Previous Month | RSS/XML | Current
WEBLOG
December 6th, 2024 (Permalink)
How to Solve a Problem*: Don't Make an Ass of U and Me
To get the most out of this entry, give the following puzzle a shot. This is a difficult puzzle, so don't be too disappointed if you can't solve it completely, but don't give up too easily. When you've tried the puzzle and read the solution, check out my comments below.
Puzzle: Cards on the Table
Instructions: You are sitting at a large table and spread out on the tabletop in front of you is a deck of cards. You know that forty-four of the cards are face down and the remaining eight are face up. However, you are blindfolded and cannot see any of the cards; nor can you tell which side of a card is up by touching it or in any other way. Your task is to separate the cards into two groups with the same number of face-up cards in each group. How do you do it?
Did you assume that the two groups must have an equal number of cards?
Did you assume that you can't turn over any cards?
Try contraction! If you're unfamiliar with contraction or need a refresher, see the first entry in this series*.
Separate eight of the cards from the remaining forty-four, then turn over each of the eight cards1. The two resulting groups will have the same number of face-up cards in each.
Explanation: Perhaps the best way to see that the above solution will actually work is to use contraction―see the first entry in this series. Suppose that, instead of eight face-up cards there were only two. Then, you would separate out two of the fifty-two cards, leaving fifty in the remaining group. It's unlikely that you would happen to pick the two face-up cards, so let's see what happens if both cards are face down. You turn both cards over, which means that both would then be face up. Since there are two face-up cards in the remaining group, that means there are the same number of face-up cards in each group: success! Did you assume that the number of face-up cards had to remain the same?
What if one of the two cards you pick out is face up, you may ask. Then, when you turn over the two cards selected one will still be face up and one face down. Since only one face-up card is left in the main group, both groups have the same number of face-up cards: success again!
Finally, suppose you just happened to choose both the face-up cards: then, when you turn them over both of the selected cards will be face down. Since there are no face-up cards left in the main group, both groups have same number of face-up cards, namely, zero: success yet again!
Therefore, no matter how many face-up cards are among the two you select, if you follow the above procedure you'll always end up with the same number of face-up cards in both groups. You can carry out this same line of thinking for four, six, and finally eight cards, and get the same result, though the reasoning is longer and more tedious.
History: The above puzzle is adapted from one that usually involves a hundred coins instead of a deck of cards, with ninety of the coins heads up on the table and ten tails up, or vice versa. The solution is logically identical. The puzzle has apparently been used as a job interview question by Apple2, but I haven't been able to find out who invented it or when.
Notes:
- Alternatively, you could turn over all forty-four cards in the other group, but that would be a lot more work―Poundstone gives this solution; see the next note. Just don't turn over all of the cards in both groups: that won't help.
- William Poundstone, How Do You Fight a Horse-Sized Duck?: And Other Perplexing Puzzles from the Toughest Interviews in the World (2021), pp. 120-122.
Comments: The first step in problem solving is to understand the nature of the problem. If you get stuck on a problem or it seems impossible to solve, go back and re-check what it is you have to do. It may be that you made a false assumption that rendered the problem unsolvable.
I selected the puzzle above as an example for this entry on problem-solving because it illustrates the danger of making false assumptions. When you began to think about how to solve it, did you assume that you had to split the deck into two equal groups? I did! But you can't solve it that way except by pure luck. Notice that the instructions place no restrictions on how many cards are in each group.
Did you assume that you couldn't turn over the cards? I did! Again, the instructions do not forbid turning the cards over, and you're unlikely to solve the puzzle without doing so.
If you make either of these assumptions, you'll find yourself stuck and the puzzle will appear to be impossible to solve. That's a sign that you should re-read the instructions to check whether you are making any false assumptions.
The title of this entry is based on the old saying that when you assume "you make an ass of you and me", that is, "ass" + "u" + "me" = "assume". However, it is often useful in reasoning to make assumptions, and some standard proof methods, such as conditional proof and indirect proof, do so. However, false assumptions can render a problem unsolvable, so be careful what you assume.
* Previous entries in the "How to Solve a Problem" series:
- Contraction, 4/6/2023
- Think Backwards, 5/5/2023
- Solving a Problem by Elimination, 6/20/2023
- Climbing Up that Hill, 7/5/2023
- Backtracking, 8/14/2023
- Divide and Conquer, 9/11/2023
December 2nd, 2024 (Permalink)
Fool or Nave?
