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 |