Monday, July 20, 2015

List of animal names sorted by syllable count

The list I said I would make is now available at

(Click to read the name of a 15-syllable animal!)

The document is open for comments. Perhaps it will grow over time.

Sunday, July 19, 2015


My kids enjoyed this puzzle—I thought my readers & their kids might enjoy it too:
  • A one-syllable animal: _________________
  • A two-syllable animal: _________________
  • A three-syllable animal: _________________
  • Etc.
The Internet can be helpful for some of the higher syllable counts. But on a quick search, I actually didn't find a super-convenient list of animal names sorted by syllable count. For my next post, I'll make a start at one.

Animal names don't have to be single words; lake trout is an animal name for the purposes of this puzzle. Also, it is not the intention here to use scientific Latin names, although that version of the puzzle would be fun too.

Wednesday, July 8, 2015

Bad Johnny and the Fractions, Cont'd

As punishment for misbehaving, Johnny's teacher made him compute \(\frac{6446266}{9669399} - \frac{6666442}{9999663}\). "That's easy," said Johnny, as he sauntered out of the classroom. "It's the same thing as \(\frac{2446666}{3669999} - \frac{2446666}{3669999}\), which is zero."

Reader jeff solved this puzzle in brief. Here's a more leisurely walkthrough, in case interesting.

If you see a fraction like \(\frac{24}{48}\), it's pretty easy to realize that it equals \(\frac{1}{2}\). How about a fraction like \(\frac{1234332}{2468664}\)? This is also \(\frac{1}{2}\). How can you tell? One way is to compare the numerator and the denominator one place value at a time. If you do that, you'll see that each digit in the denominator is twice as large as its corresponding digit in the numerator. Therefore the denominator itself is twice as large as the numerator.

The reasoning behind this "twice as large" example works for other ratios as well. For example, in the fraction \(\frac{123}{369}\), each digit in the denominator is three times as large as the corresponding digit in the numerator. That makes the denominator itself three times as large as the numerator—so the fraction equals \(\frac{1}{3}\).

In the cases above, we "sized" the denominator relative to the numerator, but you could just as well reverse the comparison. Thus, in \(\frac{123}{369}\), we can notice (along the lines of the reader's comment) that each digit in the numerator is one-third as large as the corresponding digit in the denominator. That makes the numerator itself one-third of the denominator—so the fraction equals \(\frac{1}{3}\).

Those two ways of looking at \(\frac{123}{369}\) could be expressed respectively as

\(\frac{123}{369}=\frac{123}{300+60+9} = \frac{123}{3\cdot100+3\cdot20+3\cdot3} = \frac{123}{3\cdot(100+20+3)} = \frac{123}{3\cdot123} = \frac{1}{3}\)


\(\frac{123}{369} = \frac{100+20+3}{369} = \frac{\frac{1}{3}300 + \frac{1}{3}{60}+\frac{1}{3}{9}}{369} = \frac{\frac{1}{3}(300+60+9)}{369} = \frac{\frac{1}{3}\cdot369}{369} = \frac{1}{3}\).

One last example: In \(\frac{462}{693}\), each digit in the numerator is two-thirds of the corresponding digit in the denominator. That makes the numerator itself two-thirds of the denominator—so the fraction equals \(\frac{2}{3}\).

When every digit in the numerator is a constant multiple of its corresponding digit in the denominator, that remains true even if the digits are permuted, as long as the digits in the numerator and the digits in the denominator are permuted in the same way. Thus \(\frac{462}{693} = \frac{426}{639} = \frac{246}{369}\), etc. One way to understand Johnny's cheeky response is to observe that he has permuted digits in the two given fractions in such a way as to make them manifestly identical.

Exercise: \(\frac{66336363}{88448484} - \frac{12221}{48884}\) = ?


The principles at work in this puzzle might merit a little more discussion. If so, consider this scenario about a hypothetical hotel that has one guest room, one boardroom, and one ballroom. In each of the three rooms there is a party going on.

  • In the guest room there are 3 people, 2 of whom are crashing the party. Clearly, in this room \(\frac{2}{3}\) of the people are crashers.
  • In the boardroom, there are 60 people, 40 of whom are crashing that party. So in this room, \(\frac{40}{60} = \frac{2}{3}\) of the people are crashers too. 
  • In the ballroom, there are 900 people, 600 of whom are crashing that party. So in this room as well, \(\frac{600}{900} = \frac{2}{3}\) of the people are crashers.

