The TOE process
Given any whole number, the following procedure always reaches the number 321.
Call the total number of digits x, call the number of odd digits y, call the number of even digits z.
Concatenate the numbers to create xyz.
Repeat the process until the result is 321.
Example beginning with 12345678901 we have x=11, y=6, z=5. Concatenate to 1165. Repeating gives 431 and then 321.
So 12345678901 takes 3 iterations to get to 321.
Question: What is the smallest starting number that takes 4 iterations to get to 321?
The answer is 10000000000111111111.
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1 solution
The smallest such number is 10000000000111111111.
The sequence is 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 → 2 0 1 0 1 0 → 6 2 4 → 3 0 3 → 3 2 1 .
The key is the 6 digit number 201010 that takes 3 iterations.
Does any number with fewer than 6 digits take 3 iterations?
5 digits: The first iteration of x=5 requires y+z=5. 505, 514, 513 all go directly to 321.
4 digits: The first iteration x=4 requires y+z=4 which is more promising: 440, 422, 402 all need two more iterations.
But there is no 4 digit number that itself comes from a large number of fewer than 20 digits.
440 requires a 4 digit number with all odd digits and the first two would have to form smaller than 20. 19yz can't have both y and z odd.
422 comes from something like 19yx which requires two evens, or 18yz which requires an even and an odd.
404 is right out because the four-digit number starts with 1.