**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