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Mathematical Bobby Fischer puzzle
Bobby Fischer and his fiancee, Miyoko Watai, organized a "chess-couple convention" in Iceland. They invited 64 (married or engaged) chess couples from all of the world, and it was a great event for everyone.
There was no tournament, but of course, a lot of games were played (and a lot of smalltalk as well). No one played a game with their (married or engaged) partner, no one played a game "alone", no one played more than one game with the same opponent.
When all the guests had to leave, Miyoko said good-bye to everyone and asked "How many games did you play here?" She got all different answers (=numbers). Finally she asked Bobby and got another different answer.
How many games did Bobby Fischer play at this con?
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Nothing Bobby Fischer is "a great event"!
OK I am wrong, his execution would be a great event :)
Remember who this man is and what he stands for before using such vile language as "Bobby Fischer" on this great site.
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How many guests had a $ Million match fee; was the lighting adequate, and were there any TV cameras present? Most likely Bobby played Zero games. :))
But what happened to the invitation for sugarandspice and me ??!
There's another married couple here, on the Global Bounty Hunters. Wonder if they got theirs!? I'm hurt. }8-(
Ps// CF, bet you'd give a grouchy answer too if someone ambushed you with a wakeup call asking how you liked the USA if you had a vendetta tracking you like the (unconvicted) Bobby did! He's entitled to his opinions, and hasn't been treated well by a certain administration that went gunning for him. Not to mention the Pasedena police dept. violating his rights to the extent he claims.
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Is an answer possible here?
64 couples invited => 65 couples in all counting BF & MW, means 130 people.
Since no one played solo, nor played his or her 'partner', that leaves a maximum of 128 games anyone could have played. The numbers of games available to have been played range from 0 to 128, 129 different numbers in all. Of course, Miyoko Watai didn't ask herself, so 129 different responses are possible.
Yet I feel that unless this is a trick question with 'yellow bulldozer' as the answer, there isn't enough information. Otherwise the answer is zero... I'll think about this some more...
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She got all different answers (=numbers). Finally she asked Bobby and got another different answer.
Like you said, sounds tricky !
But I wonder if the key is not in the above sentence!? That ALL ANSWERS WERE DIFFERENT.
And is it saying all were different numbers. Does that mean Bobby's would be a number different from all the others!? Or perhaps that his answer was Not A Number!? That is what I am not clear on from the presentation of (=numbers).
We know there are 129 different responses tho, it says. And 0 to 128. hmmm. Yet of the 130 players no one can play themself nor their partner. Thus everyone should have an option of only 128 to play. I assume she did not ask herself and that accounts for 129 rather than 130 "responses". Then Bobby's must be NUMBER I guess, to account for 129 responses.
What you think? Even considering this possibility, I havent gotten it yet either! But it is fun to contemplate :)
Thanks misato for the brain tease. :)
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1) She did not ask herself.
2) She got 129 different answers (including Bobby's), all answers were numbers, and it's obvious that those numbers were from zero to 128.
3) There is no word-trick in it.
I was told this one yesterday with three invited couples and hand-shaking instead of playing chess games (= numbers from zero to six). After solving it I noticed that this problem can be generalized (and 64 is a better number for chess players). So you might as well try it with three (or less?) invited couples. It's just the same.
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I think I may have it.
I think I have an idea, the person who played everybody except their partner will have played 128 games (you can't play yourself or your partner), which means that their partner must have played 0 games (since the first player cannot play their partner). If we eliminate those two people and continue then logic would suggest that the person who played 127 games will be married to the person who played 1 and so on (so the sum of games between partners is 128). This means that there will be two people that will have played the same number of games (64+64=128). Since she got different answers from everybody else but is yet to ask herself that would mean that she is the other half (no pun intended) of the 64 games couple. Meaning Bobby should have played 64 games. This might be wrong, I woke up 5 minutes ago so my brain isn't really in gear yet. :)
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... I think you have it. I did a bit of a paper exercise and it does seem that for the numbers to be different, the numbers played by each couple must be additive complements of 128. I didn't appreciate this fact when I first looked at the question. A pair of partners played 64 games each, and clearly this pair was not one Miyoko asked previous to asking Bobby, since all responses were different. So Bobby's answer will be 64, the same number of games Miyoko herself played.
