More Tiles Inside than Around?

[Crlaozwyn]

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]

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*]

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]

Thanks - next time I see him, I'll give him a piece of my mind

[Quailman]

Was this a regular homework assignment or extra credit?

[Zag]

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]

[Quailman]

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]

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]

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." - Lifeinmomland

[Jack_Ian]

Is that configuration a closed loop? I dunno, topology was never my strong-point.

[Zag]

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]

[Coyote]

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]

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]

[Coyote]

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]

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]

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]

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

[Sentran]

[groza528]

I was talking to Quail