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September 27th, 2023 (Permalink)

When is a poll an outlier?

An "outlier" is a data point that is far enough away from the others in a data set to stand out. As far as I know, there is no precise definition of how far away such a point must be to count as an "outlier"1. I mention this because a few days ago The Washington Post (WaPo) published a report on a recent poll it sponsored in partnership with ABC News. The subtitle of the article reads: "A finding that shows Trump leading Biden by a wide margin does not match other recent polling, however, suggesting it is an outlier"2. The article goes on to say:

The Post-ABC poll shows Biden trailing Trump by 10 percentage points at this early stage in the election cycle, although the sizable margin of Trump’s lead in this survey is significantly at odds with other public polls that show the general election contest a virtual dead heat. The difference between this poll and others, as well as the unusual makeup of Trump’s and Biden’s coalitions in this survey, suggest it is probably an outlier.

Given that the election is over a year away, it's a little silly to already be over-reacting to polls, but most of the establishment media have over-reacted to this one. The WaPo itself was obviously aware of the effect that publication of the results would have, which explains the unusual dismissal of its own poll as an "outlier".

As I've complained previously, news organizations that sponsor polls usually report on only their own and ignore others conducted at about the same time3. This is a bad practice since, given the large number of polls conducted, there are bound to be ones whose results are off by more than the usual margin of error4.

What's interesting and unusual in this case is WaPo calling its own poll an "outlier". Wondering whether this was the first time that WaPo had done this, I searched its website using the keywords "poll" and "outlier", and it returned no results for the last two years. Unfortunately, that's as far back as the search function appears to go. I don't recall any previous case of a news outfit deprecating its own poll, but perhaps I just never noticed it before.

Will WaPo now regularly compare the results of its own polls to those of other concurrent ones? I'd like to think so, but I suspect that if the results had been the reverse―that is, if the poll had shown Biden ahead of Trump by the same amount―it would not have called it an "outlier". If the tables had been turned, it would have been just as far away from other polls, but it would have been good news for most of WaPo's regular readers.

As I mentioned, this poll was co-sponsored by ABC News, but ABC's report on it does not use the word "outlier"5, though it does say the following:

Head-to-head in a hypothetical November 2024 matchup, Trump has 51% support while Biden has 42%―numerically up 3 points for Trump and down 2 points for Biden from an ABC/Post poll in February, shifts that are not statistically significant. There's even less change from the most recent ABC/Post poll in May, which had the race at 49-42%…. Still, with Trump inching over 50%―and other polls showing a closer contest―a close look is warranted.6

These facts indicate that the poll's recent results are probably just due to sampling error. As I mentioned above, "outlier" lacks a precise definition, but I think that a necessary condition for such status is a statistically significant difference between the alleged outlier and comparable data points. Since the current result is not a statistically significant change from WaPo/ABC's previous results, it's not an "outlier" in terms of their own polling.

Alternatively, it's been suggested that the WaPo/ABC's polls this year have been consistently biased in favor of Trump. If so, this is rather surprising since both WaPo and ABC News are establishment news media―ABC, in particular, is owned by Disney7―and they fear and loathe Trump. It's for this reason, of course, that the WaPo article tries to mollify its readership by condemning its own poll as an "outlier".

Pollster Larry Sabato suggested in an "X"8 that the poll was so "absurd on its face" that it should not even have been published9. Actually, what's absurd is Sabato's suggestion that the results be suppressed.

It's one thing to warn readers that a poll may be an outlier, and that its results should be treated with skepticism, but suppressing surprising poll results would totally discredit polling. This would create a "file drawer effect" which would make even polling aggregation unreliable. If a news outlet pays for a poll, then the results should be released, even if the outlet is skeptical about them. A poll is an experiment, and suppressing an experiment that produces an inconvenient result is a violation of scientific procedure, especially if it's suppressed for political reasons10.

That Sabato suggested this publically makes me wonder if any major polling organizations ever suppress poll results. There are pollsters who do private polls for clients that then may or may not be released to the public, depending on the results, but such polls should not be included by poll aggregators. I've never heard of a news outlet suppressing a poll, but that doesn't mean that it's never happened. I hope that WaPo, ABC News and other poll sponsors ignore Sabato's awful advice and report all their polls.