A recent article on the restoration of Notre Dame cathedral in Paris includes the following sentence: "Louise Bausiere, who spent the last two years working on the cathedral's knave, told NBC News Wednesday said [sic] she hoped people would admire what the team of craftspeople had done.1" "Knave" is an old-fashioned word for a dishonest or untrustworthy man2, as well as an alternative name for a Jack in a deck of cards. So, the Knave of Hearts, who stole the tarts in the nursery rhyme, is a Jack of Hearts3. Puzzle fans may be familiar with the type of logic puzzle in which knights always tell the truth and knaves always lie.
So, what could be meant by Bausiere "working on the cathedral's knave"? Does Notre Dame have its own designated rascal? Obviously not; instead, what Bausiere spent a couple of years working on was the nave4. "Nave" and "knave" are pronounced exactly the same, and the only difference in spelling is the silent "k" on "knave", but they mean very different things. A nave is the central part of the floorplan of a church running its length from the front entrance to the transepts5.
Since both "nave" and "knave" are English words, a pure spell-checking program won't catch the substitution of one for the other. I checked the example sentence in several online spelling or grammar checkers and some found nothing wrong with it, others at least noticed the ungrammatical "said", and a couple actually suggested dropping the "k" on "knave".
There's an old proverb that "knaves and fools divide the world"6, but naves and transepts divide a cathedral.7
Notes:
- Keir Simmons, "World gets first look at Notre Dame's new interior 5 years after devastating fire", NBC News, 11/29/2024.
- "Knave", Cambridge Dictionary, accessed: 11/30/2024.
- "The Tarts", The Real Mother Goose (1916).
- "Nave", Cambridge Dictionary, accessed: 12/1/2024.
- "Transept", Cambridge Dictionary, accessed: 12/2/2024.
- William George Smith, "Knaves", The Oxford Dictionary Of English Proverbs (1936), p. 253.
- I found this example here: Ann Althouse, "Beautiful cathedral and I'm delighted at the restoration but what on earth was Louise Bausiere doing for 2 years?", Althouse, 11/29/2024.
Casino Bonuses are not easy to find on the internet. There are simply too many and their terms and conditions makes them difficult to compare. You can find the best bonuses at casinopilot.
You can find the best casinos at MrCasinova.com as this website update online casinos and compare them on daily basis.
November 28th, 2024 (Permalink)
Thanksgiving Chez Blanc
It's Thanksgiving again at the Blanc house* and everyone is there, except Ferdinand, who couldn't make it this year. As usual, the family members have gathered around the big round table in the dining room for the traditional feast. From the following clues, determine the age of each family member and their seating position around the table, starting with Abner and proceeding clockwise.
- Claudia is ten years older than the family member who sits immediately to her left.
- The family member on her immediate right is eight years younger than Edwina.
- Claudia, who is 54, is Beatrix' mother.
- Douglas, who is nineteen, is the only family member younger than Edwina.
- Beatrix is seventeen years older than the person sitting next to her on the left.
It will probably help to draw a circular diagram of the table and the five positions around it.
The seating order, going clockwise around the table, and ages are: Abner, 44; Beatrix, 36; Douglas, 19; Edwina, 27; and Claudia, 54.
Explanation: From clue 4, Douglas is the only member of the family younger than Edwina, so he is sitting to her immediate right, by clue 2. By clue 4, Douglas is nineteen, therefore Edwina is 27, by clue 2, again. From clue 1, Claudia is only ten years older than the person sitting to her immediate left, so this cannot be Beatrix, who is Claudia's daughter, by clue 3. Also, whoever is sitting to Claudia's left must be 44 years old, so it cannot be Douglas, who is nineteen, and it isn't Edwina, who is on Douglas' left. So, the person on Claudia's left is Abner, who is therefore 44 years old. Finally, according to clue 5, Beatrix is seventeen years older than the person on her immediate left: this cannot be either Edwina, who is to the immediate left of Douglas, or Abner, who is to the left of Claudia. So, it's either Douglas or Claudia―but it can't be Claudia, who is Beatrix' mother, which makes it Douglas. Douglas is nineteen, which makes Beatrix 36. Putting these facts together, the seating arrangement is: Abner, Beatrix, Douglas, Edwina, and Claudia.
