BrainDen.com - Brain Teasers  voider

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Everything posted by voider

2. Personally I find it bizarre. I've noted that the "signal to noise ratio" (if signal means correct/true/good reasoning/answer/process, noise means bad/incorrect/false) on brainden is worse than most ad hoc forums or messages, even for the trivially easy puzzles. One of these forms is where people give answers that don't mean a thing to anyone else; the words and logic is practically gibberish in English. How does this happen? Even for most people to acknowledge a correct logical answer seems to be impossible here. If this place lacks common sense, you have to wonder what "communal" value there is given the objectives of contributing here. In this case, the fact that we have completely different solutions means most of us are completely wrong, and yet believe we are all on the right track. I have no problem accepting I could be wrong, but I'm the only one who has given a proof, and I have implicitly disproved all other answers. But in my experience, proofs and disproofs mean nothing to most people.
3. Got to say, it's __________ that five of us have completely different solutions, and no working in common. Someone find the right word for me
4. The probability of four same color (thus first chair winning): 8/8 * 3/7 * 2/6 * 1/5 = 1/35. Same answer results from using combinations. There are 70 (ordered) permutations of the 8 stamps, each with equal chance. There are 16 (ordered) permutations of 4 of the 8 stamps, not with equal chance. If this doesn't occur, then first chair cannot know the answer. Second chair now knows that first chair does not see: YYXXXX Specifically it leaves 68 permutations with equal chance. Then, if he sees XXYYXX he wins. As in the first case, there are two permutations that satisfy this. So he wins this way with 1/34 chance. I believe he learns nothing else. Third chair now knows that second chair does not see: XXYYXX This eliminates 2 possibilities, leaving 66 with equal chance. If he sees XXXXYY he wins, 2/66 chance. Also if he sees XXYY?? then he wins knowing he has XY (since the others haven't won yet). This could happen as BBWW**** or WWBB**** where **** is permutations of BBWW. 2 * 6 = 12 ways However XXYY?? intersects with XX??XX at XXYYXX so there are 8 ways left. So if he didn't win by XXXXYY he wins this with 8/64. Equivalent overall: 2/66 + 64/66 (8/64) === 10/66 = 5/33 I believe he learns nothing else. First chair: He's survived this far, so there are 56 possibilities left. YYXX?? is not possible. He applies same thing as third chair, he wins if he sees ??XXYY where he must have XY. Still 12 possibilities, minus 4 intersections. Wins with chance 8/54 = 4/27. I can't see him learning anything else. Second chair: 48 possibilities left, YYXX?? and ??XXYY are excluded. If he sees XX??YY he must have XY and he wins. 8 ways => 8/48 = 1/6 chance of winning like this. If he doesn't win like that, he knows the XX, YY, XY pairs aren't there (no one sees them). XXYYXY and XXYYXX and XXYYYY aren't there, it must mean XX YY isn't there. What about XX XY? (XX XY YY, XX XY XX can't be). One mixed pair doesn't exist, this means there are two or three mixed pairs. Thus if he sees XX??MM or MM??XX where MM = mixed pair, then he knows he has XY. XXMMMM has 8 forms: 2x for swapping X with Y, 4x for two MM pairs as XY or YX. No more multipliers because the rest must be YY. MMMMXX has 8 ways also. He wins with 16/40 chance now = 2/5 Overall: 8/48 + 40/48 (16/40) = 24/48 = 1/2 chance. Third chair: 24 left. So I think there are 2 or 3 mixed pairs. Second chair would have won if one of 1st or 3rd pair was not mixed, therefore they must both be mixed. I believe possibilities left are MMMMMM or MMXXMM. This would mewan third must have XY no matter what. Can check this by seeing how many possibilities MMMMMM and MMXXMM form: MMXXMM: 2x for XY or YX on third pair, 2x for XY/YX on 1st pair, 2x for identify of X, 1x for the rest must be YY. 8 ways. MMMMMM: 8x for three pairs XY/YX. Remaining is also a pair, 2x for that. 16 ways. Totals 24 so correct. Patterns: 1 XXXX??, 2 ways 2 XX??XX, 2 ways 3 ??XXXX, 2 ways 3 XXYY??, 8 new ways 1 ??XXYY, 8 new ways 2 XX??YY, 8 new ways 2 XXMMMM or MMMMXX, 16 ways 3 MMXXMM or MMMMMM, 24 ways By intuition or otherwise, the order of events is independent of the ways of winning (must be better way to explain). Anyhow, you can just add up the number of ways. First chair: 10 ways Second chair: 26 ways Third chair: 34 ways So the logical choice is third chair, with 34/70 chance of winning, assuming the other two students are not (color)blind and are equally logical.
5. I assume she will randomly choose the stamps because e.g. if you chose the first chair + the four stamps you see are the same color. So you want to maximise your chances.
