24 Nov '06 15:09

An absent-minded maths professor was telling his eccentric colleague about losing some of his paperwork.

"I can't find my special number lists," he complained.

"What was so special about them?" replied his colleague.

"Well," he explained, "I call a list of numbers 'special' if it has the following 7 properties:"

(1) All of the numbers on the list are positive integers, base 10.

(2) There is more than one number on the list.

(3) All of the numbers on the list have the same number of digits.

(4) No two of the numbers on the list are equal.

(5) The smallest number on the list is equal to the total number of digits of all the numbers on the list.

(6) The largest number on the list is equal to the sum of all of the digits of all of the numbers on the list.

(7) The number of items on the list is equal to the largest number divided by the smallest number.

"There must be thousands of special lists," replied his colleague.

"Hmm, I'm not sure about that, but I was only interested in four of them."

"What four?"

"What for? Well, I divide the sum of the numbers on each list by 100 and the four remainders comprise the four-number combination to my office safe. The safe is locked and I can't remember the combination without them!"

"No, I meant

"Well obviously I can't remember the lists, but I do recall their properties. First, all of the numbers on a list that lie strictly between the smallest and largest numbers I will call the middle-numbers. Now all of the middle-numbers on the first list are primes, and all of the middle-numbers on the second list are composite. Furthermore, the two largest middle-numbers on the third list do not appear on the first or second lists. Finally, the largest middle-number on the fourth list does not appear on any of the other three lists, and the smallest middle-number of the fourth list is not equal to the smallest middle-number of any of the other three lists."

"Hmm, let me think about that. By the way, what do you have in your safe?"

"I can't remember! But I usually lock up important papers that I am currently working on."

The next day the eccentric professor approached the absent-minded professor while waving a handful of papers. "I found your lists!" he shouted.

"Where in the world did you find them? I've looked everywhere!"

"I found them in your safe!"

"I can't find my special number lists," he complained.

"What was so special about them?" replied his colleague.

"Well," he explained, "I call a list of numbers 'special' if it has the following 7 properties:"

(1) All of the numbers on the list are positive integers, base 10.

(2) There is more than one number on the list.

(3) All of the numbers on the list have the same number of digits.

(4) No two of the numbers on the list are equal.

(5) The smallest number on the list is equal to the total number of digits of all the numbers on the list.

(6) The largest number on the list is equal to the sum of all of the digits of all of the numbers on the list.

(7) The number of items on the list is equal to the largest number divided by the smallest number.

"There must be thousands of special lists," replied his colleague.

"Hmm, I'm not sure about that, but I was only interested in four of them."

"What four?"

"What for? Well, I divide the sum of the numbers on each list by 100 and the four remainders comprise the four-number combination to my office safe. The safe is locked and I can't remember the combination without them!"

"No, I meant

*which*four!""Well obviously I can't remember the lists, but I do recall their properties. First, all of the numbers on a list that lie strictly between the smallest and largest numbers I will call the middle-numbers. Now all of the middle-numbers on the first list are primes, and all of the middle-numbers on the second list are composite. Furthermore, the two largest middle-numbers on the third list do not appear on the first or second lists. Finally, the largest middle-number on the fourth list does not appear on any of the other three lists, and the smallest middle-number of the fourth list is not equal to the smallest middle-number of any of the other three lists."

"Hmm, let me think about that. By the way, what do you have in your safe?"

"I can't remember! But I usually lock up important papers that I am currently working on."

The next day the eccentric professor approached the absent-minded professor while waving a handful of papers. "I found your lists!" he shouted.

"Where in the world did you find them? I've looked everywhere!"

"I found them in your safe!"

**Questions:**

(1) What is the combination to the safe?

(2) How many possible "special" lists are there?(1) What is the combination to the safe?

(2) How many possible "special" lists are there?