In conclusion, the notion of "outlier" is so vague that it's unclear whether this poll was one. The fact that it's being labelled as one by establishment observers is partly due to the direction of the deviation from other poll results; a deviation in the opposite direction would be welcomed rather than dismissed. Given that the poll's results were not statistically significantly different from those of earlier WaPo/ABC polls, whatever problem there may be with this poll appears to be due to WaPo/ABC's polling methodology. In any case, it's too soon to be obsessing over polling results: let's at least wait until next year before we over-react.

Notes:

  1. For an imprecise definition, see: Roger Porkess, The Harper-Collins Dictionary of Statistics (1996).
  2. Dan Balz, Scott Clement & Emily Guskin, "Post-ABC poll: Biden faces criticism on economy, immigration and age", The Washington Post, 9/24/2023.
  3. How to Read a Poll: Polling the Polls, 12/6/2022.
  4. How to Read a Poll: The Confidence Game, 12/6/2022.
  5. ABC did publish yesterday a lengthy and fairly good explainer stating that it's "likely" the poll is an outlier; see: G. Elliott Morris, "How outlier polls happen—and what to do with them", ABC News, 9/26/2023.
  6. Gary Langer, "Trump edges out Biden 51-42 in head-to-head matchup: Poll", ABC News, 9/24/2023. Paragraphing suppressed.
  7. Harold L. Erickson, "American Broadcasting Company", Encyclopaedia Britannica, 9/26/2023.
  8. This is what used to be called a "tweet". I dislike the silly names "Twitter" and "tweet", but "X" is no improvement.
  9. See: Joe DePaolo, "Top Pollster Trashes ABC/WaPo for Publishing ‘Laughable’ Poll Showing Trump Up 10: ‘Will Be a Lingering Embarrassment for You’", Mediaite, 9/24/2023.
  10. The article linked to in note 5, above, does a good job of explaining why suppressing outliers is a terrible idea.
September 20th, 2022 (Permalink)
Starbucks coffee shops expansion

Charts & Graphs: The Incredible Booming Coffee Shop

The bar chart shown above, which is just a detail from a full page "infographic" on the growing number of Starbucks coffee shops1, combines three types of problematic graph in one. For some unknown reason, the "bars" are actually cylinders, that is, three-dimensional―how did the designer managed to resist the temptation to turn them into Starbucks coffee cups? This is not a problem per se, but the addition of depth to the graph adds no new information and, as I explained a decade ago2, risks misleading the viewer. Here are the three problems:

  1. Adding the third dimension to the chart makes it ambiguous whether the height of the "bars" is to be measured by their cylindrical top, or by their two-dimensional edge. These are not the same because, as the chart is drawn, we seem to be looking down at an angle on the cylinders. Judging from the last cylinder's height, it appears to represent 6K stores in 2003. However, if you look closely you'll see a thin line stretching from the top edge of the cylinder over to the lefthand scale. Moreover, the top of the chart tells us that there were 6,200 shops at the time the graph was published, which seems to be verified by the position at which the thin line meets the scale. So, it seems to be the visual edge of each "bar" that represents the number of shops, rather than the top of the cylinder.

    The main effect of the extra dimension is to make it difficult to read the numbers of shops represented by the other "bars" from the scale. For instance, how many stores were there in 2000? It appears there were somewhere between 3,000 and 3,500, but it's difficult to be more precise. The chartmaker might defend it on the grounds that no one really wants to know the exact number of shops for any given year, and that the point of the chart is just to show the enormous growth of Starbucks, but a linear graph would be a better way to do it.

  2. Another problem3 with this chart is that the width of the cylinders increases with their height but, as we've seen, the data is represented only by their heights. The extra width of the taller bars can exaggerate the difference between those of different heights. For instance, the cylinder for the year 2000 represents about half as many shops as that for 2003, yet the latter cylinder is about four times larger in area. As a result, the casual viewer is likely to overestimate the increase in numbers.

