** STICKY ** League of Reason? PROVE YOURSELF!

[Pulsar]

Re: League of Reason? PROVE YOURSELF!

The doors which will be open at the end are those whose index has an odd number of divisors, and if memory serves the numbers with an odd number of divisors are the perfect squares. There are 31 perfect squares below 1000, so 31 doors.

Damnit, too easy. Well done!

 

[Marcus]

Re: League of Reason? PROVE YOURSELF!

you'll eliminate half each time you go through an evan number of people, and half a power of two is also a power of two, which in turn is even, so yes for any n wich is a power of two you want to be first.

It's a very similar line of reasoning to find that if there are 2^n - 1 in the circle, you want to be last. I'm still playing with the general problem when I get a few minutes.

 

[Pulsar]

Re: League of Reason? PROVE YOURSELF!

Here's one in a similar vein that I haven't actually sat down and solved yet myself. You and your friends (n people in total) stand in a circle, fix one person as the "starting point", and beginning from him you each take turns (going clockwise) counting up from one. The statring person says 1, the next person says 2 and so on. When someone says an even number, he immediately leaves the circle, and the last un-eliminated person wins. The problem is to figure out what "position" in the circle you want to be in to win the game.

I've found a solution, though I don't have an elegant proof why it works. Consider as an example the case n=18. Each column is a new iteration, where the remaining people keep counting:

01 19 28
02
03 20
04
05 21 29 33 35
06
07 22
08
09 23 30
10
11 24
12
13 25 31 34
14
15 26
16
17 27 32
18

So, the fifth person wins. But notice his successive numbers: 21,29,33,35.
35 - 33 = 2
33 - 29 = 4
29 - 21 = 8
21 - 05 = 16
The differences are powers of 2, from the highest power < 18 (i.e. 16) down to 2. This seems to work for all n: the person for which the difference between his current number and his following number is the highest possible power of 2, wins. And notice something else: the final number is 35 = 18+17. That's also true for all n: the last number is 2n-1. So from this we can work out the winning position:

If someone can prove it elegantly, please do!

 

[The Apathetic Despot]

Re: League of Reason? PROVE YOURSELF!

I've found a solution, though I don't have an elegant proof why it works. Consider as an example the case n=18. Each column is a new iteration, where the remaining people keep counting:

01 19 28
02
03 20
04
05 21 29 33 35
06
07 22
08
09 23 30
10
11 24
12
13 25 31 34
14
15 26
16
17 27 32
18

So, the fifth person wins. But notice his successive numbers: 21,29,33,35.
35 - 33 = 2
33 - 29 = 4
29 - 21 = 8
21 - 05 = 16
The differences are powers of 2, from the highest power < 18 (i.e. 16) down to 2. This seems to work for all n: the person for which the difference between his current number and his following number is the highest possible power of 2, wins. And notice something else: the final number is 35 = 18+17. That's also true for all n: the last number is 2n-1. So from this we can work out the winning position:

If someone can prove it elegantly, please do!

Facinating. I'll probably take a stab at proving it once exams are over, but it certainly looks good to me. Very well done indeed.

 

[Pulsar]

Re: League of Reason? PROVE YOURSELF!

The clue seems to be that for a given person, the differences between his successive numbers keep being halved. For instance, for person 13 in the example, his numbers are
13 25 31 34

25 - 13 = 12
31 - 25 = 6
34 - 31 = 3

And once the difference is an odd number, he loses. So, evidently the person for which the intial difference is a power of 2, wins. The trick is to prove this...

Good luck with your exams!

 

[ladiesman391]

Re: League of Reason? PROVE YOURSELF!

Can you figure this one out, if you haven't seen it before:

Three friends have a nice meal together, and the bill is $25

The three friends pay $10 each, which the waiter gives to the Cashier

The Cashier hands back $5 to the Waiter

But the Waiter can't split $5 three ways, so he gives the friends one dollar each and keeps 2 dollars as a tip.

They all paid $10 and got $1 back. $10-$1 = $9

There were three of them 3 X $9 = $27

If they paid $27 and the waiter kept $2:

Where did the other dollar go?

 

[Cluebot]

Re: League of Reason? PROVE YOURSELF!

Where did the other dollar go?

This apparent paradox comes from counting the $2 twice. After receiving their $3 change, the three diners have paid $27 in total - $25 to the restaurant and $2 to the waiter.

 

[ladiesman391]

Re: League of Reason? PROVE YOURSELF!

This apparent paradox comes from counting the $2 twice. After receiving their $3 change, the three diners have paid $27 in total - $25 to the restaurant and $2 to the waiter.

Thats it, it's all in the way the question is asked, I first attempted this question without paper or anything to write with, took me a while but I came to the correct conclusion.

