Complete as many of the parts of the four problems below as you can. Show your work and fully explain all of your answers. You may either type your solutions and print them to a PDF file, or write them by hand and scan them to a PDF file. Submit your solutions by uploading that PDF file along with your Online Application.
These problems are intentionally challenging and you are not expected to solve them all completely. They are designed to give you the chance to demonstrate how you wrestle with mathematical problems. Partial solutions and observations should be submitted in the event that you do not find a complete solution. All work should be entirely your own. You should not discuss the problems with anyone else, whether online or in person.
The problems are listed below. A printable PDF version of the test is also available.
Alpha: “Gamma always lies.”Which is the liar, which is the truth teller, and which is the copycat?
Beta: “I never lie.”
Gamma: “Yes you do!”
Alpha: “No, he doesn’t!”
Delta: “Sometimes I lie when I speak just after Beta.”How does Delta decide when to tell the truth?
Beta: “Delta, if you are the next one to speak, you will certainly lie.”
Gamma: “Delta, if you are the next one to speak, you will certainly lie.”
Alpha: “If Delta would lie if he were to speak next, then sometimes Delta needs to consider more than the previous two statements he has heard when deciding whether or not to lie.”
Note: Gravity pulls objects down towards the ground. We use the term “horizontal” to mean any direction parallel to the ground, and “vertical” to mean perpendicular to the ground. The walls to which the clotheslines are attached are parallel to each other, vertical, and run north-south.