The average of six numbers is 35. If each of the first three numbers increased by 4 and each of the remaining three is decreased by 8, then what is the new average?

Let the six numbers be \(a_1, a_2, a_3, a_4, a_5, a_6\).

Given that the average of the six numbers is 35, we have:

\[
\frac{a_1 + a_2 + a_3 + a_4 + a_5 + a_6}{6} = 35
\]

Multiplying both sides by 6 gives:

\[
a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 6 \times 35 = 210
\]

After the operations, the new sum of the numbers will be:

\[
(a_1 + 4) + (a_2 + 4) + (a_3 + 4) + (a_4 – 8) + (a_5 – 8) + (a_6 – 8)
\]

\[
= (a_1 + a_2 + a_3) + 3 \times 4 + (a_4 + a_5 + a_6) – 3 \times 8
\]

\[
= (a_1 + a_2 + a_3) + 12 + (a_4 + a_5 + a_6) – 24
\]

\[
= (a_1 + a_2 + a_3 + a_4 + a_5 + a_6) + 12 – 24
\]

\[
= 210 – 12
\]

\[
= 198
\]

Therefore, the new average is:

\[
\frac{198}{6} = 33
\]

So, the new average is 33.