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\title{Learning psychology from riddles: The case of stumpers}

\author{Maya Bar-Hillel\thanks{Department of Psychology, The Hebrew University of
    Jerusalem. Email: msmaya@math.huji.ac.il}
\and
Tom Noah\thanks{Department of Psychology, The Hebrew University of Jerusalem.}
\and
Shane Frederick\thanks{Department of Marketing, Yale University School of Management.}
}

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\newcommand{\jdate}{January 2018}
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%\begin{latexonly}
\vspace{-3mm}
\begin{onecolabstract}
%\end{latexonly}
Riddles can teach us psychology when we stop to consider the
psychological principles that make them ``work''.  This paper studies
a particular class of riddles that we call \textit{stumpers}, and
provides analysis of the various principles (some familiar, some
novel) that inhibit most people from finding the correct solution – or
\textit{any} solution – even though they find the answers obvious ex
post. We restrict our analysis to four stumpers, propose the
psychological antecedents of each, and provide experimental support
for our conjectures.

\smallskip
\noindent
Keywords: riddles; representation; situation models; Grice; cognitive economy; gender stereotypes
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\vspace{-1ex}
]
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%Table 1: 
\begin{strip}%\vspace{-1in}
{\rule{0mm}{0mm}\hspace{1.5in} \sc Table 1:} {\sf The stumpers. (Try
to solve them before reading on.)}\\[-1ex]
\rule{0mm}{0mm}\hrulefill\rule{0mm}{0mm}\\
\textbf{Accountant:} An accountant says: "That attorney is my brother",
  and that is true – they really do have the same parents. Yet that
  attorney denies having any brothers – and that is also true!
  How is that possible?\\[1.2ex]
\textbf{Speeding Car:} A big brown cow is lying down in the middle of a
  country road.  The street lights are not on, the moon is not out,
  and the skies are heavily clouded. A truck is driving towards the
  cow at full speed, its headlights off. Yet the driver sees the cow
  from afar easily, and avoids hitting it, without even having to
  brake hard.
  How is that possible?\\[1.2ex]
\textbf{Bus Ride:} Individual bus rides cost one dollar each.  A card
  good for five rides costs five dollars. A first-time passenger
  boards the bus alone and hands the driver five dollars, without
  saying a word. Yet the driver immediately realizes, for sure, that
  the passenger wants the card, rather than a single ride and
  change.
  How is that possible?\\[1.2ex]
\textbf{Potato Bags:} In a Bangladesh market, a small potato bag costs 5
  taka, a medium potato bag costs 7 taka, and a large potato bag costs
  9 taka.  Yet, a single potato in that market costs 10 taka.
  How is that possible?\\[-1ex]
\rule{0mm}{0mm}\hrulefill\rule{0mm}{0mm}\\
\end{strip}

{\renewcommand{\thefootnote}{}

\footnotetext{

  We wish to thank the many people who helped us tread this new
  territory. Some pointed us to several challenging new stumpers. In
  particular, we thank Herb Clark, guide extraordinaire to the
  pragmatics of language, and Jon Baron, who – like the Beatles'
  Jude – takes a paper, and makes it better.

Copyright: \copyright\ 2018.
The authors license this article under the terms of the
\href{http://creativecommons.org/licenses/by/3.0/}{Creative Commons
  Attribution 3.0 License.}
}}

\saythanks

%\renewcommand{\baselinestretch}{1.02} \large\normalsize

\section{Introduction}


\setcounter{table}{1}

%Table 2: 
\begin{table*}[t]\centering
\caption{Dominant construals, canonical solutions and general principles.}
\begin{tabular}{@{}l L{4cm}  L{4cm}  L{6.3cm}@{} }\toprule
Stumper & The dominant construal (which inhibits solution). & The alternate construal
(which yields solution). & What causes the dominant construal?\\\midrule
Accountant & The accountant is male. & The accountant is female. & 
Stereotypes:  \newline Some professions are (mentally) genderized.\\
Speeding Car & It is nighttime. & It is daytime. & Gricean norms: \newline Denying sources of illumination implies darkness.\\
Bus Ride & Passenger paid with one \newline \$5 bill. & Passenger paid with five \newline \$1 bills. & Cognitive economy: \newline It is easier to imagine a stated quantity as a single unit than as more. \\
Potato Bags & The potato bags were full of potatoes. & The potato bags were empty. & Priming: \newline Containers described by specific content are imagined with it inside.\\\bottomrule
\end{tabular}
\end{table*}

