On Friday, January 24, 2014 at 1:49:07 PM UTC-5,
alex.k...@gmail.com wrote:
Hey gents,
If anyone is still reading this thread, I would like to add some input. I recently found the Titan Test online and found it to be a great mental challenge. Now, I agree with most of the answers provided for both the verbal analogies and the spatial/
quantitative problems. However, I have some feedback that may(?) be helpful if anyone is still interested.
19. Cosmonomologo- ? I have never heard this before. I haven't even seen it, despite some astrophysics at the undergraduate level. If this is a true prefix, I'm impressed you guys found it.
20. I know initially you said some suggested Kessler as an answer, but seeing as how the "set of sets not members of themselves" or the naive set is called Russell's paradox, I think Olbers indeed fits better here as the "darkness of the night sky"
phenomenon is typically referred to as the "Olbers Parabox" and not the Kessler paradox.
24. Similarly, I believe the -swain on 'boatswain' is more a suprafix than stem. Suprafix a linguistic term referring to an affix that, in so many words, effects a tonal or stress change on a given word. While 'stem' and 'base' both fit a sort of sing-
song linguistic pattern sought in many of the "correct" (possibly) answers to other analogies, I believe that 'suprafix' fits both a definition-based and linguistic pattern better than the other two propositions.
25. I have to agree with the initial 9-square construction. Though I like the minimalist solutions posited by j.poll above, the shapes you used to construct the solution were not all squares. I have to concede to 9. I got the same answer when doing
this one independently, and though the most common answer is not always the most correct, I took the instructions literally and only used squares for my answers.
33. I found this to be the most interesting, and, subsequently, most difficult problem on the test. I had to do some really mind-stretching geometric thinking to wrap my brain around the concept of first excising the Mobius Strip, then the concept of
the two pieces (the Mobius Strip and the two-sided orientable surface left behind), then the concept of the pieces being unmoved, so that subsequent cuts sliced both pieces, despite only one yielding subsequent Mobius Strips.
Firstly, the first cut can only yield two pieces, and, interestingly enough, one is a one-sided surface, the Mobius Strip, and the other is an orientable loop. However, the second cut gets more interesting.
In order to excise another Mobius Strip, you either have to cut the same pattern into the torus, shifted laterally to provide enough room for the cut, or you have to cut the interior of the first Mobius strip in two 360 degree, circumferential motions
to excise a similar Mobius from the first. I opted for the second method of slicing for two reasons: (1) it maximizes the number of pieces from excision (and we are looking for a maximum here) and (2) while the first method cuts a second Mobius pattern
in the same way as the first cut, it doesn't actually yield a Mobius Strip because the excision of the first strip gets in the way, so to speak, interfering with a whole remaining Mobius Strip.
I then decided to 'unroll' the Torus, looking at it as a topologist would, as a solid cylinder, rather than a donut-shaped solid surface. For me, it was just easier that way. When you do this, you would see that the first method of cutting would be
possible, yielding another Mobius strip, but ultimately, after three cuts, one would only have four whole pieces, which, intuitively, seems low for this exercise.
Keeping the torus 'unrolled,' you can see that the second method of slicing the torus (on the second and third slices, at least) would render the knife perpendicular to the singular side of the Mobius on both circumferential motions. I say motions here,
because even though to cut in this manner, you are traversing the arc of the torus twice, you are doing it in one smooth cut, never tracing over the path of your previous traverse in the second or third cuts. This method cuts the interior Mobius Strip
of the first cut into two remaining surfaces, again another Mobius Strip, and an orientable loop, much like the remnant surface of the first cut.
However, because the pieces are never moved from their original positions, the double-circumferential second cut slices this surface along the way. As an aside, this is all much easier to visualize as a solid cylinder with the appropriate equivalences
on the edges of the cylinder, rather than as an intact torus. As a result of the second cut using my second method, there are now 8 subdivided pieces instead of 4, so it is, indeed, a maximum producing cut. The pieces are, using the first cut as an 'axis'
for orientation, the six subdivided pieces of the coiled orientable complement of the Mobius rendered in the first cut, (three 'above and 'below' the Mobius---again, this is easier to view in an un-coiled cylinder), the new Mobius Strip cut from the
interior of the extant Mobius Strip, and the complement orientable surface (8 total after two cuts).
Now, the third cut will follow the pattern of the second, in terms of excision via a double-traversed circumferential arc. It will however, excise the middle of the Mobius Strip rendered by the second cut. This will render two additional pieces from
the central Mobius of the second cut, and will render two of the three axial cuts from the first orientable two-sided surface, into three pieces. This leave, the new (third) Mobius, the new (third) orientable complement, and the original Mobius strip,
which has been quadruply-traversed to yield two new Mobius Strips. The 'axial' orientations withstanding, 'above' and 'below' the original Mobius, there are five remaining on each axial orientation: one initial 'slice' leaving one piece after the first
cut, the second cut leaving six, after the double-traverse rendered this single slice into 'thirds', and another double-traverse slicing the middle 'third' into three more pieces. So on each axial orientation, there are five remaining pieces, i.e. ten
altogether. These ten, coupled with the three progressively sliced Mobii (?) leaves 13.
Now, the real challenge is assessing the equivalence of my method of slicing with the question: to rehash the question's key challenge "a knife that each time precisely follows the path of such a Möbius strip. What is the maximum number of pieces that
can result if the pieces are never moved from their original positions"
The nuts and bolts are: is the knife required to make the same cutting pattern each time, in which case an oriented shift to 'offset' subsequent Mobius Strips would simply maximize the pieces yielded from their intersections. If we 'unroll' the torus,
keeping in mind the equivalence relation on the edges of our new cylinder, then we can see that what I present as the second cutting method above, is actually equivalent to the knife pattern of the first cut, it just has a perpendicular orientation to
the first cut, ie a normal orientation to the surface of the initial Mobius strip! If we focus on the Mobius and its effect on the torus as a whole, we lose the real meat and potatoes of the subsequent slices. We have to start, visually, from after the
first cut, which is where things got really interesting.
I understand this is a discussion forum and that I may be entirely off-base or wrong altogether. I did, however, get the number 13, which is interesting in and of itself. So, in the habit of good-spirited academic collaboration, discuss away!
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