Step 1: Use the oxidation number rule for oxygen.
In most compounds, oxygen has oxidation state
\[
-2
\]
There are \(8\) oxygen atoms in \(Br_3O_8\).
Therefore, total oxidation contribution of oxygen is
\[
8\times(-2)=-16
\]
Step 2: Determine the total oxidation state of bromine atoms.
Since the molecule is neutral, the total oxidation states must add up to zero.
Let the sum of oxidation states of three bromine atoms be \(x\).
Then,
\[
x-16=0
\]
\[
x=+16
\]
Step 3: Check the given options.
We need three oxidation states whose sum is \(+16\).
Option (1):
\[
5+6+5=16
\]
Option (2):
\[
6+4+6=16
\]
Option (3):
\[
7+2+7=16
\]
Option (4):
\[
6+3+7=16
\]
All options satisfy the total oxidation number condition.
Step 4: Use the structure of \(Br_3O_8\).
The molecule \(Br_3O_8\) consists of two bromine atoms in higher oxidation state connected through a central bromine atom in lower oxidation state.
The known oxidation states are
\[
+6,\,+4,\,+6
\]
Step 5: Final conclusion.
Hence, the oxidation states of the three bromine atoms are
\[
\boxed{+6,\,+4,\,+6}
\]