What is **impossible** is, given an arbitrary angle, to construct an angle whose measure is **exactly** one-third of that of the given angle, using an (ideal) compass and unmarked straight-edge. Use of those tools corresponds to application of **theorems** of Euclidean geometry, by means of which the construction (if it could be done) would be **proved** to be exact.

What is **possible** is a means of constructing an angle whose measure is **approximately** one-third of that of the given angle, accurate to **some specified maximum error**. Such an algorithm might be repeated as often as necessary to attain any desired level of accuracy, short of exactness.

These are **two separate problems**. We do not want students to be confused; we want it to be clear that solving the second problem is **not a great challenge** and so not of interest mathematically, but that the first problem has been **proved to be absolutely impossible**. What’s of mathematical interest is not any attempt to solve the first problem (which would be futile), but rather the proof of impossibility. The latter has been accomplished, and involves some pretty deep and beautiful mathematics.

Yes, this is a method I commonly use (and, in fact, used just today in face-to-face tutoring, when a student was unsure which way to move the decimal point in the usual method). I mentioned it in the first post in this series, namely Percent Change: Finding and Applying It. Here is a quote:

I also like to think of the multiplication by 100 as a unit conversion, where we multiply a number (typically a decimal) by 1, in the form of 100%, to get an answer measured in percent.

I haven’t had occasion here to show the division by 100%, but it, too, can be very useful.

]]>I gave the puzzle a try, using Heron’s formula, and found that, indeed, it doesn’t even require a calculator! With a little algebra (well, a lot for most students!), you can rearrange the formula to $$K^2=\frac{4a^2b^2-(a^2+b^2-c^2)^2}{16}\\=\frac{4\cdot74\cdot116-(74+116-370)^2}{16}\\=37\cdot58-45^2=121$$ So the area is 11 acres.

Not having quick access to the book, I had to search to find the puzzle, and when I did, what I found includes both Loyd’s own solution, and a solution that is, essentially, just what I had done:

Interestingly, there is no mention of the curious unit involved; they just use the numbers without mentioning units.

Thanks for an interesting diversion!

]]>It’s true that a < s, that is, a < (a+b+c)/2, is equivalent to a < b+c, the triangle inequality. And it is true that the fact that Heron's formula has a radicand of s(s-a)(s-b)(s-c), which must be positive. But I'm not sure I'd exactly say that the latter "instantly" leads to the former. The radicand implies only that *either* none, or two, or all four of the factors is positive, so you would have to prove that you can’t have, say, s-b and s-c being negative while s and s-a are positive. For me, at least, that requires more than a moment of pondering.

It’s an interesting observation, though.

]]>Unit conversion by unit multipliers (“dimensional analysis”) can be extended to conversions between percentages and decimals: 100% = 1, so that the unit multipliers 100%/1 and 1/100% are legitimate. Applying them can be thought of as multiplying or dividing by 100% — the former when a percentage is wanted, the latter when one is present.

0.75 x 100% = 75% (do the math, append the % sign).

47%/100% = 0.47 (do the math, cancel the % signs).

]]>An interesting aspect of the problem is that the side lengths of the triangle are best considered to have the length unit acre^1/2, which I had never contemplated before. The unit appears in the Heron formula and then disappears as it yields the triangle’s 11-acre area. (It is clear that 1 acre^1/2 = 66√10 feet. If the parcel areas were considered as being in hectares, the corresponding length unit would be the hectare^1/2, which is a hectometer — not a novel unit at all.)

*Quick, at least, with an HP-50g calculator.

]]>I don’t see that this helps factor the expression on my last line; but there’s probably a lot I don’t know about symmetric polynomials.

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