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        riddles >> putnam exam (pure math) >> More University Interview Questions
        
(Message started by: THUDandBLUNDER on Jan 19^th, 2005, 12:13am)

Title: More University Interview Questions
Post by THUDandBLUNDER on Jan 19^th, 2005, 12:13am

Three brilliant students go for separate interviews with a university admissions officer, a number theorist. He chooses 4 whole numbers and tells the 1st student their product. The student tells him the 4 numbers. He tells the 2nd student the sum of the squares of the 4 numbers and she duly tells him the 4 numbers. 'Too easy', he thinks. So he tells the 3rd student the sum of the 4 numbers. The student was unable to tell him the 4 numbers.

What 4 numbers did he choose?

Title: Re: More University Interview Questions
Post by Grimbal on Jan 19^th, 2005, 4:09am

::[hide] 1, 1, 1, 3 [/hide]:: ?

Title: Re: More University Interview Questions
Post by Eigenray on Jan 19^th, 2005, 5:59am

What is a whole number?
The first student must have known none of the numbers were negative. And we know there were no 0s, but do the second and third students know that?

Title: Re: More University Interview Questions
Post by THUDandBLUNDER on Jan 19^th, 2005, 11:41am

on 01/19/05 at 05:59:19, Eigenray wrote:

What is a whole number?
The first student must have known none of the numbers were negative. And we know there were no 0s, but do the second and third students know that?

Although Bourbaki includes zero in the set of whole numbers, I believe for the purpose of this puzzle we should use A000027 (http://www.research.att.com/projects/OEIS?Anum=A000027)

Title: Re: More University Interview Questions
Post by Icarus on Jan 19^th, 2005, 7:45pm

Bourbaki is not the only one. Traditionally,

The Natural numbers = 1, 2, 3, 4, ...
(The name comes from the Greeks, who considered other numbers as being contrived. They were not aware of the concept of 0, so 0 is not included).

The Whole numbers = 0, 1, 2, 3, 4, ...
(Since the Naturals did not include 0, when the concept of 0 was introduced into western mathematics, a new name was needed for the set with 0.)

Considering the Whole numbers to be 1, 2, 3, ... is rather ridiculous, as we already have a perfectly good name for those numbers.

Title: Re: More University Interview Questions
Post by Eigenray on Jan 19^th, 2005, 9:59pm

Most logicians I know (and certainly many others) consider 0 a natural number -- it's the equivalence class of a finite set, viz, {}.
[bbn]=[},[],[},[]},[},[],[},[]],...} ;D

Waclaw Sierpinski, the great Polish mathematician... was worried that he'd lost one piece of his luggage. "No, dear!" said his wife. "All six pieces are here." "That can't be true," said Sierpinski, "I've counted them several times: zero, one, two, three, four, five."
--- The Book of Numbers, John Conway and Richard Guy

Title: Re: More University Interview Questions
Post by Icarus on Jan 20^th, 2005, 3:40pm

It is becoming common to include 0 in the Naturals, but this is a fairly recent change. Traditionally, it was not.

And since the Naturals are being redefined to be the Wholes, I suppose it is not surprising that Wholes are being redefined to be the naturals.

However, we can settle the entire thing by saying that the professor chose 4 "positive integers". (And no, zero is not positive! >:()