Musings on mathematics and teaching.

Month: August, 2011

A Paradoxical Dissection

Is it possible to dissect an 8×8 square and rearrange the pieces to form a 13×5 rectangle? Common sense indicates that this cannot be done, because the area of the square is 64, while the area of the rectangle is 65. However, this animation seems to show that it is possible. What is going on?



Sam Loyd's Paradoxical Dissection


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How much is a trillion dollars?

The recent budget debates have forced Americans to think about truly staggering amounts of money. We talk about trillions of dollars almost glibly, but do we really understand what a trillion dollars is worth? An article on NPR’s website, titled The Trouble With Trillions, mentioned several ways to think about a trillion dollars. I thought that some of the suggestions were more helpful than others. To get to the bottom of this, I asked a survey question on Twitter.

Which of the following statements is most useful for comprehending $1 trillion?

    1. If you spend $1 million a day for a million days, you’ll reach $1 trillion.
    2. To spend $1 trillion in the average American life span of 77 years, you will have to shell out about $35,580,857 every day.
    3. $1 trillion represents $3,333 per person in the United States.
    4. One trillion seconds add up to 31,688 years.
    5. A stack of a trillion $1 bills would be 67 miles high.

There were 17 respondents including myself. Here were the survey results:

5 for A, 6 for B, 2 for C, 2 for D, 2 for E

The results were surprising to me, because I felt that there was only one good answer to this question. I will explain my reasoning, and you can argue with me in the comments.

  • The weakest answer is D, because it doesn’t even talk about money. It could be useful for conveying the idea of a trillion, but it is not useful for understanding a trillion dollars.
  • Option E does convey a vivid image of what a trillion dollar bills would look like, but it does not convey the important thing, which is the purchasing power of this stack of bills. The interpretation is too literal to be helpful.
  • Options A and B try to reduce a trillion dollars to a human scale by showing what it would be like to spend a trillion dollars. These are better choices than D or E, but it is a mistake to describe a trillion dollars in these terms. No single person could possess this much money; such quantities are reserved for describing the economies of large countries.
  • This leaves option C, which is by far the best choice. The statement gives a very specific idea of what it means when the US government spends a trillion dollars; every person’s share of that expenditure is about $3,000. This is the kind of reference point that a citizen needs in order to participate in debates about federal budget priorities. (Admittedly, a person who does not reside in the US would find this choice less compelling.)

How should we explain large numbers to others? Please share your thoughts in the comments.