$35 \cdot 29$ Using three different weights: 1 gram, 3 grams, 9 grams, we can measure the weight of different objects. Assuming the object to be weighed and the known weights can be placed on either side of the balance, how many different weights of objects can be measured?
(A) 15 .
(B) 13 .
(C) 11 .
(D) 9 .
(E) 7 .
(10th American High School Mathematics Examination, 1959)
[Solution] Using the enumeration method.
First, consider when the weights are placed only on one side of the balance:
Using one weight, items weighing $1, 3, 9$ grams can be measured;
Using two weights, items weighing $4, 10, 12$ grams can be measured;
Using three weights, an item weighing 13 grams can be measured.
Next, consider when the weights are allowed to be placed on both sides of the balance:
Using two weights, items weighing 3-1, 9-1, 9-3 grams can be measured;
Using three weights, items weighing $9-(1+3),(9+1)-3,(9+3)-1$ grams can be measured.
In summary, a total of 13 different weights can be measured.
Therefore, the answer is $(B)$.