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I see where Ralph flashes a conspiratorial grin. He has been thinking about a way to get the others to listen to them and he has come up with a plan. In the next few pages he talks about how the Navy has a lot of maps and they know where every island is so it is only a matter of time before the Navy finds them. His "conspiracy" is to have fun while helping the Navy locate ther right island by building a fire. By suggesting a plan which allows the boys to have fun, he asserts authority over Jack. His plan is also representative of the civilized nature winning over the savage nature represented by Jack's suggestion to hunt pigs and hunt the snake and kill it.