Example 5 Using weights labeled $1 \mathrm{~g}, 2 \mathrm{~g}, 6 \mathrm{~g}, 26 \mathrm{~g}$, each one in quantity, to weigh objects on a balance scale without graduations. If weights can be placed on both ends of the balance, the number of different gram weights (positive integer weights) that can be measured is ( ).
(A) 15
(B) 23
(C) 28
(D) 33
$(2006$, National Junior High School Mathematics Competition Zhejiang Province Re-test)
Explanation: If we consider each case individually, there will inevitably be omissions; if we categorize by the number of weights on each side of the balance, the approach is clear and less prone to omissions.
(1) When one weight is placed on one side of the balance and no weight is placed on the other side, the grams of the object that can be weighed are $1 \mathrm{~g}$, $2 \mathrm{~g}$, $6 \mathrm{~g}$, $26 \mathrm{~g}$;
(2) When two weights are placed on one side of the balance and no weight is placed on the other side, the grams of the object that can be weighed are $3 \mathrm{~g}$, $7 \mathrm{~g}$, $8 \mathrm{~g}$, $27 \mathrm{~g}$, $28 \mathrm{~g}$, $32 \mathrm{~g}$;
(3) When three weights are placed on one side of the balance and no weight is placed on the other side, the grams of the object that can be weighed are $9 \mathrm{~g}$, $29 \mathrm{~g}$, $33 \mathrm{~g}$, $34 \mathrm{~g}$;
(4) When four weights are placed on one side of the balance, the grams of the object that can be weighed are $35 \mathrm{~g}$;
(5) When one weight is placed on one side of the balance and one weight is placed on the other side, the grams of the object that can be weighed are $1 \mathrm{~g}$, $4 \mathrm{~g}$, $5 \mathrm{~g}$, $20 \mathrm{~g}$, $24 \mathrm{~g}$, $25 \mathrm{~g}$;
(6) When one weight is placed on one side of the balance and two weights are placed on the other side, the grams of the object that can be weighed are $3 \mathrm{~g}$, $5 \mathrm{~g}$, $7 \mathrm{~g}$, $18 \mathrm{~g}$, $19 \mathrm{~g}$, $21 \mathrm{~g}$, $22 \mathrm{~g}$, $23 \mathrm{~g}$, $25 \mathrm{~g}$, $27 \mathrm{~g}$, $30 \mathrm{~g}$, $31 \mathrm{~g}$;
(7) When one weight is placed on one side of the balance and three weights are placed on the other side, the grams of the object that can be weighed are $17 \mathrm{~g}$, $23 \mathrm{~g}$, $31 \mathrm{~g}$, $33 \mathrm{~g}$;
(8) When two weights are placed on one side of the balance and two weights are placed on the other side, the grams of the object that can be weighed are $19 \mathrm{~g}$, $21 \mathrm{~g}$, $29 \mathrm{~g}$.

It is easy to see that after removing the duplicate grams, there are 28 in total. Therefore, the answer is (C).