    In addition, if the bars really were cylinders, then the 2003 one would have approximately eight times the volume of the 2000 one. In mentally comparing them, the viewer may tend to compare their apparent volumes rather than their heights or even areas. The only way for a viewer to correctly compare the bars in this chart is to ignore the fact that they are portrayed as cylinders and compare their heights, not their areas or volumes.

  3. The above two problems we've seen before, but I've never seen the like of this third one4: the larger cylinders at the right side of the chart seem to advance towards the viewer. This is shown by the overlapping of the later bars, their advancing shadows, and their bottoms getting farther from the baseline. This creates a trick of perspective that makes the later cylinders appear closer to the viewer, and thus visually larger.

All three of these 3D effects tend to exaggerate the increase in the number of Starbucks shops. If this chart were put out by Starbucks, I would suspect that the company was trying to exaggerate its growth to impress people. Since it seems to come from a neutral organization, I suppose the only reason for the exaggeration is to create a more visually exciting chart. A plain bar chart is very boring but there ought to be a way to make it more fun to look at without misleading people.

Notes:

  1. You can see the full graphic here: "Infographic", International Networks Archive, accessed: 9/20/2023.
  2. The 3D Bar Chart, Part 1, 6/3/2013.
  3. For more on this problem, see: The One-Dimensional Pictograph, 8/1/2021.
  4. The following entry discusses a similar, but not identical, problem: The 3D Bar Chart, Part 2, 7/11/2013
September 14th, 2023 (Permalink)

Headline

Nasa jet travels 850 miles in 10 seconds1

That seems fast, but how fast is it? Does it seem credible that a jet could travel that fast? Last year, I posted a short series on credibility checking2, and this headline is a good candidate for such a check.

The headline expresses the jet's speed in a way that's difficult to evaluate, since we're used to speeds expressed in miles per hour (MPH) or kilometers per hour, and I've never before seen a speed expressed in miles per ten seconds. So, how fast would that be in miles per hour? Take ten seconds if you're fast, or a minute or two if you're more my speed, to convert the headline into MPH.

So that you can check your work, here are my calculations. Since there are sixty seconds in a minute, there are six ten-second periods per minute. So, the jet was going 6 × 850 = 5,100 miles per minute, and given that there are sixty minutes in an hour, it was travelling at 60 × 5,100 = 306,000 MPH. That is fast!

At this point, if not before, your skeptical sense should start tingling. Is that speed plausible? To find out, let's read down in the article beneath the headline:

An aircraft that can fly at ten times the speed of sound will be tested over the Pacific Ocean today—possibly leading to “hypersonic” cruise missiles that could travel from Los Angeles to Pyongyang in less than an hour. The aircraft was designed by Nasa to travel 850 miles in just ten seconds, or 7,000mph.3

Hold on! 7K MPH is a lot less than 306K MPH; in fact, the latter is almost 44 times faster. Obviously, the two claims about the jet's speed are inconsistent: either it doesn't go 850 miles in ten seconds or it's much faster than 7K MPH.

It seems to me that 306K MPH is highly implausible, but let's do some research to find out for sure. According to NASA itself4, the jet in question, which was known as the X-43A, reached the speed of 7K MPH or almost ten times the speed of sound, which is a little over 760 MPH5.

So, where did the author of the article get the notion that the jet travelled 850 miles in ten seconds? I'm not sure, but a NASA press release6 states that during a later flight the jet would travel 850 miles and its engine would fire for eleven seconds. If you assume that the engine was firing the entire time of the 850 mile flight, then you would conclude that the jet flew 850 miles in a little over ten seconds. However, given that its speed was only a little over 7K MPH, the engine must have fired only for the first eleven seconds of that flight, and the entire flight must have lasted over seven minutes.

As I pointed out in a previous entry:

When it comes to high speeds, we quickly run out of landmarks by which to judge claims for plausibility. Our experience with speeds is very limited, and anything over around a hundred MPH is just "really fast".7

How could the author or editor of the article have avoided such an egregious error? Someone should have taken the minute or two to convert the headline claim from the unfamiliar units of miles-per-ten-minutes into the familiar MPH, as we did above. Such unusual units should be avoided because they are both unintuitive and difficult to compare with the usual ones. If that had been done, the implausibility of the headline would have been obvious, as well as the difference with the later claim that the jet's speed was 7K MPH.