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Sounds right to me mattdw
I think you've Got It, by George! Congratulations.
I only have one question about it, let me run by you in a PM and see what you think. }8-)
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Just one question remains, Matt:
What about your brain power one hour after you woke up and everything is in best working condition?
My solutions were less elegant:
Bobby couldn't play 128 games because then no guest's answer could have been "zero".
So a guest was to say "128". Let it be a man, so his wife is the only one with a possible answer "zero" (because this guy played all other participants).
And then step by step downwards ("127" is impossible for Bobby as well because of the "1").
Or you start with "Bobby's answer zero is impossible because ..." and go upwards step by step.
I like the idea of the additive complement better, thanks!
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A couple of points...
One is that we don't need to suppose that each couple's games totalled 128. All we have to do, as ccmcacollister is indicating is to take complementary pairs. It is true, however, that Miyoko and Bobby play 64 games each.
Now, what if we were to pair up players whose games totalled 129? 128+1, 127+2, ..., 64+65. All different. That means Bobby played no games at all: zero. However many games Miyoko played is immaterial, though according to the parameters of the puzzle, it must have been not more than 128. Does this work? Or is this no solution because Miyoko would have in fact to have played 129 games? Back to the pencil and paper methinks....
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129 for each pair is impossible
The answers' total is 0 + 1 + 2 + ... + 127 + 128 = n * (n+1) / 2 = 8256 (except Miyoko's number because she wasn't asked).
65 (=number of couples) * 129 = 8385 would be the overall total, so - like you pointed out correctly - there is a difference of 129 (=Miyoko's number) which is a contradiction. I agree that this is no valid solution.
130 * 65 = 8450 is far too much.
Less than 128 games for each guest pair doesn't work either: 127 * 64 = 8128, so Bobby's number would be 8256 - 8128 = 128 (so he played all guest persons). Also impossible because of the one zero-answer.
And 126 * 64 = 8064 is far too few.
But why can we suppose that the couple's sums are always the same result, for each couple (at least for the guest couples)? I know that this assumption is true (with the result of 128), but I can't find an easy explanation for that.
My non-easy explanation is that the zero-player's partner is the only one with 128 possible games. So one pair is 128+0, and this must be a guest pair.
And then find the combination for the 1-player (who only played the 128-player), the 127-player must be their wife/husband (guest pair as well).
And so on ...
But this is close to my step-by-step-solution.
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I've run this one by Fritz.......
...and even he has no idea what you people are talking about. Personally, I think Mr. Green did it in the conservatory with the revolver.........:)
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I agree, I have no Idea what yous are on about.
PS: Ms Scarlet did it, in the Study with the lead pipe;)
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in that case,
what was madame murgatroyd doing in the orchard shed with the meat cleaver???
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... Bobby had to have played 1 game in order for one of his guests to have played as many as 128 (a realisation born of an early morning lie-in). Who played zero games then? As misato pointed out, this could only have been 128-game-guest's spouse/fiance(e)/partner. So the assumption I thought unnecessary turns out to be correct after all.
A fine puzzle, misato!
It were the butler wot done it.
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I Am Deeply Hurt . . .
. . . that my old friend Watai Miyoko did not invite me and my spouse (ladylinda) to participate in this event . . . *sniffle, sniffle*
*Taking Miyoko-chan off of my Ochugen gift list*
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Not only do we know now...
...how many games Bobby played, but also how many Miyoko played. Furthermore, we also know who they played! Bobby and Miyoko played exactly the same 64 people, and all of those opponents played more than 64 games each. No one who played 63 or fewer games played Bobby or Miyoko. Want proof?
Colonel Cranberry-Sauce, with the suburban utility vehicle, in the driveway.