Brain teaser thread

[Dauphin]

Originally posted by Dry

A prince conquers a territory, and decides gives one of his loyal nobles a piece of it. But the noble must trace out the boundary of the estate with his horse within a day. The horse's speed starts out at 10 km/h, but the rate of decrease of its speed is proportional to its speed with a proportionality constant of .5 /h. What's the maximum area he can get from his Prince? Prove it

If you mean that after 20h the horse is exhausted (constant decrease of speed), then the horse is able to run (20*10)/2 = 100km.
The cercle being the geometrical form with maximum surface for a given perimeter:
perim = 2*pi*r = 100
surf = pi*r*r = 2500/pi

I think he means:

a = -0.5v

Therefore

s = -20 x exp(-.5t) + 20.

At t = 24 (has to be completed in a day)
s = 19.9999 km -> r = 20/2pi
area = 100/pi km².

The maximum area is either that, or the area conquered by the Prince. Whichever is larger.

[Urban Ranger]

Originally posted by rah
The funny thing was, that I figured it would have to happen on the even number side of the street and it would be somewhere between 98 and 120, and still didn't see it when it was staring at me. So I went ahead and calculated it for the odd numbers.

There's only one word for it: brain-fart

[Ramo]

I'm very disappointed. I was expecting a calculus of variations proof for why the object with the highest area for a given perimeter the circle.

Nah, I no longer feel sadistic.

[Urban Ranger]

Any more interests?

[The Vagabond]

A group of four persons (A, B, C and D) needs to cross a bridge. It takes person A 1 min to cross the bridge, person B -- 2 min, person C -- 4 min, person D -- 5 min. One of the difficulties they face is that the bridge is too shabby and narrow, so that maximum two persons can be crossing it at a time. Another problem is that it happens at night, and they need a lantern. But there is only one lantern for the whole group. Therefore, while necessary, someone will have to keep returning back in order to bring the lantern to the remainder of the group.

Question: How should they organize the crossing to do it as fast as possible? How much time will it take them?

Note: When two persons cross the bridge, it is implied that they travel at the speed of the slowest person, of course.

[Dry]

A and B cross (2 min)
A comes back (1 min)
C and D cross (5 min)
B comes back (2 min)
A and B cross (2 min)
Total: 12 min.

[The Vagabond]

That's correct, Dry

Edit: Just wanted to add that people often mistakenly come up with the answer 13 min for that problem.

[Dauphin]

Decode this message:

Yjod djpi;f nr movr smf rsdu. @F

[Paul Hanson]

This should be nice and easy, lD

[Dauphin]

Where is the @ on your keyboard.

[Immortal Wombat]

to the right of the | I would guess.

[Dry]

Originally posted by The Vagabond
That's correct, Dry

Edit: Just wanted to add that people often mistakenly come up with the answer 13 min for that problem.

13 was indeed my first answer.
I then thought:
mmh, that seems too simple, there's a trap...
maybe the 2 slowest should cross together.
The rest was just finding out how.

[JohnM2433]

*bump*

Isn't anyone going to post another one?

I loved this thread! I don't want it to die! Noooo.....

[The Vagabond]

A car smashes into the tree. The driver gets out, looks around, and says: "How great it's halved! Otherwise I'd be dead now".

Question: What did he mean?


Disclaimer: Don't take it too seriously. It's half joke.

[Guest]

Just wanted to say that JohnM is right about the dividing of the land beetween sons.
Five is impossible, four is trivial.
The four Colour theorem, proven rather recently (in mathematical terms), states that a planar map (thats more precise than just 2D since the surface of the Earth is 2D for example) can never need more than 4 colours to color as in loinburgers question (meaning no same colour touching, not including corners etc) and was the first proof ever in mathematical history that was not checkable by hand (they used computers to do a divide and conquer method, meaning breaking all the possible cases in different categories which could each be solved) and thus is sorta classical and often referred to.