What fraction of people in the entire hotel are crashers? One way to find out is to add up all of the crashers and then divide by the total number of guests: \(\frac{600 + 40 + 2}{900 + 60 + 3}=\frac{642}{963}\). But another way is to realize that since \(\frac{2}{3}\) of every room is crashers, \(\frac{2}{3}\) of the entire hotel is crashers. The two methods must agree, so \(\frac{642}{963}=\frac{2}{3}\).


More abstractly, we can say that if \(\frac{a}{b}=\frac{c}{d}\), then for any \(k\), \(\frac{ka+c}{kb+d} = \frac{c}{d}\). (One can verify this by cross-multiplying. For simplicity I'm taking all of the variables to be positive.) Applying this principle with \(\frac{a}{b}=\frac{4}{6}\), \(\frac{c}{d}=\frac{2}{3}\), and \(k=10\), we have \(\frac{10\cdot4+2}{10\cdot6+3} = \frac{2}{3}\). Applying the principle once more with \(\frac{a}{b}=\frac{600}{900}\), \(\frac{c}{d}=\frac{40 + 2}{60 + 3}\), and \(k=100\), we have \(\frac{100\cdot6+40 + 2}{100\cdot9+60 + 3} = \frac{40 + 2}{60 + 3} = \frac{2}{3}\). And we could keep going to create larger and larger fractions, all equal to \(\frac{2}{3}\). Choosing the \(k\)'s to be powers of 10 creates a series of equivalences that can be read off directly from the digit strings of the multi-digit numbers in the numerator and denominator.

Solution to Fourth of July Puzzle: Out of Many, One

Challenge: Form 1 out of the following fractions:
3/5, 1/3, 2/3, 2/3, 2/3, 2/3, 2/3, 3/4.
In other words, create an expression, the value of which is 1, using all of the above fractions together with any or all of the symbols \(+\), \(-\), \(\times\), \(\div\), and parentheses.

My solution was

\(\frac{3}{5}\times\left(\frac{3}{4}-\frac{1}{3}\right)\times\left(\frac{2}{3}+\frac{2}{3}+\frac{2}{3}+\frac{2}{3}\right)\div\frac{2}{3} = 1\).

This uses 7 binary operation symbols and two sets of parentheses. Reader jeff's solution was

\( (\frac{2}{3}\div\frac{3}{5})+(\frac{2}{3}\div\frac{3}{4})-(\frac{2}{3}\div\frac{1}{3})+(\frac{2}{3}\div\frac{2}{3}) = 1\).

This solution also uses 7 binary operation symbols, but it's better than mine in the sense since that it requires no parentheses. (The parentheses here add clarity but could be removed leaving a well formed expression equaling 1.)

Sunday, July 5, 2015

Bad Johnny and the Fractions

As punishment for misbehaving, Johnny's teacher made him compute \(\frac{6446266}{9669399} - \frac{6666442}{9999663}\). "That's easy," said Johnny, as he sauntered out of the classroom. "It's the same thing as \(\frac{2446666}{3669999} - \frac{2446666}{3669999}\), which is zero."

Saturday, July 4, 2015

Fourth of July Puzzle

Out of Many, One

In the Constitution that was adopted in Philadelphia in 1787, there appear the following fractions:
3/5, 1/3, 2/3, 2/3, 2/3, 2/3, 2/3, 3/4.
For example, the fraction 3/5 appears infamously in Article I, Section 2, while the fraction  3/4 appears in Article V.

Challenge: Form 1 out of these fractions by using the operations of addition, subtraction, multiplication, and/or division.

In other words, create an expression, the value of which is 1, using all of the above fractions together with any or all of the symbols \(+\), \(-\), \(\times\), \(\div\), and parentheses.

Tuesday, June 30, 2015

Just Who Do We Think We Are?

