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 Post subject: 3D Brainteaser / geometric oddity
PostPosted: Fri Nov 04, 2011 9:53 am 
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Joined: Thu Oct 27, 2011 4:39 pm
Here is a new geometric brainteaser puzzle to challenge your spatial reasoning and visualization skills. This brainteaser pertains to a 3D puzzle that physically exits, so this is a puzzle within a puzzle. First some facts about the physical puzzle are in order:

A new type of ball-in-maze puzzle has been invented. It is a 3D maze in the shape of a cube. The maze can be viewed as being divided up evenly into smaller cube shaped cells. The particular maze for this discussion is called “Zig Zag Zog” and it is 5 cells wide, high and deep. The maze paths are formed by placing solid rods along the edges of selected cells of the maze. This creates a three dimensional lattice structure used to confine the ball to the paths while it rolls through the maze.

The ball has a diameter equal to the twice the width of a cell. So when the ball is sitting in the entrance of the maze it is taking up a volume of 8 cells ( 2 cells wide by 2 high by 2 deep).

The shortest amount the ball can move along any path segment is the width of a cell. The path lengths are measured in units that are the width of a cell.

The cross section of the pathway that the ball travels through is 2 cells high by 2 wide, or 4 units. The volume of a section of the path is the cross section multiplied by the path length. So if the ball moves along a section of path 3 units long, the volume of this path segment added is 3 (length) x 4 ( cross section area) = 12 cubic cell units of volume.
To get the total volume of the pathways through the maze we start with the volume taken up by the cells the ball occupies when sitting in the entrance (8) and add to that the path length multiplied by the cross section. Path Volume = 8 + 4 x Path length.

The “Zig Zag Zog” maze has a path length of 37, this includes all dead end branches. The volume of this path is: 8 + (4 x 37) = 156.

The total volume of the maze is obtained by multiplying the width by the height by the depth, in cell units. For the “Zig Zag Zog” maze the total volume is 5 x 5 x 5 = 125.

The brainteaser here is this: How can the maze have a path with a volume that is larger than the total volume of the cube that the maze fits into. Somehow a path with a volume of 156 is crammed into a cube maze that has a volume of only 125. Hmmm…

The actual “Zig Zag Zog” maze puzzle can be viewed using the link below: ... id=sg21543

To the right of the image you will see an icon for a camera and one for a 3D box. Click on the 3D box icon and a 3D image will be shown. You can drag your mouse on it to rotate the maze in 3 dimensions. You can even obtain a physical model of this maze puzzle here at Shapeways.

Post your solutions and thoughts on this geometric oddity brainteaser here. I will select the first correct answer posted and bestow upon them here the title “Spatial Reasoning Wizard”.


:D Congratulations Taus:

Your solution to this brainteaser puzzles is correct, the path that the ball travels from the entrance to the exit does overlap with itself. In fact it overlaps many times in its convoluted path.

As correctly pointed out by Taus, if we add up the volumes of all the cells that are part of the path, these can’t exceed the volume of the volume of the maze itself. What can be done though is to allow small chunks of volume from one path segment to overlap with that of another part of the path. This “reuses” small volumes in the maze. If we count all the volume that the ball passes through as it traverses the paths, even if it is reused, that is when we get the larger volume figure. The fun part about this is that it allows a longer path to be created through a maze of a given size. This overlapping also makes the maze more challenging and visually intriguing. This overlapping creates discontinuities in the edges that form the path boundaries, so you can’t simply follow the edges to solve this type of maze, as is commonly done to solve two dimensional mazes.

I hereby bestow upon Taus the title “Spatial Reasoning Wizard” for being the first to post the correct answer to this brainteaser.

Best Regards

Steve Winter
Inventor of Ethereal Maze Puzzles - A new type of 3D rolling ball maze puzzle
Available at my Ethereal Maze Puzzles Shapeways shop

Last edited by Steve Winter on Sun Nov 06, 2011 11:53 am, edited 1 time in total.

 Post subject: Re: 3D Brainteaser / geometric oddity
PostPosted: Fri Nov 04, 2011 10:27 am 
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Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
When I was a kid I used to build "ball mazes" very similar to these out of Legos. I would build the maze up as a series of 2D mazes on each level. Each level would have a hole to the level below it so when the steel ball would drop from one level to the next you could hear and feel it. Of course, with Legos it was a solid structure and you couldn't see your progress through the maze so it was a non-visual tactile and sound feedback puzzle rather than the 3D puzzle you've built here.

As I got older I built an elevator mechanism with a column that had a groove in it that the ball could fit in and the column was in one of the corners and could slide up and down freely gaining access to all of the levels. Then instead of being a linear sequence of 2D mazes I was able to make the 3D maze involve going up and down between floors to solve it. It was quite hard and quite fun.

Prior to using my real name I posted under the account named bmenrigh.

 Post subject: Re: 3D Brainteaser / geometric oddity
PostPosted: Fri Nov 04, 2011 10:54 am 
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Joined: Wed Jun 22, 2011 4:57 pm
Location: The land of dreams, coincedentally located in Alberta
Me and my little brother have done a bit of experimenting with Lego puzzles. We made an enclosed maze as well, but it wasn't complicated at all. He also made a neat 3-step puzzle box where the goal is to open a drawer :)

3x3x3 PB: 38.9 seconds
Well, I accumulated puzzles without even trying this Christmas. Whoops.
(Bermuda 8 planets, Rex Cube, Master Skewb, London Natural History Museum keychain 2x2x2, Impossiball)

 Post subject: Re: 3D Brainteaser / geometric oddity
PostPosted: Fri Nov 04, 2011 8:16 pm 
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Joined: Sat Jan 01, 2011 12:54 pm
Location: Copenhagen, Denmark
The path can overlap itself. Consider two 2x2xn boxes that overlap on a single 1x1xn box. In other words, something that, when viewed from one end, looks something like this (# denoting box and . denoting empty space):


The ball can travel through both of these 2x2xn boxes (assuming a suitable U-turn at the end), hence the volume of the path is seemingly larger than the volume of the maze, but really all that's happening is that we are miscalculating the volume of the path.

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