Surprisingly, the article has never been corrected, despite the fact that it will be two decades old next year.

Notes:

  1. Chris Ayres, "Nasa jet travels 850 miles in 10 seconds", The Times, 11/16/2004. Via: Brian W. Kernighan, Millions Billions Zillions: Defending Yourself in a World of Too Many Numbers (2018), p. 43.
  2. See:
    1. Compare & Contrast, 1/7/2022
    2. Divide & Conquer, 2/4/2022
    3. Ratios, Rates & Percentages, 3/27/2022
    4. Ballpark Estimation, 4/21/2022
  3. Paragraphing suppressed.
  4. "NASA's X-43A Scramjet Breaks Speed Record", NASA, 11/16/2004.
  5. "what is the speed of sound", Wolfram Alpha, accessed: 9/14/2023.
  6. "NASA X-43A 'Scramjet' Readied For Mach 10 Flight", NASA, 11/9/2004.
  7. Faster Than a Speeding Bullet, 1/12/2019.
September 11th, 2023 (Permalink)

How to Solve a Problem: Divide and Conquer1

Try solving the following problem.

Problem 1: A Puzzle in Woodpecker Woods

An ornithologist studying the birds in Woodpecker Woods made the following observations:

  1. All the yellowbellies are sapsuckers.
  2. Some of the blackbacks are redheads.
  3. None of the sapsuckers are redheads.

What could the ornithologist conclude is the relation between yellowbellies and blackbacks in Woodpecker Woods?

Note: This is a logic puzzle and not necessarily ornithologically correct, so knowledge about birds will not help solve it and may even mislead you. Base your answer entirely on the above clues.

If you try to solve this problem directly, you may find it difficult because there are three clues and four classes: yellowbellies, sapsuckers, blackbacks, and redheads2. What you'll probably need to do is take the problem two clues at a time. Which two clues should you use? Look for two clues that have a class in common: clues 1 and 3, which share "sapsuckers", and clues 2 and 3, which share "redheads". Since these pairs of clues have only three classes among them, you can use a Venn diagram or whatever technique you please; you may even be able to do them in your head. You could pick either of these duos to solve the puzzle, but I'll show how using 1 and 3:

  1. Clues 1 & 3: From these two clues we can conclude that no yellowbellies are redheads.
  2. Step 1 & Clue 2: Using the conclusion of the previous step together with clue 2, we can conclude that some blackbacks are not yellowbellies, which is the solution to the puzzle.

This was an inferential problem, that is, it asked you to infer something from a group of premisses. While there are advanced methods to solve such a problem directly from all the premisses, it's usually easier to break it down into two or more simpler problems. This is what I mean by "divide and conquer": solving a complex and difficult problem by breaking it down into simpler ones, because simpler is easier.

In a previous entry3, I mentioned how the primary technique for solving jigsaw puzzles is hill-climbing, that is, adding pieces to the puzzle until there are no longer any left to add. As anyone who has ever worked a jigsaw will have noticed, they tend to get easier as you go along, and the hardest part of the puzzle is getting started. The reason is that the problem space4 of the puzzle is largest at the beginning and decreases with every added piece; for instance, if the puzzle has a thousand pieces, then you have to start looking through that many pieces to find two that fit together. To do so systematically, you'd need to make almost a million comparisons―more precisely, 1,000 × 999 = 999,000―so a systematic search is impractical. What can you do?

The usual strategy for solving a jigsaw puzzle is to start out by sorting the pieces into at least two piles: border pieces and interior pieces. It's easy to tell border pieces and interior ones apart because those on the border have one straight edge. Once the pieces have been sorted into two piles, put the interior pieces aside and work on the border of the puzzle first. Assuming that a thousand piece puzzle has a hundred edge pieces, assembling the border of the puzzle first reduces the problem space by a factor of a hundred5.

In effect, this procedure turns one big puzzle into two smaller ones: the border puzzle and the interior puzzle. In this way, the problem space is reduced to more manageable sizes. In other words, you "divide and conquer" it.

As with the other problem-solving techniques we've looked at in this series, it's not always possible to divide a problem into smaller ones, but it is a possibility to consider, especially when confronted by a large and difficult problem.