*For last year's Thanksgiving at the Blanc's, see: A Puzzle to be Thankful For, 11/23/2023.
November 23rd, 2024 (Permalink)
Charts & Graphs1: Half a Graph
The bar chart shown above, which has been widely shared on anti-social media2, has two things wrong with it, one of which has to do with the data and the other with how the data is displayed. But before we get into the graph's deficiencies, let's figure out what it's trying to tell us.
The chart shows what purports to be the popular vote for the two major-party candidates in the last four presidential elections. Following the contemporary color convention, the blue bars represent the Democratic vote and the red the Republican. In addition to the basic data in the chart represented by the height of the bars, there are two indications of what the graph's creator wants us to notice: the long arrow and the question marks. The arrow draws our attention to the popular vote for the Democratic candidates in the 2012, 2016, and this year's elections, which appear to be approximately the same3. In contrast, the three question marks single out the blue bar for the 2020 election, suggesting that there's something questionable about the Democratic vote for that year, presumably because it's so much greater than the surrounding years.
Now that we understand the message of the chart, let's turn to its deficiencies.
The graph is an example of what I call, following Darrell Huff's lead4, a "gee-whiz bar chart"5. If you look closely at the faint numbers in the scale on the left of the graph, you'll see that the baseline is 50 million votes rather than zero, which has the effect of exaggerating the difference in heights between the three short blue bars and the tall one. To the eye the 2020 bar appears to be almost twice as tall as the three others, yet if you read the vote counts off the scale you'll see that the vote in each of those years was about 66 million, whereas the vote in 2020 is shown as about 81 million. So, instead of being twice as big, the 2020 vote was less that a quarter greater. For comparison purposes, to the right is what the graph would look like with the baseline starting at zero.
The purpose of a bar chart is to make it easy to compare quantities by comparing the heights of the bars, since it's easier to compare lengths visually than to compare numbers mentally. Truncating a bar chart defeats this purpose. In addition, if it is necessary to truncate a graph, it's good practice to indicate visually that the bottom has been cut off, making it less likely to mislead the viewer. That was not done with this chart so that it's only by looking closely at the small numbers in the scale that the reader can realize that it's truncated.
Now, let's turn to the problem with the data. The chart seems to have been created the day after the election, so the popular vote for this year was incomplete. If we update the graph with more recent data6―see the revised graph to the right―the arrow from the original would no longer point to the top of the last blue bar, so I've removed it. While it's clear from the revised chart that the popular vote for 2020 is an outlier, its difference from that of two years prior and this year is substantially less than the original chart suggested. Given that 2020 was the first year of the pandemic, and voting by mail was greatly increased, is it really so surprising that the vote for the winner that year was about four million greater than the vote for this year's winner?
Notes:
- Some previous "Charts & Graphs" entries:
- Charts and Graphs, 6/4/2012
- A "Gee-Whiz" Graph, 8/14/2012
- A Charts & Graphs Prize Contest, 5/8/2014
- This appears to be the initial source of the chart: Zero Hedge, "Sorry to beat a dead horse, but can we go back to what happened here?", X, 11/6/2024.
- For the 2012 and 2016 results, see: "United States Presidential Election Results", Britannica, 11/6/2024.
- Darrell Huff, How to Lie with Statistics (1954), Chapter 5: "The Gee-Whiz Graph".
- See: The Gee-Whiz Bar Graph, 4/4/2013.
- "US presidential election results"
November 19th, 2024 (Permalink)
The End of the "Trump Effect"1
I have good news and I have bad news for the pollsters. First, some good news.
The public opinion polls in this election were about as accurate as could be expected. Despite some claims to the contrary, this election was close and not a "landslide" for president-elect Donald Trump2. The most recent results show Trump's margin of victory over Vice President Kamala Harris in the popular vote as a little over 2.5 million votes, or 1.6 percentage points3, which is about half of the usual margin of error for national polls.
