THE PUZZLET PAGE


PUZZLET 172

A New Sequence

From any integer n(0) form n(1) by concatenating the number of even digits, the number of odd digits and the sum of digits like this:

129 --> (1 even) + (2 odd) + (12 sum) --> 1212.

Now form n(2) from n(1) in the same way.  Eventually, n(n) will be formed where n(n) equals n(m) and m < n.

What is the greatest length sequence that can be formed from a 3-digit integer?

The inset table demonstrates how each successive term is developed. Each term is the concatenation of the number of even digits, number of odd digits, and sum of the digits of its predecessor. New terms do not employ leading zeroes.

In this example, line 7 duplicates line 1, so the sequence repeats.

#
Term
Nr of Even
Digits
Nr of Odd
Digits
Sum of
Digits
New Term
1
128
2
1
11
2111
2
2111
1
3
5
135
3
135
0
3
9
39
4
39
0
2
12
212
5
212
2
1
5
215
6
215
1
2
8
128
7
128






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Last Updated: January 17th, 2010.