[net.math] Math riddle

bill (05/07/83)

I forgot to mention in my previous submission, that the L[infinity]
norm of the vector (x,y) is just max(|x|,|y|).  So the formula

	max(|x|,|y|) = k

also gives a square.  If you are interested, the L[1] norm and
L[infinity] norms are, like the L[2] norm, very useful for curve
fitting problems.  L[2] is Least Squares; L[infinity] is Chebyshev
fitting (minimize the maximum error); finally, L[1] is related to the
median (and is very resistant to bad data points).  See

	Branham, Richard L., "Alternatives to Least Squares",
		Astronomical Journal *87*, 928-937 (1982)

for a very illuminating discussion.

leei (05/08/83)

	In case anyone is interested, the set of curves generated by the
series of equations:

		n       n
	    a|x|  + b|y|  = k

for  1 <= x <= oo  were labelled `superellipses' by the Dutch architect Piet
Hein.  He used the solid `superellipsoids' in some of his sculptures and even
some architectural forms.  I can't remember whether I read this in one of
Martin Gardner's mathematical games columns in SA or not, but I have a feeling
that this was where I saw it. (I am supported on this by a friend sitting next
to me.  He says that the article is reprinted in Mathematical Carnival)

					-Lee Iverson
					princeton!leei

parnass@ihuxf.UUCP (06/08/83)

       The following math riddle was posted on the net recently:

       *******************************
       A principal and a teacher are talking as	several	people walk by.	The principal
       mentions	to the teacher:	"I know	the three people who just past us. The product
       of their	ages is	2450. Can you tell me their individual ages?" The teacher
       replies:	"I do not have enough information to uniquely answer the question."
       The principal then says:	"OK, the sum of	their ages is exactly twice
       your age. Now can you tell me their ages?" The teacher thinks for awhile
       and replies: "I still don't have	enough information to uniquely identify
       their ages." Then the principal states: "I will tell you	this, and this will
       be conclusive: Each of the three	people we are discussing is younger than
       myself."	The teacher says: "Now I can tell you their ages."

       Can you figure out the ages of *all* the	people involved, including the
       principal and the teacher?
       ***********************************
	       My answer is as follows:

		  o+ The	ages of	the three passersby are: 10, 5,	and 49.

		  o+ The	teacher	is 32.

		  o+ The	principal is 50.


	       My reasoning took the following form:

		 1.  I assumed only positive integer ages.

		 2.  I factored	the  product  of  the  three  ages  of	the
		     passersby	 (2450)	 into  all  combinations  of  three
		     integers.

		 3.  For each 3-tuple of ages, I found the sum of the  ages
		     in	that 3-tuple.

		 4.  I assumed the teacher knew	his/her	own  age.   If,	 at
		     this  point,  he/she still	didn't have enough informa-
		     tion to uniquely determine	the ages of the	 passersby,
		     it	 was because there were	2 or more 3-tuples that	had
		     the same sum.  Two	and only two 3-tuples had the  same
		     sum (10, 5, 49) and (50, 7, 7) both had sums of 64.

		 5.  Half the sum was equal to to the teacher's	age (32).

		 6.  The principal's age being 50 uniquely  identified	the
		     proper 3-tuple.  Any other	age for	the principal would
		     still leave the  ambiguity	 between  the  pair  of	 3-
		     tuples.


		       Robert S. Parnass, AJ9S
		       Bell Laboratories
		       IH 1B-414
		       Naperville, IL 60566
		       ihnp4!ihuxf!parnass