Coins in Twoland

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In Twoland the coins have value 1, 2, 4, 8, 16, and so on. The most interesting town there is Twoville, where the law requires you to pay with exact change, using at most two of each coin. How many ways are there to pay 1000? What patterns can we discover in the number of ways to pay in Twoville?

This lesson is due to Joshua Zucker. The video features Patty DeJarlais and Brenda Kroschel.