We want to motivate this study of algebra with three problems that we hope you will find interesting. Although we eventually solve them in this text, it might surprise you that in this class, we’re interested not in the solutions, but in why the solutions work. I could tell you how to solve them right here, and we’d be done soon enough; on to vacation! But then you wouldn’t have learned what makes this course so beautiful and essential. It would be like walking through a museum with me as your tour guide. I can summarize the purpose of each displayed article, but you can’t learn enough in a few moments to appreciate it in the same way as someone with a foundational background in that field. The purpose of this course is to give you at least a foundational knowledge in algebra.

Take twelve cards. Ask a friend to choose one, to look at it without showing it to you, then to shuffle them thoroughly. Arrange the cards on a table face up, in rows of three. Ask your friend what column the map is in; call that number α. Now collect the cards, making sure they remain in the same order as they were when you dealt them. Arrange them on a table face up again, in rows of four. You must maintain the same order; the first card you placed on the table in rows of three must be the first card you put on the table in rows of four; likewise, the last tag must remain last. The only difference is where it lies on the table. Ask your friend again what column the map is in; call that number β and free tool for scientific converter.

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