%Table 3: 
\begin{table*}[t]\centering
\caption{Design of questionnaire in Study 1.}
\begin{tabular}{ L{2cm}  L{2.75cm}  L{2.75cm}  L{2.75cm}  L{2.75cm}  L{3cm}}\toprule
 & Group 1 &Group 2 &Group 3 &Group 4 &Group 5\\
Lead stumper  & Accountant & Speeding Car & Bus Ride & Potato Bags & Farmer Joe\\\midrule
Imagine  & \mc{5}{l}{Please take a moment to imagine [the suitable inserts appear below].} \\ 
instructions & \mc{5}{l}{Form a picture in your mind. What did you see in your mind’s eye?} \\\midrule
First Imagine task*& a black asphalt road  & eggs being laid & an accountant & a gallon of water& five dollars \\
Modified vignette** & Five dollars are replaced by six dollars (in the Bus Ride). & The potato bag is replaced by a shopping bag. & No mention of lights at all (in the Speeding Car). & The attorney is replaced by an actress.& The attorney is replaced by an actress and brother is replaced by sister.\\
Second Imagine task & a shopping bag. & 4 pounds of unpackaged strawberries. & a potato bag. & a black asphalt road, with no streetlights on.& a cereal box. \\
Fifth task & "…list animals whose eggs people eat." & Imagine a transparent glass water pitcher. & Imagine a room with all the lights turned off. & Farmer Joe variant. & Imagine a car speeding with its headlights turned off. \\\midrule
Recall task & \mc{5}{p{5.5in}}{In the text box, please write down the very first question you were asked.
in this study (the one about [insert vignette's name]) as closely to the exact original wording as you can remember it.}\\\bottomrule
\mc{6}{l}{$^*$ The two options presented appear verbatim in Table 5.}\\
\mc{6}{l}{$^{**}$ We will not report further on the modified vignette.}
\end{tabular}
\end{table*}

\renewcommand{\baselinestretch}{1.03} \large\normalsize

As with pornography, a riddle is harder to define than to recognize. Riddles share no more than a "family resemblance" (Wittgenstein, 1953).  It would be hard to specify what critical features the following two questions share, but they are both clearly riddles.
\begin{itemize}
\item[i] My mother is your mother's mother-in-law.  Who am I?
\item[ii] Using just 6 identical matchsticks, construct 4 equilateral triangles.\footnote{One solution to the first riddle is: I am your father. The solution to the second riddle is to build a regular tetrahedron.}
\end{itemize}

Perhaps it is the element of surprise in the solution that makes for a
riddle – when it is funnier than expected, simpler than expected, or
just different than expected.  The surprise can lie in the very fact
that a satisfying answer is even possible.  A good riddle provides an
unambiguously correct answer that lies within one's current knowledge.
It is recognized as such with aesthetic or intellectual appreciation,
and might well elicit responses such as "lovely!", "OMG!", "but of
course!".

Riddles have a long and fruitful history in psychological research
(e.g., Garth, 1920). Scholars studying human problem-solving used
either stylized and carefully designed riddles, or ecologically valid
problems in knowledge-rich domains. Some famous riddles include the
nine-dot problem (Maier, 1930), about self-imposed solution
constraints; the candle problem (Duncker, 1945), about functional
fixedness; the fill-a-jar problems (Luchins, 1942), about solution
mechanization; the 4-cards problem (or Wason's selection task, 1968),
about intuitive logic; Tversky and Kahneman's (1983) Linda problem,
about the representativeness heuristic in probabilistic reasoning;
variants of the Monty Hall problem (e.g., Bar-Hillel \& Falk, 1982),
about conditional probabilistic reasoning; and the bat-and-ball
problem (Frederick, 2005), about cognitive reflection.  The present
paper investigates a subset of riddles that we call
\textit{stumpers}. Unlike some of the aforementioned riddles, stumpers
do not evoke a compelling, but wrong, intuitive answer. Rather,
respondents are typically unable to summon any satisfying answer at
all. Table 1 presents the ones we used.\footnote{The actual wording
  was somewhat different in Study 1 (see Appendix 1), but our claims
  pertain to both versions.}  When you attempted to solve them, were you
stumped by any of them? Can you see why they might stump others? Can
you find a common denominator?

