OFF-CENTER WHEEL

Lolinsan looked at the piece of wood before him. It is a 10” diameter round, flat piece he wanted to use as a “wheel” in his art lesson homework. Alas! While drilling the center hole (1” diameter) , he sneezed, and the hole is way off-center (although, luckily, still within the “wheel” – the center of the hole is 3” away from the center of the wheel).

And then, revelation! Willy cuts the wheel into two parts, glued them again, and got a perfectly centered hole in the wheel. How did he do it?

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Note: assume lolinsan has in his possession a magical cutter that cuts with zero width, and magic glue that can glue pieces with zero distance between them.

MATCH EQUATIONS

In each of the following three configurations of matches, move a single match to form valid equations. The = and + are composed of two matches each.

1. \/|+|=\/
2. \/|+|=\/|
3. \/|+|=|||

HOTEL INFINITY

So, an eccentric entrepreneur by the name of Alphonse Null has sent out a press release about his new, mind-blowing hotel: The Hotel Infinity. Null informs the world that this hotel has an infinite number of rooms (specifically, an infinity equal to the cardinality of the integers). A quick tour puts skeptics' claims to rest; as far as anyone can tell, this hotel has infinite rooms. The consequences are mind-boggling, and Null sets up a press conference to answer questions...

"So, Mr. Null, how will patrons get to their room, if their room number has, say, more digits than protons in the universe?"

"The elevators have an ingenious formula device instead of buttons... simply input the formula for your room number, with Ackermann numbers or somesuch... your room formula can be picked up at the front desk. There's not even any need to know what the formula means!"

"How do you produce the power and water for this hotel?"

"I have infinite generators and wells, of course. This IS an infinite hotel, you know! *chuckle*"

"What about costs? How much will it cost to stay here?"

"That's the beauty of it! Since there are as many positive even integers as there are integers, I can change the same price to only every other room and still make the same profit! I could charge only every millionth room... each guest has a one-in-a-million chance of not getting a free room, and I still get paid the same! I love the properties of infinite sets, especially when it comes to profit!"

"But, Mr. Null... I think you've made a severe mistake in your assumptions regarding profit..."

"Oh?"

The reporter then mentioned something which made Mr. Null's face turn white.

"Oh... oh goodness... THIS PRESS CONFERENCE IS OVER!" Then he ran out.

Assuming that everything Null said about the hotel is true: it really is infinite; it really is easy to get to your room; it really can generate infinite power for the guests; the cardinality of the set of multiples of a million, is the same as the cardinality of the integers...

So with what simple assumption did Mr. Null go wrong?

FRYING FOOD

The boiling point of olive oil is higher than the melting point of tin. If Italian skillets are made of tinned copper, how can they be used to fry food in olive oil?

DOMINANT FIFTH

"What's the Dominant Fifth?" asked Dr. Dingo, as his daughter Cicely came in from school.

Cicely blushed. "Just a secret society," she said. "I'm one of the vice-presidents."

"And you're meeting tonight; is that right?"

"How on earth did you know?" ask Cicely.

"You left this lying about. That's no way to keep secrets, my girl." He handed Cicely this paper:

Dominant Fifth

REASM NCNVE OTMLE SEHST TAOEI

"How did you manage to read it?" asked Cicely. "The code is known to only about eight of us."

"Change it," said Dingo. "Any fool can read that."

Where and When is the next meeting scheduled?

RIVER CROSSING

Two people come to a river. There is a boat, however it can carry one person only. How can they each get to the other side of the river using the boat?

ENVELOPE GAMBLE

I have a distribution over the Reals which you do not know. I choose two numbers from it, and write them inside envelopes. You are given one of the envelopes, and allowed to see the number inside it. Then, you are given the option to switch envelopes once. After you settle on an envelope, you win the amount inside your envelope, and you pay the amount inside the other envelope. Can you win money playing this game, with a strategy independent of my distribution?

PAST, PRESENT, FUTURE

There are three omniscient gods sitting in a chamber: Past, Present and Future. They are all truthful, but with the following caveat: Present answers the question currently being asked, Past answers the last question asked in their chamber, and Future answers the next question which will be asked in their chamber. Despite their manipulation of which question to answer, each still answers immediately as if answering the question currently being asked.

Furthermore, the gods answer in a language in which "yes" and "no" are replaced by "da" and "ya", but you do not know which is which. You only know that their answers are consistent amongst themselves.

With three questions, determine which god is which.

POP QUIZ

The professor for class Logic 315 says on Friday: "We're going to have a surprise quiz next week, but I'm not telling you what day... if you can figure out what day it will be on, I'll cancel the quiz."

The students get together and decide that the quiz can't be on Friday, as if the quiz doesn't happen by Thursday, it'll be obvious the quiz is on Friday. Similarly, the quiz can't be on Thursday, because we know it won't be on Friday, and if the quiz doesn't happen by Wednesday, it'll be obvious it's on Thursday (because it can't be on Friday). Same thing for Wednesday, Tuesday and Monday. So it can't be on ANY day, so there's no quiz next week!"

They tell the professor, who smiles and says, "Well, nice to see you're thinking about it."

On Tuesday, the professor gives the quiz, totally unexpected!

What's the flaw in the students' thinking?