Now, here's a chance for you to practice using your new tool.

Problem 2: The Three Stooges Gang

A major bank robbery is being investigated by the police, who suspect that a gang known as "the three stooges" committed it. The stooges are three thieves who rob banks and jewelry stores wearing masks representing the members of the famous comedy team they were named after. The original members of the gang were three crooks who had met in prison and started working together on their release. The three were always fighting with each other, and the police had heard rumors that two of them were on the outs. So, it was always possible that one or more of the original stooges might not have participated in the robbery.

The police interviewed their confidential informants (CIs) to find out what the word in the underworld was on the robbery and the gang. I will refer to the usual members of the gang as "Moe", "Larry", and "Curly" to protect the innocent. The police gathered the following clues from the CIs:

  1. If Moe didn't plan the robbery, then Larry participated but not Curly.
  2. Either Curly was involved in the crime or Larry wasn't.
  3. Larry didn't participate in the robbery if and only if both Moe planned it and Curly was included.

Assuming that what the CIs said is correct, who if any among the original three stooges was involved in the robbery?

Notes:

  1. For previous entries in this series, see:
    1. Contraction, 4/6/2023
    2. Think Backwards, 5/5/2023
    3. Solving a Problem by Elimination, 6/20/2023
    4. Climbing Up that Hill, 7/5/2023
    5. Backtracking, 8/14/2023
  2. If you're familiar with traditional logic, you might notice that the three clues are each categorical statements. If there were only two clues and three classes, you could treat it as a categorical syllogism, for which there are established techniques. As it is, there are three statements with four class terms among them, so it's impossible to make a single syllogism out of them. Similarly, if you know how to use Venn diagrams, you might think to represent the logical relations between the four classes in a diagram, but the standard "pretzel" diagram only relates three classes. There are diagrams for more than three classes, but they tend to be less intuitive and harder to use. See Martin Gardner's Logic Machines and Diagrams (2nd edition, 1982), chapter 2.
  3. Climbing Up that Hill, 7/5/2023.
  4. By the "problem space" of a puzzle I mean the class of all possible solutions to it.
  5. The number of comparisons is 100 × 99 = 9,900, which is one-hundredth of that for the full puzzle.
September 4th, 2023 (Permalink)

Famously Infamous

This year is the eightieth anniversary of Operation Chastise, the so-called dambusters bombing raids of World War Two1. The raids, conducted by Britain's Royal Air Force, took place on May 16th and 17th of 1943 and aimed at destroying three dams along the Ruhr river where Germany's war industry was concentrated. The operation was largely successful as two of the dams were sufficiently damaged to flood the river valley2. However, as interesting as this true story is, this is not a history lesson; rather, it's prompted by a BBC television presenter named Sally Nugent who, in commenting on the anniversary, called the raids "infamous"3.

If you only knew the word "famous" and the prefix "in-", you would probably think that "infamous" means "not famous" since "in-" is a negative prefix, but there's another way to be negative. Instead of "non-famous", "infamous" means famous for something negative4, so that calling the dambusters raids "infamous" means they are famous in some bad way. While the raids are no doubt famous in England, the fact that the BBC found it necessary to apologize for its presenter's remark shows that they are not infamous.

My guess is that either Nugent or whoever wrote the script she was reading simply didn't know the meaning of "infamous", rather than intending to suggest that the raids were well-known for being bad. Some words with negative meanings, such as "bad" and "sick", are sometimes used in the opposite sense, and I've previously come across "infamous" used in this way.

I don't know how well the dambusters raids are known in the United States nowadays, but I was aware of them from having seen the 1955 movie The Dam Busters5 on television as a boy. As you can tell from the title, this was a fictionalized film version of the famous raids. I was not the only one to see and love the movie: so did a young George Lucas, who based the final scenes of a famous 1977 movie on it6.

Substituting "infamous" for "famous" is the sort of error that neither a spelling nor even grammar checking program can be expected to catch, since both are English adjectives. To notice the substitution of one for the other requires understanding the difference in meaning between the two, and not just spelling or even grammar. I tried the full sentence spoken by Nugent in several online copyediting programs and, unsurprisingly, not a one caught it. So, if you don't want to become as infamous as Nugent, add this distinction to your mental copyeditor.

