the same weight and the 8th is a little bit lighter. You are given a balance that can measure oranges against each other and you are allowed to use the balance at most twice. How can you determine which one of the oranges is lighter than the others?
- How many people do you need to have in a room before there is a fifty percent chance that two will share a birthday? (It’s a lot lower than you would probably think!)
- You have a thousand bottles of liquid and 10 mice. 999 bottles are filled with a delicious beverage that you want to serve to your friends. The remaining bottle contains a poison which will kill the drinker. However, you most certainly do not want any of your friends to accidentally drink the poison! Fortunately, your mice can also drink these liquids, with the same effects. Unfortunately, you can only feed each mouse once, though you can feed it any mixture of liquids, and it takes a day for the effects to sink in. How can you determine which bottle contains the poison within 25 hours?
- Two people, after discussing a strategy, go into separate rooms and cease all communication. They each flip coins and then attempt to guess if the other person got heads or tails. What strategy ensures that at least one of them guesses correctly.
- You are at a party with
*n*people and someone tells you a superstar, such as a famous actor, arrives. Since you are out of the loop, you do not know who this superstar is. Your goal is to determine who the superstar is by asking only*n*questions of the form ‘Does this person know this person?’, because obviously everyone knows the superstar, but the superstar knows no one there.