 

[ladiesman391]

Re: League of Reason? PROVE YOURSELF!

How can I get the answer 24 by only using the numbers 8,8,3,3?

You can use the main signs add, subtract, multiply and divide (and brackets)....

 

[ladiesman391]

Re: League of Reason? PROVE YOURSELF!

This ones also good:

A man is stranded on an island covered in forest.

One day, when the wind is blowing from the west, lightning strikes the west end of the island and sets fire to the forest. The fire is very violent, burning everything in its path, and without intervention the fire will burn the whole island, killing the man in the process.

There are cliffs around the island, so he cannot jump off.

How can the man survive the fire? (There are no buckets or any other means to put out the fire)

 

[Cluebot]

Re: League of Reason? PROVE YOURSELF!

How can the man survive the fire? (There are no buckets or any other means to put out the fire)

Create a firebreak or refuge, either by clearing it of fuel mechanically or starting a second fire in the east that will burn out before the flames from the west arrive.

How can I get the answer 24 by only using the numbers 8,8,3,3?

You can use the main signs add, subtract, multiply and divide (and brackets)....

This has a trivial solution (8*3), unless we have to use all instances of the numbers.

 

[Ciraric]

Re: League of Reason? PROVE YOURSELF!

This ones also good:

A man is stranded on an island covered in forest.

One day, when the wind is blowing from the west, lightning strikes the west end of the island and sets fire to the forest. The fire is very violent, burning everything in its path, and without intervention the fire will burn the whole island, killing the man in the process.

There are cliffs around the island, so he cannot jump off.

How can the man survive the fire? (There are no buckets or any other means to put out the fire)

As was said, chop down trees so that the fire cannot come to his side of the island.

OR

Bury himself in the dirt until the fire burns itself out.

 

[ladiesman391]

Re: League of Reason? PROVE YOURSELF!

Create a firebreak or refuge, either by clearing it of fuel mechanically or starting a second fire in the east that will burn out before the flames from the west arrive.

That's correct.

This has a trivial solution (8*3), unless we have to use all instances of the numbers.

Sorry, yes you do have to include each number only once.

 

[ladiesman391]

Re: League of Reason? PROVE YOURSELF!

Here's a little algebra puzzle:

Solve the following:

C^L=LOGIC

 

[The Apathetic Despot]

Re: League of Reason? PROVE YOURSELF!

Good luck with your exams!

Thanks! though I'm sure you meant to say good skill :lol:

Here's a fairly easy one.

Suppose you were tiling a 2 meter by n meter path using 2*1 paving stones. All the stones are identical but you can place them so their long sides are either parallel or perpendicular to the path. So a few valid 2*3 paths would look like:

|||
|=
=|
etc.

If you can't have tiles going over the edge of the path and you can't cut the tiles, how many possible arrangements of tiles are there for an n meter long path?

 

[xman]

Re: League of Reason? PROVE YOURSELF!

I may be hangin' my ass out in such company as this, but ... 2?

||
-

 

[Fictionarious]

Re: League of Reason? PROVE YOURSELF!

How can I get the answer 24 by only using the numbers 8,8,3,3?

You can use the main signs add, subtract, multiply and divide (and brackets)....

I can get 24.3333, which, rounded down to whole numbers, is twenty four.

8*8=64
64/3=21.3333
21.3333+3=24.3333
24.3333 rounded down = 24.

I win. :geek:

 

[The Apathetic Despot]

Re: League of Reason? PROVE YOURSELF!

I may be hangin' my ass out in such company as this, but ... 2?

||
-

For n=2 (making a 2*2 path) there are two possible arrangements, if that's what you mean, but the idea behind the question is to find some method to determine the number for any value of n I might give you. So if n=4, then you have 5 possibilities:

|||| or
=|| or
|=| or
||= or
==
(I'm using | for one "vertical" stone and = for two "horizontal" stones side by side, in case that's not clear)

But what if n=10 or n=100 for example? A solution to this puzzle would be some way to calculate the answer for any n you want.

Edit: It's just occurred to me that I might not have been clear what I mean by an "arrangement". I basically mean how many different ways can you tile the entire path, not how many ways can you put down any given stone.

 

[xman]

Re: League of Reason? PROVE YOURSELF!

Well I've answered the wrong question and don't have the skills to answer the question you intended, but I do realise now that if n=1 there is only one way to tile the path. Who can enlighten me?

 

[ladiesman391]

Re: League of Reason? PROVE YOURSELF!

I can get 24.3333, which, rounded down to whole numbers, is twenty four.

8*8=64
64/3=21.3333
21.3333+3=24.3333
24.3333 rounded down = 24.

I win.

Almost, there's another solution that brings you much, much closer to 24, i'll give you a hint, only use / and - and ( ).