Problems

Ignat gives two tutors a problem. He’s thinking of two different positive integers greater than \(1\), whose sum is at most \(100\). Ignat tells Pier (\(P\)) the product of the two numbers, and Sasha (\(S\)) the sum of the two numbers. Both Pier and Sasha know all of this, and then have a little conversation:

Sasha says that Pier doesn’t know the two numbers.

Pier says that he knows the two numbers.

Finally Sasha says that he knows the two numbers.
What are the two numbers?

On a bus out of the Hattius kingdom, you sit behind two wizards. The first says to the second “I’ve a positive integer number of children, all of whose ages are positive integers. The sum of their ages is the bus number, and their product is my age."

The second wizard replies “If you told me your age and how many children you had, would I be able to work out their individual ages?"

The first wizard says “No, unfortunately not." To which the second says “Aha! Now I know how old you are." What was the bus number?

Sperner’s lemma in dimension \(2\).
Subdivide a triangle \(ABC\) arbitrarily into a triangulation consisting of smaller triangles meeting edge to edge. Define Sperner coloring of the triangulation as an assignment of three colors to the vertices of the triangulation such that:

• Each of the three vertices \(A, B,\) and \(C\) of the initial triangle has a distinct color

• The vertices that lie along any edge of triangle \(ABC\) have only two colors, the two colors at the endpoints of the edge. For example, each vertex on \(AC\) must have the same color as \(A\) or \(C.\)

Here is an example of Sperner’s triangulation

Prove that every Sperner coloring of every triangulation has at least one "rainbow triangle", a smaller triangle in the triangulation that has its vertices colored with all three different colors. More precisely, there must be an odd number of rainbow triangles.

What’s \(3\uparrow\uparrow3\)?

Is \(4^{15}\) more or less than a billion?

Approximately how many footsteps do I take in a year? (estimate to the nearest power of \(10\))

What’s \(2\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow2\)?

What’s bigger out of \(5^6\) and \(2^{14}\)?

What’s \(3\uparrow\uparrow3\) as a power?

Is \(45^{45}\) bigger or smaller than \(10^{80}\)?