Our account of these stumpers is as follows. A situation is verbally
described, and while constructing the mental model of that situation
(Zwaan \& Radvansky, 1998), some missing details are filled in
(Kosslyn, 1980). The text is akin to a movie script, and the listener
to the director who sets the stage. A given script leaves room for the
director's interpretation of the words in the script (note that a
Google-image search for words like "party" or "athlete" yields many
related, but not identical, pictures). Yet, despite the fact that a
word is worth a thousand pictures, most people will construe a scene
in much the same way, and the dominant construal can sometimes blind
them to alternatives. Stumpers exploit this by placing the solution
outside of the dominant construal. Insofar as the common construal
inhibits other construals, the respondent is stumped.

Within this general process, the dominant instantiation is caused by a
different specific principle in each stumper (Table 2). It is easy to
see that gender bias drives the Accountant vignette. The principle
driving the Speeding Car is conversational norms. The final two
principles are novel. To elaborate:

%\renewcommand{\baselinestretch}{1.01} \large\normalsize

\paragraph{Accountant:} Accountants are gender-typed as male (row 1,
Table 5). However, a variation of this riddle does not even require
gender-biased professions, since the English language itself is
manifestly sexist in how it references a generic person (e.g.,
Martyna, 1978). For example, the male pronoun “he” is used to refer to
a single person of unknown sex, and the phrase “you guys” is
indiscriminately used to refer to groups of people of both
sexes. Indeed, people are even stumped by this variant:\textit{ A is
  the son of B, but B is not the father of A. How is this possible?}
(from \url{https://www.riddles.com/u/riddle/3708}).

\paragraph{Speeding Car:} One of Grice's (1975) conversational maxims
is "Do not make your contribution more informative than is required"
(p.\ 45). Hence, when a narrative bothers to mention the absence of
light sources, darkness is evoked (since those light sources are
otherwise irrelevant). In this way, this stumper (unlike the others)
might arguably be accused of “trickiness”. This is our only stumper
where the script induces an atypical construal, as most would
ordinarily visualize a car during daytime (row 3, Table 5); our mind's
eye apparently needs light as much as our physical eye.

\paragraph{Bus Rides:} We hypothesize that it is mentally easier to
conjure a representation of a single item than of multiple
items. Thus, a five dollar bill is easier to "see" than five singles
(row 11, Table 5). This principle is a novel twist on Rosch's familiar
principle of cognitive economy (e.g., Rosch, 1978).

\paragraph{Potato Bags:} Although the purpose of containers (bottles,
boxes, jugs, bags) is to contain, we hypothesize that they will be
represented as empty unless referencing specific contents (e.g., wine
bottle, cereal box, milk jug). Hence, in one’s mind's-eye, a potato
bag will be full of potatoes, but a shopping bag will be empty (rows
27, 30, Table 5).

In our experiments, we tested our hypotheses about why these five
riddles stump in two different ways. In a direct task, we specified a
concept in the abstract (i.e., without the additional details which
the stumper narrative provides) to assess the scene that respondents
would organically summon. In the indirect task, we asked them to
recall their stumper, to assess how much the typical instantiation
corrupted their memory of the stumper’s verbatim text.

