Complete as many of the parts of the four problems below as you can. Show your work and fully explain all of your answers. You may either type your solutions and print them to a PDF file, or write them by hand and scan them to a PDF file. Submit your solutions by uploading that PDF file along with your online application.
These problems are intentionally challenging and you are not expected to solve them all completely. They are designed to give you the chance to demonstrate how you wrestle with mathematical problems. Partial solutions and observations should be submitted in the event that you do not find a complete solution. All work should be entirely your own. You should not discuss the problems with anyone else, whether online or in person.
The problems are listed below. A printable PDF version of the test is also available.
Have fun!
These are always true you see,
for positive integers $m$, $n$, and $p$!
These are always true you see,
for positive integers $m$, $n$, and $p$!
Card Exchange Property: For any two legal decks $X$ and $Y$, if $y$ is a card in deck $Y$ but not in deck $X$, then there exists some card $x$ in deck $X$ but not in deck $Y$ such that the two decks that are obtained by exchanging cards $x$ and $y$ are also legal.For instance, if the deck $abc$ (short for the deck containing cards $a$, $b$, and $c$) and the deck $bde$ are both legal, then since $d$ is in $bde$ but not in $abc$ we must be able to trade card $d$ for one of the cards from the deck $abc$ that is not in $bde$ (i.e., for at least one of $a$ or $c$, but possibly not both) and end up with two legal decks.