Interesting Puzzles, Problems, and Conundra

Is it possible to determine my number with certainty?

(Harder: if 100 is replaced with some other number of possible answers, where is the boundary between when this can and cannot be done?)

You are building your new nation in the Euclidean plane, and need to create n cities. From every city, there are two kinds of transport: a teleporter which takes you to the nearest city, and a teleporter which takes you to the farthest city. (The cities are assumed to have pairwise distinct distances.)

For large n, how can you position your cities to bound the length of the worst-case number of teleportations required to get from one city to another?

In particular, try to get o(n), or (for a slightly harder variant) O(√n). Can you do even better? (I don't know if improving on √n is possible.)

Which of the following answers is the correct answer to this question?

None of the below

All of the below

All of the above

One of the above

None of the above

None of the above

The answer is not what almost everyone who first hears this problem thinks.

Let A = {1,2,3,4,5} and let P be the set of all nonempty subsets of A. A function f from P to A is a “selector” function if f(B) is in B, and f(B union C) is either equal to f(B) or f(C‌). How many selector functions are there?

Arby: Becky did it. Becky: Ducky did it. Cindy: I didn’t do it. Ducky: Becky is lying.

Twenty children attend a rural elementary school. Every two children have a grandfather in common. Prove that some grandfather has not less than 14 grandchildren in this school.

Anne: I didn’t do it. Bill did it alone.

Bill: I didn’t do it. Caroline did it.

Caroline: I didn’t do it. Bill did it.

Detective Radstein also discovered that all three suspects are members of a club called The Halfsies. Every time they speak, they make two statements, one of which is a lie and the other one is true. Who committed the robbery?

Do there exist two powers of 2 whose base-10 digits are anagrams of each other?

In a certain mathematics class, if each boy purchases a coffee and each girl purchases a doughnut (all items costing an integer number of dollars), the class would spend a total of one dollar more than if each boy purchased a doughnut and each girl purchased a coffee. If there are more girls than boys in the class, what can one determine about the number of girls and the number of boys?

While questioning a witness, a judge is only allowed to ask questions which are to be answered ‘yes’ or ‘no’. The judge has carefully calculated that, as long as the witness answers every question truthfully, then she can solve the case in no more than 91 questions. One or more of these questions may depend on the answers to previous questions. Show that the judge can solve the case in no more than 105 questions if it is known that the witness may answer at most one question falsely.

Your questions may encompass any past events you wish, but cannot introduce paradoxes or similar sorts of logical trickery (e.g. you can't make question 1 "Will your answer to this question be 'no'?", count it as a lie no matter what they answer, and proceed with the regular questions).

Same problem as yesterday, but this time with a cuboid of dimensions k × l × m. How many straight cuts does it take in general to turn it into klm 1×1×1 cubes? 

You have a 3cm x 3cm x 3cm block of cheese, and you have a recipe that calls for 27 1cm cubes to be used instead. To accomplish this task, you have a knife which makes straight (planar) cuts, but you’re very worried about it getting too dull, so you’d like to use as few cuts as possible.

How many cuts will it take to accomplish your task? (You are allowed to rearrange the pieces after each cut, if you choose.)

In a certain room, everyone is fluent in some subset of four languages. For any two people in the room, there exists a language they are both fluent in.

Prove that there exists a language which at least 3/5 of people in the room are fluent in.

a) are the reals,

b) are exactly the integers,

c) are exactly the rational numbers?