Notes:

  1. David McKenna, "Events mark 80th anniversary of Dambusters raids", BBC News, 5/13/2023.
  2. "The Incredible Story Of The Dambusters Raid", Imperial War Museum, accessed: 9/3/2023.
  3. Charlie Parker, "BBC apologises for Sally Nugent’s ‘infamous’ Dambusters comment", The Sunday Times, 8/3/2023.
  4. "Infamous", Cambridge Dictionary, accessed: 9/3/2023.
  5. Alex von Tunzelmann, "The Dam Busters: hits its targets–and doesn't dumb down", The Grauniad, 8/7/2015.
  6. Bryan Young, "How The World War II Drama 'The Dam Busters' Influenced The Space Battles Of 'Star Wars: A New Hope'", Slash Film, 5/2/2018.

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August 31st, 2023 (Permalink)

The Old Newspeak & the New Newspeak

Notes:

  1. Jared Downing, Priscilla DeGregory & Jorge Fitz-Gibbon, "Politicians, prosecutors dropping the ball on booming NYC sex trade: advocates", The New York Post, 8/8/2023.
  2. See, for instance: Teo Bugbee, "‘Kokomo City’ Review: Dispatches From the Down Low", The New York Times, 7/27/2023.
  3. Doublespeak Headline, 8/3/2008.
  4. Frédérique Delacoste & Priscilla Alexander, editors, Sex Work: Writings by Women in the Sex Industry (2nd Edition, 1998).
  5. "Sex Work, Sex Worker", Google Books Ngram Viewer, accessed: 8/31/2023.
  6. Close Encounters with Doublespeak of the Third Kind, 9/8/2019.
  7. Anemona Hartocollis, "Colleges Celebrate Diversity With Separate Commencements", The New York Times, 6/2/2017.
  8. George Orwell, Nineteen Eighty-Four (1949), Part I, Chapter 1.

Disclaimer: I don't necessarily agree with everything in these articles, but I think they're worth reading as a whole. In abridging them, I have sometimes changed the paragraphing of the excerpts.

August 14th, 2023 (Permalink)

How to Solve a Problem: Backtracking1

As usual in these entries on problem-solving, let's start out with a puzzle.

Puzzle: A Delivery Dilemma

If you work delivering packages nowadays, you are expected to deliver a diverse assortment of items to the same address. For instance, suppose that you are tasked with delivering to a house on top of a high hill. Unfortunately, no one is home so that you'll have to leave the packages on the porch of the house. The road ends at the bottom of the hill, and the only way to get to the porch is a steep and narrow footpath up the side of the hill. Suppose, further, that you are to deliver a live goat, a hungry dog, and a pizza. Because of the steepness of the hill, you can only take up one item at a time, so it will require at least three trips to deliver all three to the porch. That's a lot of climbing! However, you can't leave the goat and the pizza alone together in the delivery van or on the porch, because the goat would eat the pizza. Similarly, you can't leave the hungry dog alone with the goat, since the dog would attack the goat. Luckily, you can leave the dog alone with the pizza, since everyone knows that dogs don't like pizza.

How can you get all three items alive and intact to the porch? What is the minimum number of trips up and down the steep hill that you will have to make?2

This puzzle looks like a prime candidate for hill-climbing3, and indeed that will play a role in solving it. Since your problem is to get all three items from the base of the hill up to the house on top, you have a clear measure of progress, namely, how many objects are on the porch. So, let's work our way through it.

The first step, of course, is to carry one of the objects up to the house and leave it on the porch, but which one? You can't take the pizza up first, since that would leave the dog and goat alone together in the delivery van. Also, you can't take the dog up first, since that would leave the goat and pizza alone. So, the only alternative left is to take the goat up first, leaving the dog and the pizza in the van4. So far, so good.

However, now you're stuck. You can't take the dog up and leave it on the porch alone with the goat since the dog will attack the goat, and you can't take the pizza up and leave it on the porch because the goat will eat it. The puzzle appears to be insoluble! In fact, it is impossible to solve if you stick to hill-climbing.

