Logic
# Logic Warmups

**magic square of order 4** is created by putting the integers 1 to 16 into a 4 by 4 square grid so that the sum of the numbers in each row, column and main diagonal is the same. What is the sum of the integers at the corners of this magic square (labeled by letters as shown above)?

Which of these statements are truths and which are lies?

- Statement 2 and Statement 5 are either both truths or both lies.
- Statement 3 and Statement 5 are either both truths or both lies.
- Exactly two of the statements are truths.
- Statement 1 and Statement 2 are either both truths or both lies.
- Statement 3 is a lie.

Enter your answer as a five digit string where the digits represent the statements in numerical order, where a 1 indicates a truth, and a 0 indicates a lie. For example, if all of the statements are truths except for Statement 4, you would enter 11101.

Solve the following cryptarithm:

$\begin{array} { l l l l l l } & & S & E & N & D \\ +& & M & O & R & E \\ \hline & M & O & N & E & Y \\ \end{array}$

and find the value of $S+E+N+D+M+O+R+Y.$

Alice, Bob, and Cathy are on trial. Exactly one of them is guilty. They each make a statement:

- Alice says, "Bob is innocent!"
- Bob says, "Cathy is innocent!"
- Cathy says one of the three is guilty (maybe herself), but we don't hear who.

The judge knows who is guilty, and heard Cathy's statement. She says, "I could tell you what Cathy said and how many true statements were made, but you still wouldn't know who is guilty."

The judge adds, "I will tell you that the guilty party told the truth. Now if I told you how many true statements were made, you would know who is guilty."

Who is guilty?

Suppose you have 8 watermelons that are exactly the same in appearance. They are equal in weight except for one, which is heavier than the others. You want to find the heaviest one by using a two-pan balance. Each pan can hold any number of watermelons, and the only items that you can place in the pans are watermelons.

Find the best strategy to determine the heaviest watermelon. In the worst case scenario, how many times do you have to use the two-pan balance?