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Usenet Newsgroups : rec.puzzles Hall of Fame


The rec.puzzles Hall of Fame is a compilation of over 500 of the most popular puzzles that have been posted and discussed in the rec.puzzles newsgroup. In most cases a detailed solution has been provided.

Many of these puzzles also appear in Braingle's own collection.

   
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Categories : pickover : pickover.09.p

Title: Cliff Puzzle 9: 3-Atoms and Growth
From: cliff@watson.ibm.com

If you respond to this puzzle, if possible please send me your name,
address, affiliation, e-mail address, so I can properly credit you if
you provide unique information.  PLEASE 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

      * * *

    Start with 3 digits: 1, 2, and 3.
Each succeding row repeats the previous three rows, in order,
as you can see from the following diagram.

1
2
3
123
23123
312323123
12323123312323123
2312331232312312323123312323123

1. What is the sum of digits in the 100th row?

2. Get rid of all the twos.  Here I've replaced each of them with a "."

1
.
3
1.3
.31.3
31.3.31.3
1.3.31.331.3.31.3
.31.331.3.31.31.3.31.331.3.31.3

In the last row of this diagram, there are three different species:  31,
331 and 3.  How many different species are there in row 30?

3.  When the sequence first hits a three, it now undergoes an enzymatic
cleavage, and the digits on the right of the 3 are swapped with the
digits on the left.

1
2
3
123
23123 now becomes 12323
312312323 now becomes 123123233
Now answer the question posed in question 2.


Solution


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