6. You can't say STOP after they've already won.
7. Consider the decision tree. You can calculate the probability that you will win, when you predetermine on which round you will say STOP (if you survive until that round). The first round probability is 4/12=0.33 Second is 0.397979... This increases, until some round to say STOP where the probability of winning will begin to drop. Obviously the "general solution" is to say STOP on the round that produces the overall highest probability of winning relative to the beginning of the decision tree. My solution would be a computation, rather than a calculation. There are multiple ways of presenting it, but none of them would look nice.
8. It looks strange but it's too easy. Might be harder if you have to construct it in your mind with your eyes closed.
9. 133 is correct, = 7C2 * 6 + 7
10. Sounds like a way to earn reputation... if you're into that...
11. abortoperation d????????????? day in monday izj sday in tuesday oizq day in days iaq Does the alphabet/number map change after using a letter?
12. Boolean algebra: 1 + 1 = 1 Galois field 2: 1 + 1 = 0 Subspaces: 1 subspace + 1 subspace = 1 subspace Sand: 1 pile + 1 pile = 1 pile Speed of light: 1 c + 1 c = 1 c Networks: power of 1 component + power of 1 component < power of (1 component + 1 component) Problems: 1 small problem + 1 small problem = RAGE Memorisation: time taken to learn something twice as long
13. I've looked at a few patterns, none of which seem to fit perfectly, but my intuition definitely favours D and rejects A and B overall. The more obvious observations are the near-symmetry along one or both diagonals, that (3, 2) is symmetrical, (3, 1) is a rotation of (3, 2), that each block has 3 of each colour, that the columns of the first row blocks have all colours.
14. Clarification: The operators are binary. I was not clear on this. What I meant was, still assuming there are 7 variables: The number of expressions with exactly one variable is 7: a, b, c, d, e, f, g The number of expressions with two or less variables (i.e. including the 7) is 133: a, b, c, d, e, f, g, a + b, a - b, a * b, a / b, ..., f / g, ... g / a. I have an answer now.
15. If you have 7 integer variables, e.g. a, b, c, d, e, f, g, and you can use each at most once in an expression with the operators +, -, *, / (integer division), how many unique expressions can be formed? E.g. (a - b) * c + f is mathematically the same as: c * (a - b) + f f + (a - b) * c f + c * (a - b) so they all correspond to a single unique expression. If there was one variable, there are 7 unique expressions. With two variables, I think there are 133 unique expressions. I don't know the answer (yet), and I very much doubt anyone can find a closed form formula for the general problem. About the context where this problem originated, I suspect it was intended to be virtually unsolvable... (and deceptively mediocre-looking)
16. I was gonna say but = W = has the right answer.
17. How does your signature work? I googled the first character and there was exactly one result: this page.
18. My interpretation of "interior diagonals": It has n sides where n is even. Under this interpretation, solve n / 2 = (1 + 1/30) * (n / 2 - 1) yields n = 62 I know I know my paint skills are pretty good
19. Turns out I was wrong. The answer is 1622125426984879983056127 = 1.622125426984879983056127 * 10^24 How? I used Anza's code and changed all longs to BigIntegers. Moral of the story? - (Generally) Don't switch to C++ to get 2^64 - 1 instead of 2^63 - 1 - Only Java, of contest languages, offers unlimited precision through BigInteger/BigDecimal - In case you still don't have a high precision calculator handy, http://www.wolframal.../?i=2%5E1000000 . Python also works as a calculator, but of course Wolfram is the ultimate genius for anything.
20. Like I said, without reading your program I can guarantee it is wrong. s[10^17] has to be greater than 2^80 + 2^78. superprismatic has a good estimate, but the exact breakdown is proving prohibitively expensive yet.
21. Assumptions are intended; he said it's a calculus problem. An example of why I'm accusing everyone of calculating volume incorrectly. It's not a standard cylinder.
22. I did not read your program and I don't know what s(10^17 - f) is considering I doubt you calculated 10^17 entries of s. Anyway you claim 4905601217378542463 = 4.9 * 10^18 to be the number of sums involving f with the sum less than 10^17. This is not correct, since the sum from f to f = f - 2, which is less than 10^17 - f. Significance? Consider the sums with f plus combinations of the set of f to f. All of them are less than 10^17 so there are 2^78 - 1 = 302231454903657293676543 = 3.0 * 10^23. This shows that your program is wrong, but more importantly how are you going to compute the real number? Obviously a full dynamic programming table is not feasible. What course is this homework for anyway?
23. http://www.suitcaseo...ox_Achilles.htm gives a brief glimpse into the depth of the paradox. Sometimes Greek philosophers get too much credit? But in this case if you think there's nothing in it then you are just reading the paradox without knowing what its subject is (if you thought about your own first paragraph?). As the link points out clearly, the discussion is not about whether Achilles reaches the tortoise (of course he does so in a finite amount of time, any statement saying "therefore he never catches up" is just for dramatic effect). I think the relativity argument is a bit over the top (plus I don't buy it). To be honest I don't know what Zeno thought about space/time discreteness/continuity and I would like to know more.
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