Alfred S. Posamentier and Ingmar Lehmann, who previously wrote the excellent book The Fabulous Fibonacci Numbers (1997), have teamed up once again to write Magnificent Mistakes in Mathematics (Prometheus Books, 2013).

It’s a book with a self-explanatory title, and you only need a high-school-level background in math to understand it.

The following are excerpts from the book, reprinted with permission of the publishers:

Stumbling Blocks: Numbering Mistakes

There are oftentimes counting errors that occur and remain unnoticed for a long time. On January 1, 2000, the New York Times corrected an error that had been made more than one hundred years earlier. On February 6, 1898, an employee of this newspaper noticed that the current day’s issue was number 14,499. And so he mistakenly gave the following day’s issue number 15,000, instead of 14,500. It was not until Saturday, January 1, 2000, that this mistake was corrected. On that day, issue number 51,254 was published, while the previous day’s issue was numbered 51,753. In case you’re wondering, issue number 1 of the New York Times was published on September 18, 1851.

Some printing errors are not as easy to correct as that of the New York Times’s numbering system. We have been shown through the Popeye comics that spinach brings a person extra strength. This is largely a result of some misunderstandings — or perhaps mistakes — about the iron content in spinach. Spinach has approximately 3.5 mg of iron per 100 g of spinach; while in its cooked version, it contains about 2 mg, which turns out to be a lot less than bread, meat, or fish. This misunderstanding of spinach’s iron value stems back to the 1930s, when there was a printing mistake. A decimal point was mistakenly moved one place to the right, which, of course, gives a value ten times that of the intended value. One might see this mistake as one where the value of iron in spinach was given ten times the correct amount, namely, 35 mg per 100 g of fresh spinach. With Popeye’s reinforcement, many children grew up imagining spinach as an instant source of power.

Rounding off Can Cause a Mistaken Answer

There are times when rounding off correctly leads to a wrong answer. Consider the following example: At an airport there are 963 stranded passengers. Busses are ordered to take these passengers on their way. Each bus can hold 59 passengers. The question is, how many buses are required to take these passengers on their way? Typically, a student would do the following calculation. 963 ÷ 59 ≈ 16.32203389. Since the number of buses must be a whole number, the student would round off the answer correctly to 16, since the 3 following the decimal point is less than 5. This answer clearly does not solve the problem. And although the calculation was correct, the problem was solved incorrectly. Obviously seventeen buses will be required, where the seventeenth bus will not be completely full. Here is an example of a mistaken answer despite the fact that correct calculations were made.

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A Mistake Based on Prematurely Jumping to Conclusions

Suppose you take a circle, put some dots along the outside, and then connect them. If only two lines cross at any point, into how many regions will the circle be divided?

As we close this chapter on arithmetic mistakes, we should notice that sometimes what is seen as a mistake may, in fact, not be a mistake at all. Consider the following sequence and ask that for the next number: 1, 2, 4, 8, 16. Most people would assume that 32 will be the next number. Yes, that would be fine. However, when the next number is given as 31 (instead of the expected 32), cries of “wrong!” are usually heard.

Much to your amazement, this is can also be a correct answer, and 1, 2, 4, 8, 16, 31 can be a legitimate sequence, and not a mistake!

The task now is to be convinced of the legitimacy of this sequence. It would be nice if it could be done geometrically, as that would give evidence of a physical nature…

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To be further convinced that this sequence is legitimate and not resulting from mistakenly replacing the “32” with a “31,” we shall consider the Pascal triangle. This triangle is formed by beginning on top with 1, then the second row has 1, 1, and the third row is obtained by placing 1s at the end and adding the two numbers in the second row (1 + 1 = 2) to get the 2; the fourth row is obtained the same way. After the end 1s are placed, the 3s are gotten from the sum of the two numbers above (to the right and left), that is, 1 + 2 = 3, and 2 + 1 = 3.

The horizontal sums of the rows of the Pascal triangle to the right of the bold line drawn: 1, 2, 4, 8, 16, 31, 57, 99, 163. This is again our newly developed sequence.

A geometric interpretation should further support the legitimacy of this sequence and support the beauty and consistency inherent in mathematics. To do this, we shall make a chart [see below] of the number of regions into which a circle can be partitioned by joining points on the circle, where no three lines meet at one point; otherwise a region would be lost.

Let’s focus on the case where n = 6. [See below]

We notice there is no thirty-second region.

Now that you see that this unusual sequence (1, 2, 4, 8, 16, 31, 57, 99, 163…) appears in various other contexts, you should be convinced that even though there appeared to be a mistake at the original introduction of the “31,” there was, in fact, no mistake. Thus, even mistakes can be deceptive — or mistakenly identified as mistakes!