My friend Eduardo Torres Davila and I made an ArXiv game that allows you to test your ability to guess which random paper seems more novel!

Here are some random problems that I think are fun and/or cool. If you email me a valid solution to any of these, I'll Include your name below the problem as a document of your bragging rights in perpetuity. 

1) (un)Lucky Luke: This is a great problem I learned from Sándor Dobos at the Budapest Semesters in Mathematics (BSM) program. A lottery sells tickets where participants can choose k numbers from 1 to n (inclusive) for each ticket. Then to find the winner(s), k distinct numbers are drawn from the 1, ... , n. If any participant has a ticket whose k numbers match the k drawn numbers, they win. Instead of trying to win, unLucky Luke is trying to have a ticket that doesn't have any of the k drawn numbers. In fact, unLucky Luke wants to create a set of tickets that guarantees that at least one of the tickets will not have any of the k drawn numbers no matter what. Let L(n; k) be the minimum number of tickets for which this is possible for a given n and k. 

Warm up: L(99; 6). That is, we want to create a set of tickets, each with 6 numbers from 1-99, that will always have at least one ticket with none of the 6 winning numbers. Solution. 

Main problem: What is L(36; 6)? If the lottery draws numbers from 1-36 and each ticket consists of 6 distinct numbers, what is the smallest number of tickets for which we can create a selection of tickets that will guarantee that at least one of the tickets does not include any of the 6 winning numbers? 

2) Flora and fauna: Another BSM problem, introduced to me by Gabór Simonyi. Say there are n families living in a territory. The Department of Agriculture has divvied up the territory into n farming zones of equal area (the individual zones are not necessarily connected and may consist of many pockets). Independently, the Department of Wildlife has also drawn borders for n hunting zones of equal area. Each family is to be assigned one farming zone and one hunting zone. Is it possible to do so such that every family has a place to build their house that is contained in both their farming zone and their hunting zone? 

3) Coloring and coins:

• 3a: "Inscribe" a cube inside a sphere so that all eight vertices are touching the sphere. Paint the sphere however you like as long as 90% of the sphere is red and 10% is blue. The colored regions don't have to be connected. Go wild with dots, lattices, etc. Is it always possible for me to rotate the cube inside the sphere so that no vertex touches blue?

• 3b: Place ten dots on the plane. If I have ten coins of equal size, can I place them in a non-overlapping way so that all ten dots are covered? 

4) Catalan Counting: Let C_n be the n-th Catalan number. Give a combinatorial argument that for all t > 1, 

\sum_{i=0}^{\lfloor t/2 \rfloor} (-1)^{i}C_{t-i-1}\binom{t-i}{i} = 0

Solver(s): Naomi Burks.