Now, the bad news: even though the polls were reasonably accurate this year, that doesn't mean that they were of any use in predicting who would win. The best bet of doing that is through averaging the results of all the polls taken within a week or two of the election. Here's a summary of such averages from a recent article:
The national polling averages…all showed Harris with an advantage: She led by 1.2 percent per FiveThirtyEight, one percent per Nate Silver, 2 percent according to the Washington Post (which rounds numbers), and one percent according to the New York Times (which also rounds numbers). RealClearPolitics…showed Harris leading by 0.1 percent.4
Averaging the averages gives very slightly over one percentage point in favor of Harris. While all of these are near misses, it's interesting that all are misses in the same direction, though just barely in the case of RCP. If these were coin flips, the probability of their all erring in the same direction is only one out of sixteen. Thus, it seems likely that, in an election so close―in which the electorate is divided roughly 50-50―this must be a systematic rather than a random error, that is, a bias. Specifically, it's evidence for the "Trump effect" in this election.
As long as our presidential elections are close―and this has been the case since the turn of the century―even the most accurate polls are not going to help predict the results. Unless and until we return to the years of genuine landslides, such as those in 19645 or 19846, polls and even their averages are simply not precise enough to predict a presidential election.
Let's end with some more good news for the pollsters. While the polls suggested that Harris would win by a nose, the "Trump effect"―that is, the underestimate of the former president's support―was less than in previous contests: between two and three percentage points rather than as much as four points7. Given that Trump is constitutionally prohibited from running for president again, this should be the end of the eponymous effect. We saw in 2022 that when Trump himself was not running, the "effect" seemed to have little or no effect8, and Republicans in general don't appear to be affected by it. Even this year, in some of the states that Trump won, Trump-endorsed senate candidates lost9. So, four years from now we won't have to worry about Trump skewing the polls.
Notes:
- This entry is partly based on the following previous entry: "It ain't over till it's over.", 9/23/2020.
- See: Did Trump Win in a "Landslide"?, 11/14/2024.
- "US presidential election results", Reuters, 11/19/2024.
- Ed Kilgore, "What the Harris vs. Trump Polls Got Wrong", New York Magazine, 11/6/2024.
- "United States presidential election of 1964", Britannica, 10/27/2024.
- "United States presidential election of 1984", Britannica, 10/30/2024.
- G. Elliott Morris, "2024 polls were accurate but still underestimated Trump", ABC News, 11/8/2024.
- See: A Post-Election Trump Effect Post-Mortem, 11/19/2022.
- Riley Beggin, "Why did Democrats win Senate races in so many states Trump won? Ticket splitters", USA Today, 11/9/2024.
November 14th, 2024 (Permalink)
Did Trump Win in a "Landslide"?
Some commentators on the recent election are claiming that former president, and now future president, Donald Trump, won in a "landslide". Here, for instance, is a headline from USA Today:
Trump beat Harris in a landslide.
Will his shy voters feel emboldened?1
More predictable is the following headline from the conservative Washington Times:
Trump's landslide win redraws electoral map,
shatters Democratic strongholds2
"Landslide" is, of course, a vague word, so there is no exact number or percentage of votes that defines it. It's also ambiguous in American presidential politics as to whether the alleged "landslide" occurred in the popular vote or in the electoral college. Let's look first at the popular vote.
That vote is still not entirely counted3, but the remaining votes will probably not be enough to change the following analysis. According to the most recent results4, Trump received a little less than 76 million votes and Vice President Kamala Harris got a bit less than 73 million, so that Trump won by about three million votes.
In percentage terms, Trump received a bit more than 51% of the vote and Harris a little less than 49%, so that Trump not only won the popular vote but also a majority of it. However, Trump's margin of victory over Harris was only a bit more than two percentage points, and it's possible that that margin will decrease as additional votes come in from innumerate states.
Conveniently, an article also from USA Today provides some historical context5: it includes both the popular vote margins of victory and electoral college totals for the previous ten president elections. In those ten contests, the only candidate who actually won both the electoral college and the popular vote, but whose margin of victory was less than Trump's, was George W. Bush in 2004 with just over three million votes. Joe Biden beat Trump in the previous election by a margin about twice as large as Trump's this year. So, if Trump's victory was a "landslide", then Biden's was surely an even bigger one.
Similarly, in the electoral college, Trump won 312 votes, which is safely above the 270 needed, and six votes greater than his 2016 win. However, Barack Obama received 365 and 332 in 2008 and 2012, respectively. If at least 312 electoral votes is a "landslide" in the electoral college, then six of the previous ten elections were landslides.
So, if Trump's victory was not especially large in either the popular vote or electoral college, why do some journalists seem to think it was a "landslide"6? One factor is probably the result of youth and inexperience, together with historical illiteracy. Another is shock that Trump not only won but that, unlike 2016, he won both the electoral college and the popular vote.
