Scotty Tilton

stilton
Office: HSS 5062
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Disclaimer: These aren't original problems of mine, but I can't remember where I got them all from. I tried to give people/creators credit where I could remember. If you see this and want credit, e-mail me and I'll edit this page to give you credit. Also, if something is wrong here, e-mail me and I'll fix it.

Manim for March 2024 from the AMS Epsilon of Math Calendar

Click here to see all the videos!

More Riddle-y Types of Problems that are good for a long hike or if you run out of things to talk about with your friends.

Let's say you're holding a rock in a boat that is in a swimming pool. Let's also say that your good pal is standing at the edge of the swimming pool and has a waterproof sharpie that they use mark the height of the water on the wall of the swimming pool. Then say that you toss the rock out of the boat and it sinks to the bottom of the pool. Your buddy yells at you that the water level has changed. Did the water level rise, sink, or did your buddy lie to you, and the water actually stayed at the same level?
Six good pals are surrounding a basket that contains six eggs. They decide that these eggs are good to have around, and they decide to share them. Each person takes an egg home with them after the gathering, but there is still an egg in the basket. How the heck?
You are making a cake in an oven, but the darned timer is busted. All you have are three 1-hour candles that each have wicks coming out both ends that burn nonuniformly for an hour. What I mean by this is that the candle will be gone after an hour, but you can't measure how much time is left on the candle with a ruler. This is because the person making the candles did a terrible job with molding the wax uniformly around the wicks, but they did a stupendous job of making them burn for exactly an hour. Moving forward, you have a cake in the oven. The recipe said that the cake will explode if you cook it for any time longer than 1 hour and 45 minutes. You have guests coming over, and you really don't want this cake to explode. Using your three 1-hour candles, can you time when to take the cake out perfectly without disaster?
Punctuate the following so it is coherent:
John while Jim had had had had had had had Had had had had a better impact on the teacher than had had had
(Note that had refers to the verb >had in every instance above.)
My friend Peter C-K told me about this problem. He makes some cool art.
Punctuate the following so it is coherent
Before was was was was was is
Punctuate and captitalize the following so it is coherent
buffalo buffalo buffalo buffalo buffalo buffalo buffalo buffalo
Note: Buffalo is the name of a town in Wyoming and a town in New York. The word buffalo, however, can be either a noun describing a large four-legged animal or a verb that means to bully.
Climbing Question, but good for anyone: You have a 60m rope, a sharp knife, a harness, and an ATC (a great tool to help you maneuver down the rope). You also happen to be on top of an 80 foot cliff which has anchors at the top with you and 40m below you. You need to get to the ground safely, and you can't fall more than half a foot at the risk of twisting your ankle. There is no one else near to help you. How do you get down safely?

More Math-y Types of Problems

From MindYourDecisions on Youtube: There are 25 mechanical horses and a single racetrack. Each horse completes the track in a pre-programmed time, and the horses all have different finishing times, unknown to you. You can race 5 horses at a time. After a race is over, you get a printout with the order the horses finished, but not the finishing times of the horses. What is the minimum number of races you need to identify the fastest 3 horses? Problem and Solution.
$\displaystyle \fbox{$\int (\ln(x))^nx^s \, ds$}$ where $n\in \mathbb{N}, s\in \mathbb{R}$ (with potentially more conditions on $s$.) A buddy Cody Morrin at my REU at Chico State told me about this problem.
$\displaystyle\fbox{$\int_0^1 x\sqrt{x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{x\cdots}}}}dx$}$
$\displaystyle\fbox{$\int_{-\infty}^\infty e^{ax^2+bx+c}dx$}$ (First, try when $a=-1$ and $b=0$.)
$\displaystyle\fbox{$\int_0^\infty \frac{e^x-1}{x}dx$}$
Say there is an evil submarine at a fixed rational coordinate in an infinite planar ocean ($\vec p\in \mathbb{Q}^2$). The submarine moves in a fixed rational direction every day (aka there is a fixed direction and distance, given by a vector $\vec{v}\in \mathbb{Q}^2$, in which the submarine moves every day). You are in posession of an airplane that can teleport exactly once a day to any fixed rational coordinate in $\mathbb{Q}^2$ and drop a bomb at that coordinate. Come up with a strategy that guarantees you to hit the submarine in finite time.
Given any number written in base $10$, there is some power of $2$ whose first digits are this number. I.e. if I write $123,456$ in base $10$, there is some $N$ where $2^N$ looks like $123456\cdots$ in base 10.
Not a problem, but on Valentine's Day, send this to your crush $x^2+(y-\sqrt[3]{x^2})^2\leq 1$ (Increase 1 to a larger number depending on how much you care about the person)

Super Math-y Problems

(That I had a lot of fun with)

From This UCSD Qual: Let $$K:=\left\{[x:y:z:w]\in \mathbb{CP}^3\mid x^4+y^4+z^4+w^4 = 0\right\}.$$ I'll tell you that $\pi_1(K) = 0$, $K$ is compact without boundary, and $K$ has Euler characteristic $\chi(K) = 24$. Compute $H_\ast(K;\mathbb{Z})$.
From my friend Adam Holeman, but he said he didn't come up with it:
Are there topological spaces $X,Y$ where $X\not\cong Y$, but $X\times [0,1] \cong Y\times [0,1]$? ($[0,1]$ is the unit interval with the topology inherited from $\mathbb{R}$ with it's usual topology). Give proof.
What is the explicit group structure of $\mathbb{CP}^\infty$? A satisfying answer would describe how to explicitly multiply two elements of the set $\mathbb{CP}^\infty$.
(This question came up becauses $\mathbb{S}^1$ is a $K(\mathbb{Z},1)$, and then someone told me this should mean that $\mathbb{CP}^\infty$ should have a group structure since it is a $K(\mathbb{Z},2)$.
This post gives an answer, but didn't make it explicit enough for me to feel satisfied.)
From my friend Arseniy Kryashev:
Are there topological spaces $X,Y$ such that there is a continuous bijection $X\to Y$, a continuous bijection $Y\to X$, but $X$ is not homeomorphic to $Y$?

More, More, More!

To study for my Real Analysis Qual, I made Real Analysis-le. It is supposed to change the problem every day.
Scotty Tilton
4th year Ph.D. - UC San Diego

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