\section{Method}

\renewcommand{\baselinestretch}{1.05} \large\normalsize

We ran two similar studies on Mturk, 18 months apart, both on a
Thursday between 9:00am and 12:00pm. The first had 499 subjects, 55\%
of whom were male, with an average age of 36. The second had
516\footnote{Forty six additional respondents dropped out before
  completing the computerized questionnaire, and are not counted. One
  subject was removed for giving gibberish answers to the free form
  questions.}  subjects, 56\% of whom were female, with an average age
of 38. All but seven were native English speakers. Respondents were
randomly assigned to the experimental groups described below, and were
paid a dollar each for their participation. They answered a
multi-screen questionnaire, administered individually through
Qualtrics.\footnote{We gathered more data than we report on in this
  paper. (1) We omit a fifth “stumper” about Farmer Joe (see Appendix
  1) because it didn’t sufficiently stump people (though you can still
  see its residues in parts of Tables 3 and 4). (2) We omit analysis
  of a task called "Modified Stumper" (see Tables 3 and 4), in which
  critical words from the original were replaced, with intent to
  facilitate solution – which they generally did. (3) We omit an
  analysis of confidence ratings respondents provided for their
  offered solutions to the standard and the modified stumpers. These
  were omitted to streamline an already data-heavy paper. None change
  the reported picture, and the interested reader can find them in the
  accompanying raw data.}

We now describe the flow of this questionnaire, screen by screen
(respondents could not return to previous screens). It is also
summarized in Tables 3 and 4. The uninterested reader can skip
directly to the Results (and refer back as needed).

%Table 4: 
\begin{table*}[t]\centering
\caption{Design of questionnaire in Study 2.}
\begin{tabular}{@{}L{2cm}  L{2.8cm}  L{2.8cm}  L{2.8cm}  L{2.8cm}  L{2.8cm}@{}}\toprule
{} & Group 6 &Group 7 &Group 8 &Group 9 & Group 10\\
Lead stumper  & Accountant & Speeding Car & Bus Ride & Potato Bags& Farmer Joe\\\midrule
Imagine instructions & \mc{5}{p{14cm}}{Imagine [the inserted words are shown below]. Spend some time forming a picture in your mind. Embellish it with detail.}\\\midrule
First Imagine task$^*$ & a white man all dressed in white on a white gravel road. & five keys & a bag of potatoes & ornamental painted eggs. & a water pitcher. \\
Second Imagine task & "Please list contemporary U.S. currency bills in ascending order." & a nurse. & a black man all dressed in black on a black asphalt road. & four ounces of pure gold. & five ounces of Camembert cheese.\\
Modified vignette$^{**}$ & The potato bag is replaced by a shopping bag. & Five dollars are replaced by two dollars (in the Bus Ride). & Second Farmer Joe variant. & No mention of lights at all (in the Speeding Car).& The attorney is replaced by a nurse.\\
Third Imagine task & a bell ringing. & a wine bottle. & an ounce of pure gold. & a water bucket.& a parked car. \\
Fourth Imagine task & a bottle of wine. & a pound of pure gold. & a wine decanter. & two dollars.& a bottle. \\\midrule
Recall & \mc{5}{p{14cm}}{In the text box, please write down the first vignette you read (the one about [insert vignette's name]). Use words as close to the original words as you possibly can.}\\\bottomrule
\mc{6}{l}{$^*$ The two options presented appear verbatim in Table 5.}\\
\mc{6}{l}{$^{**}$ We will not report further on the modified vignette.}
\end{tabular}
\end{table*}

\subsection{Questionnaire design}

The opening screen gave the instructions (reproduced in full in
Appendix 2). Subjects then indicated their age, sex, and native
language, before proceeding to the study. First they received the lead
stumper\footnote{In Study 2, the stumper was prefaced with some words
  meant to encourage respondents to admit to being stumped. These
  appear in Appendix 2.} (referred to throughout the questionnaire
only as a "vignette"). On the following screen they were told: "If you
can think of a good answer, please enter it in the text box [they
could take their own time entering their free form answer]. If you
can’t come up with what seems like a satisfying answer, then please
just write the words “I am stumped.” They then indicated whether they
had ever heard this riddle before.\footnote{Overall only 20
  subjects said they had. We elected to retain their data
  nonetheless, because it turned out that stated familiarity did not
  guarantee that they would solve correctly.}

This was followed by "Imagine" tasks or some ad hoc task, as specified
in Tables 3 and 4. "Imagine" tasks pertained to a different vignette
than the lead stumper.\footnote{With the sole exception of Group 8,
third Imagine task.}  Subjects were told: "Imagine [target
item – e.g., "an accountant", or "five dollars", etc.]. Form a picture
in your mind." They then indicated which of two mutually exclusive
possibilities they had envisioned. The presentation order of these two
options was counterbalanced, as was the order of the Imagine tasks.

