Problems

A tourist walked 3.5 hours, and for every period of time, in one hour, he walked exactly 5 km. Does this mean that his average speed is 5 km/h?

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?

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?

Show that a knight’s tour is impossible on a \(3\times3\) grid.

Show that two queens together can attack every square on a \(4\times4\) grid, but one queen on her own cannot do it. This type of problem is called ‘queen’s domination’.

How many queens can you place on a \(4\times4\) grid so that none of them attack each other?

Show an knight’s tour on a \(5\times6\) chessboard. That is, a path where a knight starts at one square, and then visits every square exactly once, making only moves legal to a knight.