That was the jarring question posed by Chief Justice John Roberts in his vehement Obergefell dissent (one of the court cases I discussed in my earlier post). The question punctuated his argument that since he could not find for the plaintiffs on constitutional grounds, their fate ought to be left with the states. In his conclusion, Roberts forcefully distinguished between public policy and constitutional law:
If you are among the many Americans—of whatever sexual orientation—who favor expanding same-sex marriage, by all means celebrate today’s decision. Celebrate the achievement of a desired goal. Celebrate the opportunity for a new expression of commitment to a partner. Celebrate the availability of new benefits. But do not celebrate the Constitution. It had nothing to do with it.
Those last two sentences appeared gratuitous to some observers, and I have to agree, if only because it seemed to me that Roberts never really tackled Kennedy's core argument head-on. Instead of dismantling the majority's approach, he championed alternatives to it and raised concerns about its implications. Points taken—and the dissent may live on in some fashion in Fourteenth Amendment jurisprudence. But without a more direct assault on Kennedy's reasoning, the dissent doesn't fully earn such a categorical conclusion.

While I've been following the legal back-and-forth pretty closely over the past year or two, I realize that same-sex marriage isn't just a question of the finer points of law. Obergefell has major life implications for real people. As far as that goes, I understand why progressives are happy about the decision. What I don't understand is why more conservatives aren't happy about it. Andrew Sullivan has been making the conservative case for marriage equality for decades. And there is a "live and let live" aspect to the Obergefell decision that makes it a victory for small-government conservatives, too; see, for example, this essay on the libertarian website (Maybe anti-government conservatives actually have a soft spot for government power after all, as long as it's state and local goverments that are wielding it.)

The elephant in the room for the Obergefell decision is bias. Bigotry against gay people is widespread, even among elected officials. A state senator in North Dakota, Rep. Dwight Kiefert, recently posted this to his Facebook page:

The article continues:
Kiefert said in the interview that his post was not a "public statement," but merely a request that people to look into the issue.
"There's a lot of question marks," he said."Was what I read true? I don't know.
"The bottom line is: Through my faith, I have to oppose it," Kiefert, who is Christian, said of homosexuality.
He added that his viewpoints have earned him threats in the past.
"I just hope people respect me for who I am," he said
Ironic, that last. In any case, I haven't addressed hatred or bias at any length because the Supreme Court in Obergefell didn't address it either. The plaintiffs weren't alleging bias, the states weren't admitting to bias, and the justices didn't explore the issue in any depth. However, some of the lower court decisions did consider the issue of bias; here is the blunt language of the Seventh Circuit:
Our pair of cases is rich in detail but ultimately straightforward to decide. The challenged laws discriminate against a minority defined by an immutable characteristic, and the only rationale that the states put forth with any conviction—that same-sex couples and their children don’t need marriage because same-sex couples can’t produce children, intended or unintended—is so full of holes that it cannot be taken seriously.
(The author of this decision was judge Richard Posner; he offered his take on the Obergefell case here.)

As Rep. Kiefert helpfully reminds us, the other major consideration here is religion. Probably a fair number of conservative politicians are saying what they're saying about same-sex marriage not because same-sex marriage particularly bothers them, but because they can't afford to lose the support of people who vote on the basis of religious views.

In the wake of Obergefell, I suspect we'll be seeing more court cases about the limits of religious freedom under the Constitution. Like all rights, First Amendment rights are not absolute, but enumerated rights in the constitution do receive a lot of deference in law. Some county clerks are currently refusing to issue marriage licenses for reasons of religious conscience. The First Amendment guarantees the free exercise of religion, and many of the people quoted in the linked article sound sincere. Nevertheless, it seems to me that those clerks aren't freely exercising their religion—they're forcing their religion on others using the power of the State. I don't think that will hold up legally under the First Amendment.

Some people seem to feel that we Americans means us Christians. That concept doesn't make us a country, though—it makes us a folk. For myself, I think that what we are as countrymen has less to do with cultural things like religion or language, which we inherited from Europe, and everything to do with the Constitution, which set us apart from Europe.

Who do we think we are? The question Roberts posed to his brethren applies to the rest of us as well. Since the Obergefell decision was handed down, I haven't been able to get this question out of my mind. With due respect to the Chief Justice—and applying only the capacities for reason and reflection that all citizens must use to make up their minds—I'm pretty well convinced on the merits that the Supreme Court has just overthrown an unconstitutional law. That is worth celebrating.