Following the Imagine tasks, subjects were requested to "please write
down the very first question you were asked in this study (the one
about the [lead stumper's reference here]), as closely to the exact
original wording as you can remember it."\footnote{By mistake the
  Accountant stumper in Study 1 was mislabeled in this task, and so we
  have no data for it in Table 7.} The hope (which was fulfilled; see Results)
was that respondents would recall aspects of that riddle that were not
present, providing additional evidence for the dominant visualization.

\renewcommand{\baselinestretch}{1.03} \large\normalsize

\section{Results and Discussion}

%Table 5: 
\begin{table*}[p]\centering
\caption{Imagine tasks, their binary options, and their popularity.}
{\small
\begin{tabular}{@{}L{5cm}L{5.5cm}L{5.5cm}r@{}}\toprule
Imagine & \mc{2}{c}{"What did you see in your mind's eye?"} & N\\\midrule
\mc{4}{@{}l}{\textbf{Accountant Stumper}}\\
1. An accountant &A male accountant  71\% &A female accountant   29\% &102\\
2. A nurse & A female nurse   95\% & A male nurse   5\% &99\\\midrule
\mc{4}{@{}l}{\textbf{Speeding Car Stumper}}\\
3. A parked car &A view by day   91\% &A view by night   9\% &100\\
4. A bell ringing &It happened in the light   82\% &It happened in the dark   18\% &99\\
5. A black asphalt road &A day-time view of the road  82\% &A night-time view of the road  18\% &96\\
6. A black man all dressed in black on a black asphalt road &A view by day   61\% &A view by night   39\% &101\\
7. A white man all dressed in white on a white gravel road &A view by day   93\% &A view by night   7\% &99\\
8. A black asphalt road with no streetlights on &A night-time view of the road  81\% &A day-time view of the road  19\% &93\\
9. A room with all the lights turned off &A dark room   95\% &A bright room   5\% &102\\
10. A car speeding with its headlights turned off &A night-time scene   68\% &A day-time scene 32\% &99\\\midrule
\mc{4}{@{}l}{\textbf{Bus Ride Stumper}}\\
11. Five dollars &A five dollar bill   88\% &Five single dollar bills   12\% &99\\
12. A gallon of water &A single large container filled with water 100\% &Several small containers filled with water 0\% &94\\
13. An ounce of pure gold &A single lump of gold   90\% &More than one lump of gold 10\% &101\\
14. A pound of pure gold &A single lump of gold   84\% &More than one lump of gold  16\% &99\\
15. Four ounces of pure gold &A single lump of gold   60\% &More than one lump of gold 40\% &95\\
16. 4 pounds of unpacked strawberries &One large pile of strawberries  78\% &Several small piles of strawberries  22\% &108\\
17. Five keys &A key ring with all the keys on it 63\% &Loose keys   37\% &99\\
18. Five ounces of Camembert cheese &A single piece of cheese   65\% &More than one piece of cheese 35\% &100\\
19. Two dollars &A couple of singles   83\% &One \$2 bill 17\% &95\\
\mbox{20. List contemporary U.S. currency bills in ascending order:} &\mc{2}{c}{\textit{53\% forgot to list \$2 bill}} & 99\\\midrule
\mc{4}{@{}l}{\textbf{Potato Bag Stumper}}\\
21. A bag of potatoes &A bag with potatoes in it  99\% &An empty bag 1\% &101 \\ 
22. A bottle of wine &A bottle with wine in it 97\% &An empty bottle 3\% &99\\ 
23. A cereal box &A full box with cereal in it  94\% &An empty box with nothing in it 6\% &98\\ 
24. A water pitcher &A pitcher with water in it 85\% &An empty pitcher 15\% &100\\ 
25. A wine bottle &A bottle with wine in it 84\% &An empty bottle 16\% &99\\ 
26. A water bucket &A bucket with water in it 83\% &An empty bucket 17\% &95\\ 
27. A potato bag &A full bag with potatoes in it 82\% &An empty bag with nothing in it 18\% &101\\ 
28. A transparent glass water pitcher &A full pitcher with water in it  74\% &An empty pitcher with no water in it  26\% &108\\ 
29. A wine decanter &A decanter with some wine in it  68\% &Just the decanter, nothing in it  32\% &101\\ 
30. A shopping bag &An empty bag with nothing in it  67\% &A full bag with things in it  33\% &96\\ 
31. A bottle &Just the bottle, nothing in it 61\% &A bottle with some liquid in it 39\%& 100\\\bottomrule
\end{tabular}
}
\end{table*}


