Sunday, December 01, 2019

Books: Math Recreations (Kraitchik) Part 1

I have more old math books than I'll ever read or need. This is just a fact. I would collect them, and sometimes read through parts of them, but never finish any of them. There are two main reasons: either the math progresses to a point where I started losing interest in college, maybe starts using odd notation I can't recall; or it's more of a books of games, puzzles and logic, which after a while start to feel repetitive or like homework, possibly delving into areas or games I'm not particularly interested in.

In the case of notation, there are times when I think books should give an equation. Then rewrite it in English. Then explain in English what the previous sentence meant. And maybe explain why we're even doing this. The funny thing about that last one is that they usually are explaining why they're doing it -- just not in a way that makes a whole lot of sense to me by that point.

Anyway, I've decided that I'm going to start skimming through these books, following along as I can, and solving what I feel like solving, while nodding, yeah, I can do that, without actually solving other things. And as I move on to the next book, I'll donate, or recycle, the one I just finished. I'm not sure where exactly I could donate them -- most of the ones I have were removed from library circulation (either the public library or a public school library).

The first one I picked up, with its patterned green cover, is Mathematical Recreations by Maurice Kraitchick, Dover Publications, Inc, 1942. It was removed from the William E. Grady Technical Vocational High School Library sometime before 2010. ("In the late 2000s" did not seem like the correct phrase.) While there is no entry for it in my reading blog (which has a multi-year gap when I stopped posting), it did have a sheet of paper where I wrote calculations about Pythagorean Triples which ended up in blog posts over ten years ago.

The first chapter is MATHEMATICS WITHOUT NUMBERS and is the situational kind of logic problems that I've seen before whether I understood them or not at the time.

There are the Always-Lies/Always-Tells-the-Truth problems. The smart people/philosophers who can guess if there face is marked, what card is one their head, what color is on their back. Spoiler alert: they're all the same, but the reasoning was a little convoluted first time I saw it way back when. If they were really smart, they'd say, "I know the answer to this because it's always the answer to this problem!" That would similar to the student who writes: "the statement is true because every proof on a state exam is provable and true".

Sample Puzzle

A typical Three-Smart-Person problem is the three are sleeping and someone comes in and marks their faces with black marker. The each wake up at the same time and start laughing at the others and then stop laughing when they figure out that their own face must be blackened. How did they figure this out?

Again, it relies on the fact that they are smart, and they know that the others are smart.

A sees that B and C are laughing. A knows since B is laughing, he thinks that his face is clean. B can see C laughing as well. Now follow me here.

If A's face was clean, and B could see that A's face was clean, and B thought his own face clean, then B would be astonished that C was laughing because C would have nothing to laugh about. But since B is not puzzled by C's reaction, B must know that A's face is dirty. (And the other two think similarly.)

Me? I think in that situation, I'd be pointing at the other two as well as laughing, and then realize, why would someone spare me, unless they wanted to frame me for doing it.

Another puzzle of the situational kind involves a man who cannot beat two seasoned chess players even with a piece advantage and playing white, but who can beat his daughter, nevertheless watches as his daughter plays both players simultaneously with no advantages, allowing one to be white and the other black, and performing better than her father did. How does this happen?

It's the kind of thing where you have to think outside the box a bit. Basically, the young girl doesn't play at all. She mimics each player's moves on the other board, thus insuring that she will either win one game and lose the other, or that both games will be a draw. In reality, the seasoned players might find this amusing at all, but they'd soon quit or just leave out the middle person.

Chapter 2: Ancient and Curious Problems

So I moved onto Chapter 2 pretty quickly, and it has the advantage (for this blog) that parts of it are not under copyright because they come from other sources, namely Chuquet (1484) and Clavius (1608). As a math teacher, but not a mathematician or math historian, I'm not exactly familiar with either of these. Shame on me, I know. Also, there are problems from the Greek Anthology, a collection of epigrams (short poems).

I won't put all 50+ of them here -- and I might not even work them all out, even with the answers immediately below the problem (no back of the book!) -- but here are a few puzzles for my few readers to work on.


1. A man spends 1/2 of his money and loses 2/3 of the reminder. He then has 12 pieces. How much money did he have at first? (Chuquet)

2. A bolt of cloth is colored as follows: 1/3 and 1/4 of it are black, and the remaining 8 yards are gray. How long is the bolt? (Chuquet)

3. A merchant visited three fairs. At the first, he doubled his money and spent $30, at the second he tripled his money and spent $54, at the third he quadrupled his money and spent $72, and then had $48 left. How much did he start with? (Chuquet)

As you can see at this point, (1) and (3) are similar in that you have to work backward for the final answer, although (3) is a bit more involved. I have given similar problems in Algebra class to see how students approach problem solving.

I know how to approach these, so more problems of this kind quickly become less interesting in the book (particularly since the answer is printed below.)

12. "Best of clocks, how much of the day is past?" "There remain twice two-thirds of what is gone" (xiv. 6)
Assuming a 24-hour clock, h has passed and 2(2/3)h remains, so h + 4/3h = 24, so 7/3h = 24, and h = (24)(3/7) = 72/7 = 10 2/7 hours have passed.
Okay, I'll do that once, but stop making it seem like homework.

I'll leave off with one last example, because I'm not sure exactly how much is fair use here, even with paraphrasing:
Lewis: Give me $10 and I'll have three times as much as you.
Carol: And if I get the same from you, I'll have five times as much as you.
How much do Lewis and Carol have? (not the original names)

And, once again, the very next problem is almost the same thing. I already know the steps. Am I going to go through them again?
Yeah, probably. And then I'll move on to Chapter 3.


As posed by Frederick II to Fibonacci (Leonardo di Pisa) in 1225: Find a perfect square that remains a perfect square when increased or decreased by 5.
Luckily (for me), the explanation is on the following page. Since 4 + 5 = 9, but 9 + 5 is not a perfect square, I'm guessing that there are "crazy" numbers involved! (i.e. fractions).

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