Same or different paths?

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Dave Davison
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Same or different paths?

Post by Dave Davison »

Question: Part of this maze has been masked by a piece of card with a square cut out, the question is can Mr A meet Mr B without crossing any hedge? In other words, are they both on the same path? There is a simple theorum for this cunundrum which if known, makes the answer easy. [Hope this one does not spoil your day Al!] Cheers Dave.
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John Gray
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Re: Same or different paths?

Post by John Gray »

It seems pretty easy. Mr A just goes east along the grassy path until he reaches the yellow (sandy?) path, walks south along it to the bottom, then west for a short distance, then north up the grassy path again back to the hedged area to meet Mr B (who we assume hasn't sneaked off somewhere sneakily, to make things difficult).

Next problem/conundrum?
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HansV
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Re: Same of different paths?

Post by HansV »

Apart from John Gray's wittiness, there's no way to know. Perhaps the maze looks like this:
P1.jpg
in which case mr A and mr B cannot meet. But perhaps it's like this:
P2.jpg
and they can meet easily...
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Best wishes,
Hans

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Jezza
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Re: Same of different paths?

Post by Jezza »

Can I assume that this is a 3D puzzle and I can fold the maze?
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pi-eater
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Re: Same or different paths?

Post by pi-eater »

I think that there are several assumptions about the maze that we need to make that are not given. Is the maze made of nothing but circular arcs as shown? Are there no breaks in the lines? I'm going to go on that assumption and guess that since there are an even number of lines between A and B, then it MAY be possible. An odd number would not have been.

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Dave Davison
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Re: Same or different paths?

Post by Dave Davison »

Hello chaps...I think I must chip in and clarify things a bit. I have tried to locate the original question on the web where I first saw it so I could ask the correct question but SORRY, can't remember which site it was and I fear I may have misled you all by not asking quite the right question by suggesting it is a kind of maze. I paste below the unmasked puzzle and the answer given which I'm sure most of you will grasp immediately but for me not so clear without some perusal. Apolgies!!!!! :heavy: Cheers Dave.

Region B is inside the curve. This can be said because of an interesting theorem about simple closed curves. All "inside" regions of such a curve are separated from each other by an even number of lines. The same is true of all "outside" regions. And any inside region is separated from any outside region by an odd number of lines. Zero is considered an even number, so if there are no lines between two regions, then of course they will be part of the same "side," and our theorem still holds. When we pass from any part of region A to any part of region B, along any path, we cross an even number of lines. In the illustration one such path is shown by the dotted line. As it can be seen the line crosses four lines, an even number. So we can say with certainty that no matter what the rest of this curve looks like, region B is also inside.
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pi-eater
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Re: Same or different paths?

Post by pi-eater »

I just thought about it as if I had the entire maze imported into a paint program. If I use the fill tool and start it filling at A and then B also ends up getting filled with the same color, it follows that you can get from A to B. Just a little thought shows that joined locations will have even numbers of crossings. But that wouldn't be true if there were breaks in the walls as the paint would leak through.