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westworld
 Post subject: Who killed Sharky
PostPosted: Sun, 29 Jan 2012 00:32:21 UTC 
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1. A big-time mob boss named Sharky was killed by one of his own band of four henchmen. Detective Sharp interviewed the men and determined that all were lying except for one. The results of the interviews are as follows: Socko: Lefty killed Sharky. Fats: Muscles didn t kill Sharky. Lefty: Muscles was shooting craps with Socko when Sharky was killed. Muscles: Lefty didn t kill Sharky. Who killed Sharky?
Answer : Socko, Fats, Lefty, Muscles,or There is not enough information.

I cant find enough to tell the liar :confused:

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Soroban
 Post subject: Re: Who killed Sharky
PostPosted: Sun, 29 Jan 2012 01:04:27 UTC 
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Hello, westworld!

Careful . . . We are looking for the one truthful henchman.

Quote:
1. A big-time mob boss named Sharky was killed by one of his four henchmen.
Detective Sharp interviewed the men and determined that all were lying except for one.

The results of the interviews are as follows:
. . (1) Socko: Lefty killed Sharky.
. . (2) Fats: Muscles did not kill Sharky.
. . (3) Lefty: Muscles was shooting craps with Socko when Sharky was killed.
. . (4) Muscles: Lefty did not kill Sharky.

Who killed Sharky?

. . (a) Socko . . (b) Fats . . (c) Lefty . . (d) Muscles . . (e) not enough information

Statements (1) and (4) are contradictory.
. . Hence, one must be true, the other must be false.

If (1) is true: "Lefty killed Sharky", then (2) is also true . . . not possible.

Hence, (1) is a lie: Lefty did not kill Sharky
. . and (4) is true: "Lefty did not kill Sharky."

Then (2) is a lie . . . Muscles did kill Sharky.

And (3) is a lie . . . Muscles did not have an alibi for the time of the murder.


Answer: .(d) Muscles
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outermeasure
 Post subject: Re: Who killed Sharky
PostPosted: Mon, 30 Jan 2012 00:40:27 UTC 
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Topic moved from Algebra to Miscellaneous.

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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