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# Hoeflin's Object II.

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Hoeflin's Object II. - Back to the Geometry Puzzles

The five figures shown below represent the appearance of a solid, opaque object as seen from five of its six sides. Each line shown depicts a side of the object that is perpendicular to the plane of this page. The object was constructed by gluing together a number of identical cubes so that at least one face of each added cube precisely and entirely covers and is everywhere contiguous with one face of a previous cube. Draw the sixth view of the object.

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Hoeflin's Object II. - solution

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• 1 month later...

I came up with this solution:

hoeflin2-solution2.GIF[/attachment:fe3d7]

There are 12 blocks:

Assuming the first picture is the "top view",

Block positions at...

* top level: (1,2), (2,1), (2,3), (3,2) [a plus without a center]

* middle level: (1,2), (2,1), (2,2), (2,3), (3,2) [a plus with a center]

* bottom level: (1,1), (1,2), (2,1)

How can your solution have the center block at it's own level???

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• 1 month later...

I agree with Paul (except my answer is flipped horizontally)

I used Blender (a 3D model/render program) to make the model and I checked it from each angle and it was undeniably correct.

I also checked again and can't see a possibility where the center square could be on a third layer... it would have to be all the way at the end making the opposite view a 'plus' with a center...

So yeah, the answer is actually wrong... and now I feel like a geek.

And because I'm keen I'll add the picture

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I might have forgotten to delete those lines in my solution ... I did this one many years ago and had the pictures drawn on a few papers with pencil (of course, with a few strikethroughs )

Edit: I have just found my old dusty paper with the drawn solution and it is indeed as you have written ... my bad

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No problem. It's always more fun when the answer is wrong =P

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• 1 month later...
The five figures shown below represent the appearance of a solid, opaque object as seen from five of its six sides. Each line shown depicts a side of the object that is perpendicular to the plane of this page. The object was constructed by gluing together a number of identical cubes so that at least one face of each added cube precisely and entirely covers and is everywhere contiguous with one face of a previous cube. Draw the sixth view of the object.

I'd almost argue both are correct, as based only on the five views given either solution is correct (it is impossible to determine if the center is empty or not). However, I interpret "the object was constructed by gluing together a number of identical cubes so that at least one face... is everywhere contiguous with one face of a previous cube" to mean that the cube must be a single solid body (i.e. won't fall apart). The originally posted answer with no center cube would be correct without this requirement (but would fall apart pretty easy) but is not correct with my interpretation as it is a conglomerate of 3 unconnected (at least not by an entire face) solid bodies.

See the image below. Note for the orthogonal views that the lines of the original 5 are the same. Nice catch Paul.

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• 1 year later...

Well it looks like this solution has been entirely well-explained. But because I spent the last hour solving my own version of the solution in Excel before I read this topic, you are going to see it anyways.

Note the axis marker.

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• 1 year later...

I solved this without any maths or engineering. If the object is a solid hexahedron then the silhouette of any side will be a mirror image of its opposite side. Thus figures 2 & 4 and 3 & 5 in the diagram are mirror images so figure 6 will be a mirror of figure 1.

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