%Table 6:
\begin{table*}[t]\centering
\caption{Types of stumper responses, in percents. (Rates may not sum to 100 due to rounding.)}
\begin{tabular}{ l  cc  C{2cm}  C{2cm}  C{2cm} C{2cm}}\toprule
Stumper &  Study & N & Solved the riddle & Gave non canonical “solution” & Gave 
unacceptable responses & Admitted to being stumped\\\midrule
Accountant&1 &96 &35 &33 &6 &25\\
Accountant&2 &99 &48 &0 &26 &25\\
Speeding Car&1 &108 &34 &24 &26 &16\\
Speeding Car&2 &99 &39 &20 &20 &20\\
Bus Ride&1 &102& 1& 33& 54& 12\\
Bus Ride&2 &101 &12 &19 &41 &28\\
Potato Bags&1 &94 &13 &56 &3 &28\\
Potato Bags&2 &95 &38 &26 &20&16\\\bottomrule
\end{tabular}
\end{table*}

Table 6 shows the types of answers given to the stumpers. Though most
respondents failed to come up with correct solutions, not all admitted to
feeling stumped; a substantial fraction offered unexpected answers of
varying quality. We should note that, since our subjects were deprived
of the normal exchange that usually accompanies the oral presentation
of riddles, as Mturk "workers" they might have regarded it as their
job to provide \textit{some} type of answer,\footnote{Indeed, almost
  all respondents turned out to be serious, diligent, workers, who
  gave us trustworthy data.} rather than admit defeat (see Appendix 3
for an amusing list of these “inventive” accounts).

\renewcommand{\baselinestretch}{1.0} \large\normalsize

\subsection{Direct test – The "Imagine" tasks}

The psychological accounts for the dominant instantiations lie at the
heart of this study. We aimed to prove that respondents who were
stumped were victims of a common instantiation determined by a
predictable psychological principle. In the Introduction, we proposed
specific accounts for why each of our vignettes stumps respondents.

The various Imagine tasks were meant to test these
accounts.\footnote{Since we dropped Farmer Joe, we shall also not
  report the data pertaining to our account of Farmer Joe – except to
  say that, not surprisingly, it indeed confirmed that "eggs" are
  visualized as "chicken eggs".} Respondents were asked to imagine
some target (Table 5, Imagine column), without any context or setting,
without action, and without being asked to explain anything. The two
possibilities presented for their choice are shown verbatim in Table
5's next two columns. The results consistently confirm our hypotheses:
the hypothesized option, in the middle column, is always the more
popular one.

Specifically:

\textbf{Accountant:} Most respondents imagined "an accountant" as male (row 1). 

\textbf{Speeding Car:} The results show that, although by default
scenes are seen in daytime (rows 3--7),\footnote{Note that three
  mentions of "black" in a row were visualized "by day" (61\%, row 6)
  at a lesser rate than three mentions of "white" (93\%, row 7), or
  than a single mention of "black" (82\%, row 5). This could be
  subject to further research.} explicit denial of light sources (rows
8--10) overcomes the default, triggering dark.

