Sam Loyd writes,
"I recently came upon an old travel book that described some of the primitive methods once used for conducting business in the Philippines.
The traveling merchant shown in the picture uses a balance scale and four metal rings that serve as weights. The rings are of different
weights and sizes and he carries them on his arm like bracelets.
With these four rings, the trader can weigh anything from a quarter of a pound up to ten pounds. The trader can get within a quarter of a pound
of any weight within the limits mentioned above.
What are the weights of the four rings?"
Hint #1: The sum of the 4 weights is exactly 10 pounds.
Hint #2: What must the smallest weight be?
Hint #3: Two of the weights are less than one pound.
If you are not familiar with a balance scale, think of the problem like this:
Can you find four numbers which may be added or subtracted that will yield all the numbers from .25, .50, .75, 1.00, 1.25, ... 9.5, 9.75, 10.00?
Not all of the numbers need to be used for each weighing.
Solution to the Problem:
The four rings weigh 1/4 pound, 3/4 pound, 2 1/4 pound, and 6 3/4 pound.With these four weights you can weigh any item from 1/4 pound to 10 pounds.
Here are all the possibilities:
| Weight | Rings |
|---|---|
| 1/4 | 1/4 |
| 1/2 | 3/4 - 1/4 |
| 3/4 | 3/4 |
| 1 | 1/4 + 3/4 |
| 1 1/4 | 2 1/4 - 3/4 - 1/4 |
| 1 1/2 | 2 1/4 - 3/4 |
| 1 3/4 | 2 1/4 - 3/4 + 1/4 |
| 2 | 2 1/4 - 1/4 |
| 2 1/4 | 2 1/4 |
| 2 1/2 | 1/4 + 2 1/4 |
| 2 3/4 | 2 1/4 - 1/4 + 3/4 |
| 3 | 3/4 + 2 1/4 |
| 3 1/4 | 1/4 + 3/4 + 2 1/4 |
| 3 1/2 | 6 3/4 - 2 1/4 - 3/4 - 1/4 |
| 3 3/4 | 6 3/4 - 2 1/4 - 3/4 |
| 4 | 6 3/4 - 2 1/4 - 3/4 + 1/4 |
| 4 1/4 | 6 3/4 - 2 1/4 - 1/4 |
| 4 1/2 | 6 3/4 - 2 1/4 |
| 4 3/4 | 6 3/4 - 2 1/4 + 1/4 |
| 5 | 6 3/4 - 2 1/4 + 3/4 - 1/4 |
| 5 1/4 | 6 3/4 - 2 1/4 + 3/4 |
| 5 1/2 | 6 3/4 - 2 1/4 + 3/4 + 1/4 |
| 5 3/4 | 6 3/4 - 3/4 - 1/4 |
| 6 | 6 3/4 - 3/4 |
| 6 1/4 | 6 3/4 - 3/4 + 1/4 |
| 6 1/2 | 6 3/4 - 1/4 |
| 6 3/4 | 6 3/4 |
| 7 | 6 3/4 + 1/4 |
| 7 1/4 | 6 3/4 + 3/4 - 1/4 |
| 7 1/2 | 6 3/4 + 3/4 |
| 7 3/4 | 6 3/4 + 3/4 + 1/4 |
| 8 | 6 3/4 + 2 1/4 - 3/4 - 1/4 |
| 8 1/4 | 6 3/4 + 2 1/4 - 3/4 |
| 8 1/2 | 6 3/4 + 2 1/4 - 3/4 + 1/4 |
| 8 3/4 | 6 3/4 + 2 1/4 - 1/4 |
| 9 | 6 3/4 + 2 1/4 |
| 9 1/4 | 6 3/4 + 2 1/4 - 3/4 |
| 9 1/2 | 6 3/4 + 2 1/4 + 3/4 - 1/4 |
| 9 3/4 | 6 3/4 + 2 1/4 + 3/4 |
| 10 | 6 3/4 + 2 1/4 + 3/4 + 1/4 |
Correctly solved by:
| 1. James Alarie | Flint, Michigan |
| 2. Kelly Stubblefield | Mobile, Alabama |
| 3. Kimberly Howe | Vienna, Virginia |
| 4. Sapna Dominic | Champakulam, Alleppey, Kerala, India |
| 5. Alexander Thomas |
Mountain View High School, Mountain View, Wyoming |
| 6. Alexis Hunter |
Mountain View High School, Mountain View, Wyoming |
| 7. Brijesh Dave | Mumbai City, Maharashtra, India |
| 8. Tyler Stoddard |
Mountain View High School, Mountain View, Wyoming |