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 Forum index » Diversions » Perplex City Puzzle Cards » PXC: Black Puzzle Cards
[SOLVED] #217 Black Pixel Upper Left - Tower of Cubes
Moderators: AnthraX101, bagsbee, BrianEnigma, cassandra, Giskard, lhall, Mikeyj, myf, poozle, RobMagus, xnbomb
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Random_Hajile
Guest


 		I worked it out to be:

Spoiler (Rollover to View):
68,719,476,735 minutes

equates to:

1,131,104 years - 2 days - 20 hours - 15 minutes

And to think some people call me pedantic...



PostPosted: Wed Aug 10, 2005 8:29 pm
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skenmy
Veteran


Joined: 22 Aug 2005
Posts: 96
Location: Essex, England
Spoiler (Rollover to View):
68719476735


Confirmed. I have my points from this answer
_________________
PerplexCityTrades - Because the puzzle is only just beginning!
045/100

PostPosted: Mon Oct 10, 2005 3:05 pm
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TekkiBreki
Boot

Joined: 23 Apr 2006
Posts: 16
Location: Los Angeles

 		This is another of those "sum of the series" math problems where the simplest mode of attack is to list the first few elements and hope you see a pattern!

1 cube, 1 move
2 cubes, 3 moves
3 cubes, 7 moves
4 cubes, 15 moves

PostPosted: Sun May 07, 2006 2:53 am
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griph
Greenhorn

Joined: 12 Jun 2006
Posts: 3

 		For the curious, Wikipedia has a pretty canonical explanation of Towers of Hanoi.

This was my first black card, I was all syched up to be really stumped. But I majored in computer science, so this was like child's play. Time to go buy booster pack #2 Smile

PostPosted: Mon Jun 12, 2006 10:22 am
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e_nygma
Decorated

Joined: 17 May 2006
Posts: 247
Location: Maryland, US
griph wrote:
This was my first black card, I was all syched up to be really stumped. But I majored in computer science, so this was like child's play. Time to go buy booster pack #2 Smile


Griph, while this card was easy for you and I (being both CS majors), don't get cocky. Some cards, like the linguistic ones, I found extremely difficult but others found trivial. This is why UF exists, so we can pool our collective knowledge to advance everyone.

PostPosted: Mon Jun 12, 2006 2:55 pm
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griph
Greenhorn

Joined: 12 Jun 2006
Posts: 3
e_nygma wrote:
Griph, while this card was easy for you and I (being both CS majors), don't get cocky.


My apologies, I definitely wasn't trying to be cocky. I know it's often hard to convey disposition on Internet forums, guess I failed Smile I'll say that before solving this black card I spent probably a couple hours on a yellow that I'm sure a lot of people got really quickly. I am not a good puzzle solver, I was just expressing my delight at getting such a lucky break on my first black.

PostPosted: Mon Jun 12, 2006 4:41 pm
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e_nygma
Decorated

Joined: 17 May 2006
Posts: 247
Location: Maryland, US
griph wrote:
e_nygma wrote:
Griph, while this card was easy for you and I (being both CS majors), don't get cocky.


My apologies, I definitely wasn't trying to be cocky. I know it's often hard to convey disposition on Internet forums, guess I failed Smile I'll say that before solving this black card I spent probably a couple hours on a yellow that I'm sure a lot of people got really quickly. I am not a good puzzle solver, I was just expressing my delight at getting such a lucky break on my first black.


My apologies for the misinterpretation. Wink

(for what it's worth, Milo (a red for Pete's sake!) completely stumped me Embarassed ... I couldn't make heads or tails of it (sorry for the bad pun))

PostPosted: Mon Jun 12, 2006 6:15 pm
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Ifurita
Boot

Joined: 17 Apr 2006
Posts: 51
Location: California, USA

 		it's been very interesting to observe how people process different cards. There are blacks and silvers which I've breezed through only to get stumped on reds and oranges, while fellow puzzle solvers get stumped on those same silvers/blacks, but breeze through the ones I've been beating my head over.

PostPosted: Mon Jun 12, 2006 6:58 pm
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LilPickle-Amplizine
Boot

Joined: 13 Aug 2006
Posts: 17

 		LOL
LOL, I am a Cosc student as well! I was kinda even expecting this problem to show up. Maybe even a dining philosophers problem, too? (somehow)

Anyway, I thought I'd give my little rendition of why the answer is what it is.

Spoiler (Rollover to View):

This puzzle demonstrates a recurrence relation problem. But the program to simulate the Tower is surprisingly simple being only 5 lines of code. In the code, there are two recursive calls (the program calls itself twice, that perverted little bastard) and only one "real" bit of work.

We need to find the scale, or more properly "the order (big-O)," on which this program runs. The time (T) it takes is going to be dependent on the number (N) of cubes. Stay with me now, this is gonna get moving kinda fast.

Assuming T with a problem set of size 0 will also be 0, we can say:
T(0)=0

According to our program, we have 2 recursive calls with set size N-1, but there is also that 1 bit of real work it takes to move a cube. This translates mathematically to:
T(N)=2T(N-1) + 1

To make this a little easier later, we will also take a moment to define other values of T with different set sizes. These are found simply by substituting "N-1" for "N" in the last equation:
T(N-1)=2T(N-2) + 1
T(N-2)=2T(N-3) + 1

K, now for the dirty stuff. Square brackets represent a substitution:
#0: T(0)=0
#1: T(N)=2T(N-1) + 1
#2: =2[2T(N-2) + 1] + 1
#3: =2[2[2T(N-3) + 1] + 1] + 1
...yada yada yada...
#k: =2^k T(N-k) + SUM(i=0,k-1){2^i}

let that soak in a moment, go get another mountain dew, have some cheetos.... and we're back.
I'm sorry if that doesn't all make sense to you, but I can't really take the time to explain all the math. The purpose of that though was to generalize the order of time taken during any point in the process. You're doing good, we're almost there!

Let k=N, so we get
2^N T(0) + SUM(i=0,N-1){2^i}

T(N-N)=T(0) of course, and we earlier decided it also equals 0.
Then that summation happens to be a finite series (similar to #046 Sum Shortcut's Gauss thinger) and actually equals 2^(n+1) - 1, where n is the upperlimit of the summation (N-1 in our case).

One last reworking gives us our solution
T(N) = 2^(N-1+1) - 1 = 2^N - 1

So a problem set of N=36 cubes, we hopefully get 68719476735 minutes

Now go take a break, you deserve it Cool

Oh, and the big-O notation is O(2^N), wikipedia it if I haven't already scared you off.


PostPosted: Thu Aug 17, 2006 1:58 am
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The_Cube
Veteran


Joined: 07 Aug 2006
Posts: 126
Location: Sitting at my computer...DUH!!!

 		Website
could some go to http://www.brotherhoodofthesix.com/
And send them an email mine isn't working.........Please

PostPosted: Sat Sep 02, 2006 6:29 am
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