Hey gents,quantitative problems. However, I have some feedback that may(?) be helpful if anyone is still interested.
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/
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.phenomenon is typically referred to as the "Olbers Parabox" and not the Kessler paradox.
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"
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 doingthis 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 ofthe 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.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 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
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 bepossible, 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
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 equivalenceson 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'
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 fromthe 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,
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 MÃ¶bius strip. What is the maximum number of pieces thatcan 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
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!
If you spot an error or would like to add answers, please reply to thisDoes anyone know of a similar conversation about the Langdon Adult Intelligence Test? I came across it in an old magazine and took it, but it is no longer scored.
post or send me an e-mail, or both.
Don't forget to explain your answer, the more detail the better.
The test questions can be found here: http://www.eskimo.com/~miyaguch/titan.html
1. STRIP : MoBIUS :: BOTTLE : KLEIN
2. THOUGHT : ACTION :: OBSESSIVE : COMPULSIVE
3. LACKING MONEY : PENURIOUS :: DOTING ON ONE'S WIFE : UXORIOUS
(definitions)
4. MICE : MEN :: CABBAGES : KINGS (from literature)
5. TIRE : RETREAD :: PARCHMENT : PALIMPSEST
6. ALL IS ONE : MONISM :: ALL IS SELF : SOLIPSISM (a doctrine in
philosophy, nothing exists except the self, or alternately the
existence of everything else depends on the existence of the self)
7. SWORD : DAMOCLES :: BED : PROCRUSTES
8. THING : DANGEROUS :: SPRING : PIERIAN
(A little learning is a dangerous thing; drink deep, or taste not the
Pierian spring - Alexander Pope. An
Essay on Criticism)
9. HOLLOW VICTORY : PYRRHIC :: HOLLOW VILLAGE : Potemkin
10. PILLAR : OBELISK :: MONSTER : BASILISK (OBELISK from Greek
obeliskos, BASILISK from Greek basiliskos)
11. 4 : HAND :: 9 : SPAN (a hand is 4 inches, a span is 9 inches)
12. GOLD : MALLEABLE :: CHALK : FRIABLE (rhyming pattern)
13. EASY JOB : SINECURE :: GUIDING LIGHT : CYNOSURE (the rhyming
pattern is the key here)
14. LEG : AMBULATE :: ARM : BRACHIATE
15. MOSQUITO : MALARIA :: CANNIBALISM : KURU (a disease limited to the
Fore tribe of New Guinea, who eat human brain in their religious
rituals)
16. HEAR : SEE :: TEMPORAL : OCCIPITAL (Lobes of the brain that deal
with hearing and sight.
17. ASTRONOMY AND PHYSICS : ASTROPHYSICS :: HISTORY AND STATISTICS : CLIOMETRICS (Cliometrics is the use of statistics or economics in
historical studies.)
18. JEKYLL : HYDE :: ELOI : MORLOCKS (from the books by Robert Louis Stevenson and H.G. Wells)
19. UNIVERSE : COSMO- :: UNIVERSAL LAWS : COSMONOMOLOGO-
20. SET OF SETS NOT MEMBERS OF THEMSELVES : RUSSELL :: DARKNESS OF THE
NIGHT SKY IN AN INFINITE UNIVERSE : OLBERS
(KEPLER has also been offered as an answer)
21. TEACHING : UPLIFTING :: PEDAGOGIC : ANAGOGIC (from Greek
for spiritual uplift)
22. LANGUAGE GAMES : LUDWIG :: PIANO CONCERTI FOR THE LEFT HAND : PAUL
(So obscure it is silly. Ludwig and Paul Wittgenstein were brothers.
One was a pianist, the other was the famous philosopher who used the
phrase "language games" in his book Philosophical Investigations.
Knowing this meaningless bit of trivia can hardly be an indication of
great intelligence.)
23. IDOLS : TWILIGHT :: MORALS : GENEALOGY (Titles of Nietzsche's
books)
24. SWEET*NESS* : SUFFIX :: BOAT*SWAIN* : STEM ("ness" is a suffix,
"swain" is the stem of the compound word "boatswain."
Alternate answer is BASE, which fits the rhyming pattern better
29. The answer is 20.
31. The answer is 22
(this answer may be wrong)
37.
Initially, we need to consider 11 possibilities where we choose 10
white marbles.
1) After the insertion, the box contains 10 whites. The chances of
this are 1 in 2^10, i.e. 1 in 1024. If this is the case, in our
subsequent drawing we're bound to pick out 10 whites. Total probability
of this possibility = 1/1024
2) After the insertion, the box contains 9 whites. The chances of this
are 10!/9!/1! = 10 in 1024. If this is the case, in our subsequent
drawing our chances of picking out 10 whites is (9/10)^10 because we
have a 9/10 chance each time. Total probability of this possibility =
10/1024 * (9/10)^10
3) After the insertion, the box contains 8 whites. The chances of this
are 10!/8!/2! = 45 in 1024. If this is the case, in our subsequent
drawing our chances of picking out 10 whites is (8/10)^10 because we
have a 8/10 chance each time. Total probability of this possibility =
45/1024 * (8/10)^10
4) Box = 7 whites. Total = 120/1024 * (7/10)^10 i.e. nCr * (r/10)^10
5) Box = 6 whites. Total = 210/1024 * (6/10)^10
6) Box = 5 whites. Total = 252/1024 * (5/10)^10
7) Box = 4 whites. Total = 210/1024 * (4/10)^10
8 ) Box = 3 whites. Total = 120/1024 * (3/10)^10
9) Box = 2 whites. Total = 45/1024 * (2/10)^10
10) Box = 1 whites. Total = 10/1024 * (1/10)^10
11) Box = 0 whites. Total = 1/1024 * (0/10)^10 = 0!!
So, the chances of drawing 10 whites is the sum of these which is 0.013913029625. Approximately. However, we know that we actually drew
10 whites, so this 1 in 72 (ish) chance has happened. The chance that
it happened because we had possibility 1) above is the chance of 1)
relative to this 0.0139... chance. So, the answer is
(1/1024)/0.013913029625 which is 0.070190499... i.e. 7%.
So there!
RESPONESE TO THIS ANSWER:
I suggest that the answer you have given is to the question "What is
the probability of you pulling out 10 white marbles one at a time?"
The question is, however, "what is the probability of all marbles being white(or black)? The answer to that, I believe, is 1/1024, which to
nearest % is '0'.
REPLY TO RESPONSE:
I just re-read it, and I think I got it right. The question is, what is
the probability of them all being white, given that you randomly
withdrew and returned 10 white marbles consecutively.
38. 2 / 27
39. 8 / 3^7 or 4 / 3^27 (two solutions have been offered)
See an explanation here of 38 and 39 here: http://groups.google.com/group/rec.puzzles/msg/8b1875e5406828ff?dmode=source 45. -4697
x = n^3 - n!
2^3 = 8, 2! = 2 so 8-2 = 6
3^3 = 27, 3! = 6 so 27-6 = 21
4^3 = 64, 4! = 24 so 64-24 = 40...
46. 95,041,567
the numbers are all products of increasingly large numbers of
successive primes.
2 = 2 (or 2x1?),
15 = 3x5,
1001 = 7x11x13,
215,441 = 17x19x23x29
and I think the next in the series should be
95,041,567 = 31x37x41x43x47.
47. 3 (The list is of the digits of pi/4)
48. pi^2 r^4 / 2 The hyper-volume of a 4-dimensional hyper-sphere.
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