Problems

Suppose you have a coffee mug made of stretchy and expandable material. How do you mold it into a donut that has a hole inside?

Label the vertices of a cube with the numbers $1,2,3,\dots ,8$ so that the sum of the labels of the four vertices of each of the six faces is the same.

On the questioners’ planet (where everyone can only ask questions. Cricks can only ask questions to which the answer is yes, and Goops can only ask questions to which the answer is no), you meet 4 alien mathematicians.

They’re called Alexander Grothendieck, Bernhard Riemann, Claire Voisin and Daniel Kan (you may like to shorten their names to $A$, $B$, $C$ and $D$).
Alexander asks the following question “Am I the kind who could ask whether Bernhard could ask whether Claire could ask whether Daniel is a Goop?"

Amongst the final three (that is, Bernhard, Claire and Daniel), are there an even or an odd number of Goops?

On the questioners’ planet, there are two types of aliens, Cricks and Goops. These aliens can only ask questions. Cricks can only ask questions to which the answer is yes, Goops can only ask questions to which the answer is no.

There are 19 aliens standing in a circle. Each of them asks the following question “Do I have a Crick standing next to me on both sides?" Then one of them asks you in private “Is 57 a prime number?" How many Cricks were actually in the circle?

The letters $O$, $P$, $S$ and $T$ represent different digits from $1$ to $9$. The same letters correspond to the same digits, while different letters correspond to different digits.

Find $O+P+S+T$, given that $SPOT+POTS=15,279$.

In the diagram below, I wish to write the numbers $6,11,19,23,25,27$ and $29$ in the squares, but I want the sum of the numbers in the horizontal row to equal the sum of the numbers in the vertical column. What number should I put in the blue square with the question mark?