If eight apples cost the same as four bananas, and two bananas cost the same as three cucumbers, how many cucumbers can Tyler buy for the price of 16 apples?
Since 8 apples cost the same as four bananas, we see that 16 apples cost the same as 8 bananas.  Similarly, 2 bananas cost the same as 3 cucumbers, so 8 bananas cost the same as 12 cucumbers.  Hence, 16 apples have the same price as $\boxed{12}$ cucumbers.