Math Encounters - Intuition Gone Awry: Puzzles that S-T-R-E-T-C-H Your Mind

Humans are not born with perfect mathematical intuition, yet we make most decisions based on "feel." Expand your thinking as mathematician Peter Winkler uses mind-boggling puzzles to help you adjust your intuition when it's about to run off the rails. Peter had a special introduction by Joel Spencer, Professor of Mathematics at the Courant Institute.

[mazenmokhtar]

To gonnabphd: When you flip a slice you also flip the side that contains the icing, which lines up on the next run around the circle. Some details:
- Define the angle of the cut to be A and Tau = the angle of a full circle (equal to 2 PI)
- Cut N slices around the circle until you get close to a full circle.
- Consider the slice F that crosses the full circle. It can be divided into two potions: A_1 before the full circle and A_2 after
- Therefore Tau = (N . A) + A_1 and A_1 + A_2 = A and ((N . A) + A) mod Tau = A_2
- Once that slice is flipped, A_1 is upside down, but is positioned after A_2, which is now right-side up <<-- This is the critical observation
- Cut another N slices, flipping portions of the cake back up. These cuts start at angle A_2, which means that they are identical to the first N cuts but moved up by an angle of A_2
- The last of the second set of N cuts will intersect slice S at an angle of A_2, bringing the whole cake right side up.

[numbernexus]

It's surprising, but it works for any angle. The first hint about why is that, when you cut out and reverse a slice, you don't just toggle iced and non-iced; you also reflect.

To understand how the puzzle works, I'd draw pictures. You can represent the cake by a circle, and the question is, which angles are iced? Actually do the experiment for some choice of angle (say, 2pi/3 + a little bit), in pencil so you can draw and erase circle arcs.

Once you do the experiment, you'll see how it works.

[gonnabphd]

Anyone have an explanation/proof for the "icing on the cake"-puzzle? It still seems like the angle has to be a rational multiple of pi to me...

[kunalmandalia]

Really great talk. The 100 prisoner box solution is stunning.

[Elias_Halloran]

Do the people in dot town prefer the dot product to the scalar product, I ask.

[infinitesimotel]

Enjoyed this, but probability is just a convoluted way of expressing that we still dont know until it happens. Its still 50/50 cuz you never know if your going to be on the 99 side or the 1 side.

[davidjames]

I have a valid solution to the 100 prisoners puzzle. Have all the male prisoners change their names to John Doe and have all the female prisoners change their names to Jane Doe. Then there is a much more reasonable chance they will all find their name. There is nothing in the rules that should prevent this so it is a valid solution.

[scopeless]

Some nice stuff here, but the speaker is very bland, could do with making the presentation more enjoyable instead of putting people to sleep.

[davidjames]

Here is another valid solution to the 100 prisoners puzzle. The first person entering the room has about a 50% chance of finding his or her name in one of the 50 boxes. Let's assume his name is Steve Smith. Steve will also remember what box 1 other prisoners name is in (let's say Sally Jones) and tells her the box # (the ordinal position) after exiting the room but before it is Sally Jones' turn to go into the room. This is allowed because the rules say that communication with fellow prisoners is not allowed but prisoners is plural so if you communicate with only 1 prisoner after exiting the room, that is not breaking any rule. Sally would then look in her box and then pick the name of another person and then tell that person upon exiting the room... The name of the other person will be the name in the box immediately to the right (with wraparound) of the box of the person who found their own name. For example, if Steve found his name in box 19, he would look in box 20 and then inform that person (Sally Jones in this example) that her name is in box 20. Sally would then open box 20 and also open box 21, then inform the person who's name is in box 21 that it is there. This is a valid solution because it does not break any of the stated rules.

[Moronvideos]

This guy needs massive doses of COUGH DROPS .... Throat clearing is very annoying.

[davidjames]

The probability of each prisoner finding his or her own name by looking in 50 random boxes is NOT 50% There are 2 reasons for this. 1) They could choose the same box more than once which would hurt their chances. 2) 2 or more prisoners might have the same name, thus increasing their chances. This person giving this presentation is not very good at basic Math to get something that simple wrong.