Problems

Liam saw an unusual clock in the museum: the clock had no digits, and it’s not clear how the clock should be rotated. That is, we know that \(1\) is the next digit clockwise from \(12\), \(2\) is the next digit clockwise from \(1\), and so on. Moreover all the arrows (hour, minute, and second) have the same length, so it’s not clear which is which. What time does the clock show?

Due to a mistake in the bakery, a cake that was supposed to be shaped as two concentric pieces (like on the right diagram below) came out like the left diagram below. Find the smallest number of pieces the cake should be cut into in order to rearrange the pieces into the cake on the right side of the picture.

Note that the cake is \(\textit{not}\) tiered like a wedding cake, but is shaped like a cylinder with a flat top. Curved cuts are allowed.

One cell was cut out of a \(3\times6\) rectangle, as seen in the diagram. How should you glue this cell in a different place to get a figure that can be cut into two identical ones? If needed, the resulting parts can be rotated and reflected.

Could you meet a person inhabiting this planet who asks you “Am I a Goop?"

On this planet you meet a couple called Tom and Betty. You hear Tom ask someone: “Are Betty and I both Goops?"
What kind is Betty?

You learn that one of the aliens living on this planet is a wizard. You learnt that by overhearing a certain question being asked on the planet. What question could that have been?

Suppose you meet a person inhabiting this planet and they ask you “Am I a Crick?" What would you conclude?

You meet two friends, Katja and Anja. Katja once asked Anja “Is at least one of us a Goop?"
What kinds are Katja and Anja?

You later learn that there is exactly one wizard on this planet of Cricks and Goops. You would like to find out who that is.
You meet an alien called Andrew. He asks you “Am I the kind that could ask whether I am not the wizard?"
Do you have enough information to tell for sure who the wizard is by now?

You meet another alien, whose name is Teddy. He asks you “Am I the kind who could ask whether I am a Goop?"
Can anything be deduced about Teddy?