# The Grey Labyrinth is a collection of puzzles, riddles, mind games, paradoxes and other intellectually challenging diversions. Related topics: puzzle games, logic puzzles, lateral thinking puzzles, philosophy, mind benders, brain teasers, word problems, conundrums, 3d puzzles, spatial reasoning, intelligence tests, mathematical diversions, paradoxes, physics problems, reasoning, math, science.

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Crlaozwyn
Icarian Member

 Posted: Fri Apr 08, 2011 5:15 pm    Post subject: 1 A friend presented me with this puzzle, but I've been unable to solve it. It's driving me crazy, so please help me out! You have a 6x6 grid. How can you draw a closed loop that contains more tiles inside it than around it? All my solutions come up with as many tiles around as inside. For example, take the following incorrect solutions: (~ are "unused" tiles, + are the tiles used to make the loop, and = are tiles inside the loop) ~++++~ +====+ +====+ +====+ +====+ ~++++~ So there are 16 tiles used to draw the loop, but only 16 inside. The other solution I came up with also uses as many tiles to draw the loop as are inside: ~~++~~ ~+==+~ +====+ +====+ ~+==+~ ~~++~~ Now there are 12 tiles used but 12 tiles inside. Is my friend pulling my leg? Thanks in advance if you can help!
Jack_Ian
Big Endian

 Posted: Fri Apr 08, 2011 11:58 pm    Post subject: 2 Draw the following on a ball and declare O to be outside. X X X X X X X X + X X X X + O + X X X X + X X X X X X X X X X X X X X X
Lepton*
Guest

 Posted: Sat Apr 09, 2011 2:45 am    Post subject: 3 I think it is impossible. The first solution seems pretty likely to be the maximum. Incidentally, a 6x6 grid is the largest square grid where the task is impossible. Using the technique from the first example, you could enclose 25 squares in 20 blocks on a 7x7 grid. The formula is x^2 blocks in 4x blocks on an (x+2) grid. For x =< 4, the enclosed area is equal or smaller than the loop. There could yet be a trick, but I think it would be dissatisfying to this interpretation of the puzzle.
Crlaozwyn
Icarian Member

 Posted: Sat Apr 09, 2011 5:41 am    Post subject: 4 Thanks - next time I see him, I'll give him a piece of my mind
Quailman
His Postmajesty

 Posted: Sat Apr 09, 2011 1:42 pm    Post subject: 5 Was this a regular homework assignment or extra credit?
Zag
Tired of his old title

 Posted: Sat Apr 09, 2011 2:44 pm    Post subject: 6 You're so suspicious, Q. What course would this be homework for? It seems more like a genuine puzzle. I rather like Jack_Ian's solution, even if it is derived from the joke about the accountant, the engineer, and the mathematician. I had the same conclusion as Lepton, but wasn't quite confidant enough to post it.
Crlaozwyn
Icarian Member

Posted: Sun Apr 10, 2011 5:56 am    Post subject: 7

My mommy doesn't let me talk to strangers.

I've been out of school for quite a few years, but the vote of confidence is overwhelming
Quailman
His Postmajesty

 Posted: Sun Apr 10, 2011 2:26 pm    Post subject: 8 Sorry. We occasionally get new members who register only to post a difficult homework problem. With minimal information in your profile, I became suspicious.
Jack_Ian
Big Endian

 Posted: Sun Apr 10, 2011 10:40 pm    Post subject: 9 Hmmm... So only possible for a grid of size X, Y where where X & Y > 1 and X + Y - XY/4 - 3 < 0 Which rules out 6x6, since 6+6-9-3=0 but a 5x9 should work and needs 4 tiles fewer than a 7x7
Sentran
Ray of Sucking Funshine

 Posted: Mon Apr 11, 2011 5:22 pm    Post subject: 10 I actually figured out a valid answer to this based on Jack_Ian's comment about the ball. Grid a piece of paper 6x6, then draw the border across the top and bottom and 1 side. Then roll the paper so that the bordered edge is touching the non-bordered edge. This would leave 16 squares as a border around 20 squares._________________Sentran "Speaking of double negatives, I haven't read greylab yet today." - LifeinmomlandLast edited by Sentran on Mon Apr 11, 2011 9:06 pm; edited 1 time in total
Jack_Ian
Big Endian

