Understanding the Problem

Problem

A $19^{th}$ century naval commander devised an ingenious means of communicating with other ships in their fleet. They employed light signals by devising a set of special bulbs, each capable of toggling between red and green states. Intrigued by their potential, they explored how different arrangements could convey different messages. Initially, with only two bulbs, they found that they could transmit four distinct messages by altering the bulbs’ states.

As they delved deeper, experimenting with larger arrays of bulbs, they pondered the potential messages that were achievable with a 10-bulb array. Counting arrangements with up to four bulbs was manageable, but they sought assistance as the complexity grew with each added bulb. Their future endeavors hinged on the number of messages they could convey with these bulbs. Can we assist them in deciphering this puzzle?

Note: “R G” and “G R” are two different combinations.