Math is cool!

Hiya all! I am your friendly neighbourhood mathematician, here to spread the joy of maths with you guys.

I am a Ph.D. in Denmark, currently procrastinating my project, and I thought it was a wonderful idea to celebrate my 10th cake-day with some math-related jokes and riddles.

You get two jokes per riddle, lucky you!

J1) Why is π sad? Cos' π is negative.

J2) "The number you have dialed is imaginary. Please rotate your phone 90 degrees and try again."

And now for the riddle:

R1) Can you hang a picture frame from two nails using a string with both ends tied to the picture frame, such that the picture frame falls down if you remove any of the two nails?

Solution: ˙sʇuǝɯɯoɔ ǝɥʇ uᴉ uǝʌᴉƃ sᴉ uoᴉʇnlos ǝlqᴉssod ɐ ɟo ƃuᴉʍɐɹp ∀ ˙sǝ⅄

How is this math related? Well, it is a type of question that is actually relevant for topology, a certain branch of mathematics. Formulated into mathematics, showing that you can actually hang the picture frame in such a way is a proof that the fundamental group of the twice punctured plane is non-abelian, while the once-punctured plane has an abelian fundamental group. This is of course just mathematics nonsense, but we quite enjoy it.

J3) One day, Jesus said to his disciples: "The Kingdom of Heaven is like 3x squared plus 8x minus 9." St. Thomas looked very confused and asked St. Peter: "What does the teacher mean?" St.Peter replied: "Don't worry - it's just another one of his parabolas.

J4) Some insist that the pope is the greatest cardinal. But mathematicians knows this cannot be so, as every pope has a successor.

And the riddle:

R2) Using exactly four 4's and the operations addition (+), subtraction (-), multiplication (x) and division (%), along with parenthesis, make equations that are equal to all the integers 0 to 9.

Here is an example equal to 12: 4x(4-4%4) = 12. There are multiple solutions for many of the numbers!

Solution: See my comment below.

Final set of jokes and riddles:

J5) A math professor is teaching a class, and at a certain point stops, and says "now this is obvious".
A student asks if he's sure it's obvious.
The professor pauses, then leaves the room.
Half an hour later he returns and says, "yeah, it's obvious ".

J6) Q: What's a polar bear?
A: A cartesian bear after a coordinate transform.

And the riddle:

R3) An alien abducts a group of mathematicians and forces them to play its deadly game. Any rule broken means that they will all be killed. The game is as follows: The mathematicians are lined up in a row, all facing the same direction like a conga line. A hat is placed on each person's head, either white or black. They must now guess which color hat is on their own head, but they are not allowed to take it off or talk with each other.
The mathematician in the back is now asked for the color of his hat. If he guesses correctly, he is spared, but if his guess is wrong, he is executed. Everyone can hear his guess. Afterwards, the alien proceeds to ask the next person in the line, and so on and so forth.
The mathematicians are allowed to discuss strategy beforehand, and determine their order in the line.
What strategy can the mathematicians come up with that saves the most lives possible?

Solution: ˙ʇɐɥ ɹᴉǝɥʇ ɟo ɹoloɔ ǝɥʇ ʇno ǝɹnƃᴉɟ oʇ poɥʇǝɯ ǝɯɐs ǝɥʇ ǝsn uɐɔ ʎǝɥʇ 'sǝssǝnƃ snoᴉʌǝɹd ǝɥʇ ɟo uoᴉʇɐɯɹoɟuᴉ ǝɥʇ ƃuᴉsn 'suɐᴉɔᴉʇɐɯǝɥʇɐɯ ƃuᴉʍolloɟ llɐ ɹoℲ
˙,,ʞɔɐlq,, sɐʍ ssǝnƃ snoᴉʌǝɹd ǝɥʇ ɟᴉ sʞɹoʍ poɥʇǝɯ ǝɯɐs ǝɥ┴ ˙ʞɔɐlq ǝq ʇsnɯ ʇɐɥ uʍo ɹᴉǝɥʇ 'sʇɐɥ ɟo ɹǝqɯnu uǝʌǝ puɐ sǝǝs ʎǝɥʇ ɟI ˙ǝʇᴉɥʍ ǝq oʇ sɐɥ ʇɐɥ uʍo ɹᴉǝɥʇ 'sʇɐɥ ǝʇᴉɥʍ ɟo ɹǝqɯnu ppo puɐ sǝǝs ʎǝɥʇ puɐ ',,ǝʇᴉɥʍ,, sɐʍ ssǝnƃ ǝɥʇ ɟI :ǝʇᴉɥʍ sᴉ ʇɐɥ uʍo sᴉɥ ɟᴉ ʇno ǝɹnƃᴉɟ uǝɥʇ uɐɔ ʎǝɥʇ 'ssǝnƃ s,uɐᴉɔᴉʇɐɯǝɥʇɐɯ snoᴉʌǝɹd ǝɥʇ ƃuᴉsn puɐ 'ɯǝɥʇ ɟo ʇuoɹɟ uᴉ sʇɐɥ ǝʇᴉɥʍ ɟo ɹǝqɯnu ǝɥʇ sʇunoɔ ǝuᴉl ǝɥʇ uᴉ uɐᴉɔᴉʇɐɯǝɥʇɐɯ ʇxǝu ǝɥʇ 'ʍoN ˙pǝɹɐds ƃuᴉǝq ɟo ǝɔuɐɥɔ 0ϛ-0ϛ ɐ sɐɥ uǝɥʇ ʎǝɥ┴ ˙ǝsᴉʍɹǝɥʇo ,,ʞɔɐlq,, puɐ 'ɯᴉɥ ɟo ʇuoɹɟ uᴉ sʇɐɥ ǝʇᴉɥʍ ɟo ɹǝqɯnu uǝʌǝ uɐ sᴉ ǝɹǝɥʇ ɟᴉ ,,ǝʇᴉɥʍ,, sǝssǝnƃ ʍou uɐᴉɔᴉʇɐɯǝɥʇɐɯ ʇsǝllɐʇ ʎɹǝʌ ǝɥ┴ ˙ɯǝɥʇ ɟo ʇuoɹɟ uᴉ ǝuoʎɹǝʌǝ ɟo sʇɐɥ ǝɥʇ ɟo ɹoloɔ ǝɥʇ ǝǝs uɐɔ ʎǝɥʇ 'snɥ┴ ˙ʞɔɐq ǝɥʇ uᴉ ʇsǝllɐʇ ǝɥʇ 'ʇɥƃᴉǝɥ oʇ ƃuᴉpɹoɔɔɐ ǝuᴉl ǝɥʇ uᴉ pǝɹǝpɹo ǝq suɐᴉɔᴉʇɐɯǝɥʇɐɯ ǝɥʇ ǝʞɐW ˙pǝɹɐds ǝq uɐɔ ǝuo ʇnq ll∀

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Thanks for reading, and happy Spooktober! If you have any math-related or ph.d.-related questions you would like answered, I would be more than willing to do so in the comments.