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The Question ( http://www.ecst.csuchico.edu/~kend/potw/index.html )

1.Take a 2-digit number and multiply the two digits together. Repeat with each product until a single digit is the result. For most starting numbers, no more than three of these steps are needed to arrive at the final single digit. (For example: 57: 5*7=35: 3*5=15: 1*5=5). Find one that takes four steps.
 
Answer:

The ONLY two digit number that takes four steps is 77.
7*7=49; 4*9=36; 3*6=18; 1*8=8

Believe me, it was only the fifth number I tried. However, it inspired me to write a small tcl program to find such numbers with higher number of digits.

Let us denote the number of steps it takes to reduce a n digit number to a single digit as "hardness"
Conjecture: Any n digit number has a hardness less or equal to  n+2.

The above conjecture holds good for n<=6 (that was as far as I could push tcl).
 

Number of digits (n)  hardness Digit sets*
2 4 {7,7}
3 5 {6,7,9}  {6,8,8} 
4 6 {6,7,8,8} 
5 7 {6,8,8,8,9}
6 8 NONE
Table 1: The digit sets that require highest number of reductions in their class. Any number formed by the permutation of digits in a digit set will have the same hardness. For example, all numbers 679, 697, 769, 796, 967, 976 have the same hardness of 5. 

Some interesting findings:

As seen in the above table, There are two sets of three digit numbers which have a maximum hardness of 5
Surprisingly, there was no six digit number with a hardness of 8
 
 

Clockwise from top left: a) Graph showing the hardness of numbers between 10 and 99, b) numbers between 100 & 999 and c) numbers between 1000 & 9999. Note that the max hardness numbers are clustered around 6,7,8,9 in  each level. Anyone see any other pattern? 

2.Start with a 3-digit number. Choose any one of the digits and remove it to make a 2-digit number. Multiply the 2-digit number by the digit you removed. Eventually, you will get a 2-digit number, which will yield a single digit, as above. What number should you start with, and what steps should you take to find the longest series of steps you can take to arrive at that final single digit?

Answer:

I'm not quite sure if I understood the question correctly, but will try.

The product of one of the digits and the remaining 2 (merged to form a 2 digit number) should be equal to any of the numbers formed by the digit sets of hardness=5. From the two digit sets in hand, we have:

688 = 88x6. Hence, the numbers {688, 868, 886} are good candidates. If 6 is removed and the above process executed, it will take 5 steps to reduce. Other permutations will place thd digit 8 in the 100's place and we cannot factorize to the required specs.

679 = 97x7. The numbers {977, 797} are good candidates.

However, you cannot remove "any" one of the digits in the above numbers. e.g, removing 9 from 977 will give 9x77 = 693, which does not have hardness 5.