To solve the puzzle, you must backtrack, which is the topic of this entry. Here's how to do it: take the dog or the pizza―it doesn't matter which―up to the porch and leave it, then take the goat back down to the van! This step goes against common sense, which is what makes the puzzle hard. It also violates the hill-climbing algorithm, since all you've done is switch one object on the porch for another. The hill-climbing algorithm tells you to always take an action that increases the number of items delivered, but you can't do so and solve the puzzle.

To continue the solution: you return to the van with the goat, leave it in the van, and take the pizza or dog, as the case may be, up the hill. You can safely leave the dog and pizza together on the porch as you return to the van for the goat. Finally, you take the goat back up and place it on the porch. It took four trips up and down the hill to deliver the three items!5

Hill-climbing alone is usually not enough to solve a problem. Instead, you need to combine it with backtracking, that is, when you get stuck and can't seem to make further progress towards the goal, backtrack to an earlier stage and take a different route. For instance, in trip-planning, we often use hill-climbing by choosing among routes that lead in the general direction of where we want to go. However, sometimes the shortest route to a destination may require that you backtrack to gain access to it, or it may take a road that sometimes heads in the wrong direction.

In the above puzzle, you must backtrack because you have two different goals that sometimes conflict: one goal is to get all three items on the porch―and if this was your only goal you could accomplish it with hill-climbing alone―but your other goal is to deliver the items undamaged, and for that you have to backtrack.

In the previous entry, I mentioned three different obstacles faced by hill-climbing: plateaus, ridges, and multiple peaks. If, in the course of hill-climbing, you get stuck on any of these obstacles, then backtracking will be necessary in order to get off. The above puzzle is a different type of obstacle―namely, a detour6―which comes about when you have more than just the goal of getting to the top of the hill. When you have more than one goal, you will often need to backtrack in order to accomplish all of them.

Now that you have another problem-solving tool in your kit, try the following puzzle for practice. Note that this puzzle takes place in a world of wizards and little people where magic is real, but you won't need magic to solve it.

Puzzle: To the Far Shore

The Great Gray Mage and his two traveling companions, both halflings, needed to cross a bottomless and nameless river. They dared not swim it for fear of monsters rising from its murky depths. The mage's magic was losing its power, and the three were on a quest to discover why. Luckily, there was a small rowboat pulled up on the bank that the three could borrow to make the crossing. However, the boat was of a size that could not hold all three travelers at the same time since it could carry only the weight of the mage without sinking. Of course, the halflings each weighed half as much as the mage so the boat could carry both across. If the mage rowed the boat across the river, the halflings would be left behind; and if the halflings rowed across, then the mage would be left. The mage's powers were so weak that he could not use them to draw the empty boat back across the river, and it appeared that only one crossing could be made. How did the Great Gray Mage get himself and the two halflings across the river without using magic?7

Notes:

  1. For previous entries in this series, see:
    1. Contraction, 4/6/2023
    2. Think Backwards, 5/5/2023
    3. Solving a Problem by Elimination, 6/20/2023
    4. Climbing Up that Hill, 7/5/2023
  2. This is a version of a puzzle that goes all the way back to Alcuin in the eighth century. It's usually presented as the story of a farmer who buys a cabbage, goat, and wolf (?) at a market, and has to cross a river in a small boat that holds only one of the three items at a time. See: Marcel Danesi, The Puzzle Instinct: The Meaning of Puzzles in Human Life (2002), pp. 153-155.
  3. See: Climbing Up that Hill, 7/5/2023.
  4. Notice that we solved the sub-problem of which object to deliver first by elimination; see: Solving a Problem by Elimination, 6/20/2023.
  5. You might wonder what happens to the three items after you leave them on the porch, but that's not your problem!
  6. Wayne A. Wickelgren, How to Solve Problems: Elements of a Theory of Problems and Problem Solving (1974), pp. 85-88.
  7. This is a simplified version of another classic puzzle from Alcuin: as it's usually presented there are two adults and two children, but the solution simply repeats twice the process for getting one adult across the river. See: William Harston, A Brief History of Puzzles: Baffling Brainteasers from the Sphinx to Sudoku (2019), Puzzle 2.

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