You need several weights of different sizes in order to balance any weight from 1 to 17 pounds (in whole numbers). If you can use any weight only once for each separate weighing of 1 to 17 pounds, what is the minimum number of weights you will need? What are the weights?
Solution to Problem:
The answer is 5 weights.
There are many possibilities.
(1) You could use a 1-pound weight, a 2-pound weight, a 4-pound weight,
an 8-pound weight, and a 16-pound weight to make all the combinations
from 1 to 31 pounds (more than the 1 to 17 pounds which was asked for).
(2) You could use two 1-pound weights, a 2-pound weight, a 4-pound weight,
and a 9-pound weight to make all the combinations from 1 to 17.
(3) You could use a 1-pound weight, two 2-pound weights, a 3-pound weight,
and a 9-pound weight to make all the combinations from 1 to 17.
(4) You could use a 1-pound weight, a 2-pound weight, a 4-pound weight,
a 5-pound weight, and a 6-pound weight to make all the combinations from
1 to 17.
And there are others...
Two people sent in solutions with weights being used on both sides, and one of
those persons,
Keith Mealy, found that it is possible to do the weighings with only four weights!
Here is Keith's solution:
1, 3, 7, 15 works by putting a weight on either side of the scale (+ indicates opposite the object and - on the same side as the object).
1 = +1
2 = 3 - 1
3 = 3
4 = 7 - 3
5 = 7 + 1 -3
6 = 7 -1
7 = 7
8 = 8 + 1
9 = 7 + 3 -1
10= 7 + 3
11= 7 + 3 +1
12=15 - 3
13=15 + 1 - 3
14=15 - 1
15=15
16=15 + 1
17=15 + 3 - 1
(EXTRA CREDIT)
16-26 are just 15 + (numbers formed from 1 to 11), as
above.
Correctly solved by:
| 1. Rick Jones | Kennett Square, Pennsylvania |
| 2. Kurt Fischer | Crest Hill, Illinois |
| 3. Kirstine Wynn | Winchester, Virginia |
| 4. Walt Arrison | Philadelphia, Pennsylvania |
| 5. Renata Sommerville | Austin, Texas |
| 6. Mr. Bennington's grade 8 class | High River, Alberta, Canada |
| 7. Keith Mealy | Cincinnati, Ohio |
| 8. George Gaither | Winchester, Virginia |
| 9. Bob Hearn | Winchester, Virginia |
| 10. Ricki Stern | Highland Park, New Jersey |
| 11. ---------- | United Kingdom |
| 12. Gusti Oggenfuss | Montet, Switzerland |