Jane Fairfax, I wanted non primes only.
Sometimes they are also referred to as composite numbers. I was just trying to be clear in my question what non primes are. Okay, I shall make it non primes alone. Thanks for correcting me.
Sometimes, in my over enthusiasm I make mistakes. Thanks for correcting me, Jane Fairfax

Johnny, great work! You got it right!
Congratulations!!!

#41. Create a 4X4 Magic Square using the numbers 1-16 without referring. If you can do it in less than five minutes, thats a very good performance.

Last edited by TheDude (2007-11-10 02:16:14)

Surely in that proof, x needs to be restricted to the naturals? That complicates things a bit.

Solution to #42.

Outstanding, TheDude! I wholeheartedly accept your proof.

The solution is,
into as many parts as when the parts are closest to e.
The proof, the solution are given supra.

Example:- Question :Divide 10 into equal parts such that the produce is the maximum.
Solution :

Two parts
10/2 = 5
Product 5x5=25.

Three parts
10/3=3.3333=10/3 approximately.
Product 10/3 x 10/3 x 10/3 = 1000/27. = 37 approximately

Four parts
10/4=2.5
Product = 10/4 x 10/4 x 10/4 x 10/4 = 10000/256 = 39.0625

Five parts
10/5=2
Product = 2x2x2x2x2 = 32.

Six parts
10/6 = 1.66666
Product = 10/6 x 10/6 x 10/6 x 10/6 x 10/6 x 10/6 = 1000000/46656 = 21.43 approximately.

The number of parts close to e, that is the natural logarithm base, of value 2.71828 roughly, are 2.5 and 3.33333 when divided into 4 equal parts and 3 equal parts respectively. Of the two of them, it is clear 2.5 is closer to 2.71828 than 3.3333.

Therefore, the product is the greatest when the number 10 is divided into 4 equal parts. As given above, the product is 39.0625, visibly, which is the highest of all.

The number e appears where we expect it least in Mathematics. This question and proof is intended to show the fact. To a novice or a layman, it would be immensely suprising to know that an irrational number would have anything to do with the seemingly simple question. As a matter of fact, the obvious reply to the question would have been 2 parts. Mathematics is a subject which requires a degree of certainity to a much greater level. I am pleased that the question has been solved in such a short duration.
Commendabl performance, TheDude and Identity!

#43. I have 1000 one cent coins. These have to be packed in ten plastic bags in such a way that I should be able to give away any number of cents from 1 to 1000 in bags alone. For example, after I have put the 1000 one cent coins in bags, I may be required to give away 1 cent, or 100 cents or 433 cents etc. The number may vary from 1 to 1000. I should not open the bags once the cents are put in the plastic bags and I have marked how many cents each bag contains. I should give away bags only. How would I distribute the 1000 on cent dimes in the ten plastic bags?

Solution to #43.
Excellent, TheDude!!! Keep it up!!!

#45. The average age of 10 members of a committee is the same as it was 4 years ago, because an old member has been replaced by a young member. Find how much younger is the new member ?

#46. A farmer built a fence around his 17 cows, in a square shaped region. He used 27 fence poles on each side of the square. How many poles did he need altogether?