Title: Cliff Puzzle 1: Can you beat the numbers game? From: cliff@watson.ibm.com

If you respond to this puzzle, if possible please include your name, address, affiliation, e-mail address. If you like, tell me a little bit about yourself. You might also directly mail me a copy of your response in addition to any responding you do in the newsgroup. I will assume it is OK to describe your answer in any article or publication I may write in the future, with attribution to you, unless you state otherwise. Thanks, Cliff Pickover

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At a recent trip to the Ontario Science Center in Toronto, Canada I came across an interesting puzzle. The center is located minutes from downtown Toronto and it's a vast playground of science with hundreds of exhibits inviting you to touch, try, test, and titillate your curiosity. The puzzle I saw there can be stated as follows. In the 10 boxes below, write a 10-digit number. The digit in the first box indicates the total number of zeros in the entire number. The box marked "1" indicates the total number of 1's in the number. The box marked "2" indicates the total number of 2's in the number, and so on. For example, the "3" in the box labeled "0" would indicate that there must be exactly three 0's in the 10-digit number.

| 0| 1| 2| 3| 4| 5| 6| 7| 8| 9| | 3| | | | | | | | | |

Stop And Think

1. Is there a solution to this problem? Are there many solutions to this problem?

2. A more advanced an interesting problem is to continue to generate a sequence in a recursive fashion such that each row becomes the sequence for the previous. For example, start with the usual 0 through 9 digits in row 1:

Row 1: 0 1 2 3 4 5 6 7 8 9

Assume Row 2 is your solution to the puzzle. I've just inserted random digits below so as not to give away the solution:

Row 1: 0 1 2 3 4 5 6 7 8 9 S(1) Row 2: 9 3 2 3 3 1 6 7 8 9 S(2) Row 3: S(3)

Row 2 is now the starting point, and your next job is to form row 3, row 4, etc. using the same rules. In the previous example, a digit in the first box would indicate how many 9's there are in the next 10-digit number, and so forth.

Contest: I am looking for the longest sequence of numbers users can come up with using these rules. Can you find a Row 2 or Row 3? Is it even possible to generate a "row 2" or "row 3"?