Answer to Problem of the Week for 03/12/2001

Answer to March 12, 2001 Problem

by Bob Stanton in Games Magazine March 2001

The Forgotten Password Problem

When Joan Q. Handley went to college, she was assigned a password to use on the computer system. The password consisted of a series of five different digits. She was told to memorize it and not write it down. Joan decided to invent a reminder, but all she could come up with was: (1) The sum of the first two digits was exactly twice the sum of the last four, and (2) the sum of the three digits in the middle was twice the sum of the last two. On the first day of classes, Joan forgot her password. From her reminder, can you reconstruct the password?
Solution to Problem: The password is 93102. The sum of the first two digits is exactly twice the sum of the last four. The smallest number that is the total of four different digits is 6 (0 + 1 + 2 + 3). The sum of the first two digits, therefore, must be at least 12. The largest possible total for two different digits is 17 (8 + 9). Since the total of the first two digits is twice the sum of the last four, this total must be an even number. So the possible totals for the first two digits are 12, 14, and 16. But 16 (7 + 9) is impossible, because the total of the last four digits would have to be 8, and four different digits can't total 8 if one of them is 7 or 9. Likewise, the total of the first two digits can't be 14 (5 + 9 or 6 + 8) because four different digits can't total 7 if one of them is 5, 9, 6, or 8. The sum of the first two digits must therefore be 12. Since the sum of the last four digits is 6, none of the digits can be greater than 3. So the first two digits must be 9 and 3, in that order. The remaining digits must be 0, 1, and 2, in some order. The only order in which the sum of the middle three digits is twice the sum of the last two digits is 93102.

When Joan Q. Handley went to college, she was assigned a password to use on the computer system. The password consisted of a series of five different digits. She was told to memorize it and not write it down.

Joan decided to invent a reminder, but all she could come up with was:
(1) The sum of the first two digits was exactly twice the sum of the last four, and
(2) the sum of the three digits in the middle was twice the sum of the last two.

On the first day of classes, Joan forgot her password. From her reminder, can you reconstruct the password?

Solution to Problem:

The password is 93102.

The sum of the first two digits is exactly twice the sum of the last four. The smallest number that is the total of four different digits is 6
(0 + 1 + 2 + 3).

The sum of the first two digits, therefore, must be at least 12. The largest possible total for two different digits is 17 (8 + 9). Since the total of the first two digits is twice the sum of the last four, this total must be an even number.
So the possible totals for the first two digits are 12, 14, and 16.
But 16 (7 + 9) is impossible, because the total of the last four digits would have to be 8, and four different digits can't total 8 if one of them is 7 or 9.

Likewise, the total of the first two digits can't be 14 (5 + 9 or 6 + 8) because four different digits can't total 7 if one of them is 5, 9, 6, or 8.

The sum of the first two digits must therefore be 12. Since the sum of the last four digits is 6, none of the digits can be greater than 3. So the first two digits must be 9 and 3, in that order. The remaining digits must be 0, 1, and 2, in some order. The only order in which the sum of the middle three digits is twice the sum of the last two digits is 93102.

Correctly solved by:

1. Bill Hall	Wellington, Florida
2. Sasha Joseph	Hebron, Maine
3. Kirstine Wynn	Winchester, Virginia
4. David Dixon	Bennettsville, South Carolina
5. David Powell	Winchester, Virginia
6. Ryan Charest	Winchester, Virginia
7. George Gaither	Winchester, Virginia
8. Walt Arrison	Philadelphia, Pennsylvania
9. Jim Kennedy	Limerick, Ireland
10. Richard Johnson	La Jolla, California
11. Kurt Fischer	Joliet Illinois
12. Keith Mealy	Cincinnati, Ohio
13. Kasturi Rajendran	----------
14. Erin McGinnis	Winchester, Virginia
15. John C. Funk	Ventura, California
16. Carla Burris	St. Louis, Missouri