 Posted: Mon Apr 11, 2011 8:16 pm    Post subject: 11 Is that configuration a closed loop? I dunno, topology was never my strong-point.
Zag
Tired of his old title

 Posted: Mon Apr 11, 2011 8:38 pm    Post subject: 12 Brilliant, Sentran. That's certainly the answer that the puzzle was looking for. You said to fold it, but I assume you meant to roll it into a cylinder.
Sentran
Ray of Sucking Funshine

Posted: Mon Apr 11, 2011 9:06 pm    Post subject: 13

 Zag wrote: Brilliant, Sentran. That's certainly the answer that the puzzle was looking for. You said to fold it, but I assume you meant to roll it into a cylinder.

Correct, I meant roll. I corrected my previous post.
_________________
Sentran
"Speaking of double negatives, I haven't read greylab yet today." - Lifeinmomland
Coyote

 Posted: Mon Apr 11, 2011 9:54 pm    Post subject: 14 Since diagonal connections are allowed, you really only need to use 14 squares. But why stop at a cylinder? If you connect the top and bottom edges too, you can use ten squares to loop around 25 squares. I'll admit that at this point, the definition of 'loop' is getting a little loopy.
groza528
No Place Like Home

 Posted: Tue Apr 12, 2011 7:27 pm    Post subject: 15 Sentran - I believe it would actually be 18 tiles enclosing 20, but that's academic. Quail - That's how I started! And 11+ years later I'm still here.
Sentran
Ray of Sucking Funshine

Posted: Tue Apr 12, 2011 10:17 pm    Post subject: 16

 groza528 wrote: Sentran - I believe it would actually be 18 tiles enclosing 20, but that's academic.

++++++
+=====
+=====
+=====
+=====
++++++

Coyote, a cylinder is far easier to deal with on paper than folding it into an oddly-shaped ball. If you omit the corner connections, you don't have a clean single loop.
_________________
Sentran
"Speaking of double negatives, I haven't read greylab yet today." - Lifeinmomland
Coyote

 Posted: Tue Apr 12, 2011 11:06 pm    Post subject: 17 I went to Krispy Kreme and tried to order a dozen oddly-shaped balls, but unfortunately all they had were donuts. I'd be careful about gauging the validity of a solution on how easy it is to manage, After all, as easy as rolling the paper into a cylinder may be, it's even easier to just leave the paper lying on the table, which would invalidate your solution. Just be grateful I didn't go with a projective plane. As for the cleanliness of the loop--I was going by the definition in the opening post. You'll note that both the given examples shown there rely on diagonal connections. Posts #3 and #9 follow this definition as well. In any case, whether diagonal connections are allowed or not, I think both your solution and mine require a little laxness in the definition of 'loop'. Not to mention the definitions of 'inside' and 'outside'.
groza528
No Place Like Home

 Posted: Wed Apr 13, 2011 12:18 am    Post subject: 18 Right, I know where I made my mistake. I started with the 16 enclosing 16 model and added 1 to top and bottom without subtracting four for the edge. That being said, you could improve it to 14 by cutting the corners, if diagonals still constitute a closed loop.
Quailman
His Postmajesty

 Posted: Wed Apr 13, 2011 12:44 am    Post subject: 19 Sentran - for your cylinder solution, why not just put a border along the top and bottom. When you roll it into a cylinder the touching edge disappears.
groza528
No Place Like Home

 Posted: Wed Apr 13, 2011 4:07 am    Post subject: 20 It's difficult to argue that as a closed loop though... Looks like two closed loops to me.
Sentran
Ray of Sucking Funshine

Posted: Wed Apr 13, 2011 4:20 am    Post subject: 21

 groza528 wrote: It's difficult to argue that as a closed loop though... Looks like two closed loops to me.

That would be true if I had not included the line down the side. That turns it into a closed circuit.
_________________
Sentran
"Speaking of double negatives, I haven't read greylab yet today." - Lifeinmomland
groza528
No Place Like Home

 Posted: Thu Apr 14, 2011 6:16 pm    Post subject: 22 I was talking to Quail
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