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.

# 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:
(Note that "had" refers to the verb "had" in every instance above.)
• 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)

# Other things

• An implicit equation for 3D plotting a torus: $\displaystyle\left(\sqrt{x^2+y^2}-R\right)^2+z^2 = r^2$ where $R$ is the radius from the donut hole to the middle of the dough, and $r$ is the radius from the middle of the dough to the frosting.
• An implicit equation for 3D plotting an $n$ torus: $\displaystyle\left(\left(\prod_1^n (x-(i-1))(x-i)\right)+y^2\right)^2+z^2 = r^2$ where $r$ is small. Credit to user856.