Source: Original and previous puzzle experience.
1. PaulWogJ@aol.com and Ken Burres sent the solution that only the numbers 0 and 1 meet the requirements. Ken Burres checked this with a program. PaulWogJ gives this rationale:
First I found that all solutions had to be three digits or less. Assume that the solution has 4 digits. Even if all the digits were nine, the sum of the squares would only be 4 * 81, which isn't four digits. I also found that if a solution had three digits it had to start with a 1. If you use all nines the sum of the squares is 243, The max you could get with 3 digits, so it has to start with a two or a one. If you use a 2 and two nines, the sum of the squares is 163, the max you can get when you have a two as the first digit. So all solutions must have 1 2 or 3 digits, and all 3 digit solutions would start with a 1.
[KD: I stated the problem inacurrately and I believe this is the correct solution as I stated it. I have restated it in a later puzzle: Productive Numbers Revisited .]There are 99 solutions! 9 solutions x010, where x is a single digit 1 through 9, and 90 solutions xy100, where xy is a two digit number 10 through 99. All solutions must be between 3 and 6 digits inclusive , since the problem requires "the first three digits and the last three digits" and the largest product of three digit numbers is 999*999 = 998001. Therefore, I used the following awk program to find the solutions: BEGIN { for(num=100;num<999999;num++) { first=substr(num,1,3); last=substr(num,length(num)-2,3); if (num==first*last) print num,first,last; } }I [Ken] think these solutions can also be found through logic:
Each number describes the number of digits in the previous number. 1 <-- We have to begin somewhere 11 <-- '1' appears 1 time 21 <-- '1' appears 2 times 1112 <-- '1' appears once, '2' appears once 3112 <-- '1' appears 3 times, '2' appears 1 time 211213 <-- '1' appears 2 times, '2' appears 1 time, '3' appears 1 time Continuing with a more readable format: 21 12 13 31 22 13 21 22 23 11 42 13 31 12 13 14 41 12 23 14 31 22 13 24 21 32 23 14 21 32 23 14 That's it, 21322314 !