Notes:
- Charles Trepany, "Trump beat Harris in a landslide. Will his shy voters feel emboldened?", USA Today, 11/6/2024.
- Susan Ferrechio, "Trump's landslide win redraws electoral map, shatters Democratic strongholds", The Washington Times, 11/6/2024.
- I don't know why it takes some states so long to count ballots. It's now over a week since election day, and there's no reasonable excuse for it taking so long, especially when we have computers.
- See: "2024 General Election Results", The Hill, accessed: 11/13/2024.
- Rachel Barber, "How big was Trump's win? How it compares with the past 10 presidential victories", USA Today, 11/6/2024.
- At least one journalist knew better, see: Zachary B. Wolf, "Trump's win was real but not a landslide. Here's where it ranks ", CNN, 11/9/2024.
November 10th, 2024 (Permalink)
Subsiding or Subsisting?
Most of the pairs and triplets of words that we've looked at in this series of entries on easily confused words have been homophones, that is, words that are spelled differently but sound alike. However, there is a smaller class of pairs that neither sound alike nor are spelled alike, yet are still confusible, for example, "perpetrate" and "perpetuate"1: the problem with such pairs is that they are so similar looking as to be easily misread, one for the other, by a quick or casual glance.
The following sentence from a recent newspaper article introduces another such confusion: "The six Israelis were apparently kept in this tunnel system the entire remainder of their lives, subsiding mostly on energy bars, according to the report."2 "To subside" means to sink or lessen3. While I've no doubt that the six hostages who were forced to live on glorified candy bars for months did figuratively sink lower, and their health and strength lessened, nonetheless I don't think that this is what the article's authors meant.
"To subsist" means to survive4, and when used in the phrase "subsist on" it means to barely survive, as in "to subsist on nuts and berries". While I can't prove it, I suspect that the authors of the news report meant to write that the hostages subsisted on energy bars, rather than that they "subsided" on them.
None of the usual reference books I check warns of confusing "subside" for "subsist", which suggests that perhaps it isn't a common confusion. Yet, a quick internet search reveals another example from a webpage for a seller of hiking equipment: "Contrary to popular belief, hikers who engage in extended hiking expeditions are not forced to subside on nuts and berries.5"
Both of these examples go one way, that is, "subside" takes the place of "subsist", rather than the other way around. Specifically, it's the phrase "subsist on" something that seems to cause the confusion. Other than the fact that the words look alike, I can't imagine why this happens.
Since "subsiding" and "subside" are both spelled correctly in the example sentences, a pure spell-checking program won't catch the mistake and, since both words are verbs, even a grammar-checking program is not likely to do so. I tried both of the example sentences in several free online checkers, and not a single one flagged "subside" as an error, so don't count on such programs to save you. In contrast, I had the following exchange with ChatGPT:
Me: Hi, can you help me with an English language problem?
ChatGPT: Of course! I'd be happy to help. What's the language problem you're working on?
Me: Is there anything wrong with the following sentence: The six Israelis were apparently kept in this tunnel system the entire remainder of their lives, subsiding mostly on energy bars, according to the report.
ChatGPT: Yes, there is a small issue with word choice in this sentence. The word "subsiding" is not appropriate in this context. "Subsiding" means to sink, diminish, or decrease, which doesn't fit the intended meaning here. The correct word would be "subsisting," which means to survive or sustain oneself, typically with minimal resources.
Here's the corrected sentence: "The six Israelis were apparently kept in this tunnel system the entire remainder of their lives, subsisting mostly on energy bars, according to the report." Let me know if you need further clarification!6
I'm skeptical about ChatGPT7 but, at least in this one case, it was more useful than dedicated spell-checking programs.
Notes:
- See: Perpetrate or Perpetuate, 7/2/2021.
- "Six slain hostages were likely Sinwar's shields, lived for months on energy bars", The Times of Israel, 10/20/2024.
- "Subside", Cambridge Dictionary, accessed: 11/9/2024.
- "Subsist", Cambridge Dictionary, accessed: 11/9/2024.
- "Backpacking Basics", Vargo, accessed: 11/9/2024.
- "Private Exchange", ChatGPT, 11/10/2024.
- See:
- Are you smarter than an artificial intelligence?, 6/1/2024
- Child's Play?, 7/1/2024