Fun with Math

Fun you say?

Soooo I took a really badass math class about a year back. Instead of mindlessly grinding equations all day, you'd often just get a single story problem and get a few days to solve it, then around a week to write a paper explaining the answer in a way that anyone with a very basic understanding math can comprehend.

I sent Navarog this problem a while ago, and like most situations he let me down miserably. Now I challenge the rest of you! I can give hints at some point if you want, though they will give away the solution pretty quick I think. Use spoilers for any legitimate guesses and I'll tell you if you are correct or not. If you guys like this kind of math I have a few more. If not, you need to step your nerd game up.

It is not a trick question or a "gotchya" answer, nor is it impossible to solve. I will put all hints in spoiler tags. GLHF

The Handshake Problem​

Mr. & Mrs. Anbouba recently attended a party at which there were three other couples. Various handshakes took place. No one shook hands with their spouse. No one shook his or her own hand. No one shook hands with the same person twice. At the end of the evening, Mrs. Anbouba asked the seven other people how many hands he or her had shaken. Each person gave her a different answer. How many hands did Mr. Anbouba shake?

Gauss' Problem​

Find the sum of all the integers from 1 to 100 without the use of a computer or calculator. 1+2+...+99+100 = ?

The Fly Problem​

A rectangular room measure 30 feet in length and 12 feet in height, and the ends are 12 feet in width. A fly, with a broken wing, resets at a point one foot down from the ceiling at the middle of one end. A smudge of food is located one foot up from the floor at the middle of the other end. The fly has just enough energy to walk 40 feet. Show that there is a path along which the fly can walk that will enable it to get the food.

Crossing the River​

A farmer, a goat, a wolf, and a cabbage have to cross a river. A boat nearby only has enough room for the farmer and one other thing. What is the fewest number of trips he mus take so that the goat does not eat the cabbage, and so the wolf does not eat the goat?

Cutting a Cake​

Cut a round cake so that you get eight equal size pieces using exactly 3 straight cuts.

The Blue-Eyed Islanders​

There is an island upon which a tribe resides. The tribe consists of 1000 people, with eyes that are either blue or brown. Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, each resident can (and does) see the eye colors of all other residents, but has no way of discovering his or her own (there are no reflective surfaces). If a tribes person does discover his or her own eye color, then their religion compels them to commit ritual suicide at noon the following day in the village square for all to witness. All the tribes people are highly logical and devout, and they all know that each other is also highly logical and devout (and they all know that they all know that each other is highly logical and devout, and so forth).
For the purposes of this logic puzzle, "highly logical" means that any conclusion that can logically deduced from the information and observations available to an islander, will automatically be known to that islander.

Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics (each of them can of course only see 999 of the 1000 tribespeople).

One day, a blue-eyed foreigner visits to the island and wins the complete trust of the tribe. One the evening of his departure, he addresses the entire tribe to thank them for their hospitality. However, not knowing the customs, the foreigner makes the mistake of mentioning eye color in his address, remarking “how unusual it is to see another blue-eyed person like myself in this region of the world”.

What effect, if any, does this have on the tribe?
​

Pythagorean theorem: a^2 + b^2 = c^2

 

Is it something to do with the fact that she herself did not have to say who she shook hands with and therefor all other were able to say a different person? I can't think about it in detail I really have to go :3

 

Nope.

 

oh god I severely misread that. I thought it was to do with "How is it possible that none of them said they shook hands with the same person" Not sure why. Even though that still works if she herself had said who she shook hands with too.

 

Is it a specific value we're looking for? Or can it be he shook hands with *inert number*, *insert number* or *insert number* people.
I would tell you how far I think I've got in working it out, but I have absolutely no idea how to use spoilers.

 

"[spoiler']" Message "[/spoiler']" without quotations

 

The answer is a single integer.

 

Solution for the handshake problem:

 

Shock fucking nailed it. I'm honestly impressed, that was fast.

I assume you cheated. Want more?

Also if anyone is curious, here is a picture that makes this problem fairly easy to understand. Spoilers, yo:

Mr. and Mrs. Anbouba would be the top couple.

 

Do another Something that only I can get.

 

moar pls

 

The handshake problem is my personal favorite. I think I can find a few more though. Adding two now since one is very short and simple.

 

also, fast? that took me like an hour :l

 

It takes some people days believe it or not.

The gauss problem is kind of lame, but it is a classic.

 

Isn't Pythagoras' theorem a^2 + b^2 = c^2?

 

That's what it says, not sure what you are talking about. You can't prove shit.

 

Gauss Problem

 

Gauss Problem

A lot of problems resemble the gauss problem. It is just a good one for helping to notice patterns within numbers and what not.

Have fun with the fly problem.

 

Fly Problem (Incorrect)

 

I recommend not looking at my spoilers if you want to do it for yourself