\textbf{Bus Rides:} Imagining a single item is easier than imagining
multiple items, even holding overall quantity of the variable constant
(rows 11--18). This is true for liquids and solids alike; for small
quantities and larger ones; for units that integrate into a "new"
form, like a \$5 bill; or congeal into a single piece, like cheese or
gold; or stay loose, like strawberries; or which, like keys, require
an imaginary ring to bind them; and regardless of frame – 4 ounces or
a pound.\footnote{One exception was predicted. The \$2 bill is the
  rarest of US bills, to the point where when asked to "list
  contemporary US currency bills", 53\% of our respondents did not
  even mention it. Correspondingly, 83\% of our respondents saw two
  singles. Apparently, at the extremes, statistical considerations
  also weigh in. }

It seems that we're on to a valid new psychological principle. Further
exploration (e.g., its boundary conditions) falls outside the present
scope of this paper, which is limited to discovery and demonstration of
existence.

\textbf{Potato Bags:} The table confirms that, if a container is
described along with its potential content (rows 23--29), the dominant
visualization will include that content – but otherwise
not. "Shopping" is not a specific content, and, hence, a "shopping
bag" is mostly seen empty. One might ask when content is specific
enough to be seen (e.g., "Laundry bag"? "Sewing box"?), but this, too,
is subject for future research. The form "container of content"
(e.g., bag of potatoes, bottle of wine), is necessarily full, for
syntactical reasons (rows 21--22).
 
\subsection*{Indirect test – The Recall task}

Respondents were asked to recall the lead stumper
verbatim.\footnote{In Study 1, our instructions said "Please write
  down the very first question you were asked" and nearly 25\%
  interpreted this literally as referencing only the final question
  ("How is that possible?"), rather than the riddle. In Study 2, we
  changed the phrasing of this inquiry and only 6.5\% misinterpreted
  it this way.} We searched their protocols for words missing from the
original stumper that might betoken the influence of the dominant
instantiation that is impeding solution, such as masculine pronouns in
the Accountant; "night" or "dark" in Speeding Car; “five dollar bill"
or "fiver" in Bus Ride; or "bag of potatoes" in Potato Bag. The
incidence of such overt “tells” is shown in Table 7 (these are, of
course, underestimates of the dominant instantiations, since
respondents don’t articulate everything that they visualize.)

%Table 7:
\begin{table*}[t]\centering
\caption{Results of the "Recall the first vignette" task. Cell entries are numbers (except in parentheses).}
\begin{tabular}{ L{2.2cm} c C{3cm} C{3cm} C{3.7cm}}\toprule
Stumper & Study & How many did not give the canonical answer? & Of those, how many recalled the stumper?  &Of those, how many included dominant visualization "tells"?\\\midrule
Accountant & 1 &  62 & \mc{2}{c}{\small (Data missing due to technical error.)}  \\
Accountant & 2 &51 &44 &15 (34\%) \\
Speeding Car & 1 &71 &52 &23 (44\%) \\
Speeding Car & 2  &60 &55 &35 (64\%) \\
Bus Ride & 1&101 &75 &19 (25\%) \\
Bus Ride & 2 & 89 &81 &29 (36\%) \\
Potato Bag & 1  &82 &55 &24 (44\%) \\
Potato Bag & 2 &59 &58 &44 (76\%) \\\bottomrule
\end{tabular}
\end{table*}

\renewcommand{\baselinestretch}{1.0} \large\normalsize

\section{General Discussion}

The premise of this paper is that riddles can teach us psychology by
focusing attention on the psychological principles that make them
"work". We chose a particular class of riddles, \textit{stumpers},
which all follow a similar format. A verbal scenario is presented,
which typically evokes a visual scene. Among the possible scenes
compatible with the text, one predominates. The stumper is designed so
that the dominant instantiation does not contain the answer. As a
result, those who cannot break free of the first scene they imagine
remain stumped.

We explore the psychological principles that determine the dominant
instantiation in each of a set of four stumpers. The Accountant relies
on representing an unspecified protagonist as a male. The Speeding Car
relies on Gricean norms, according to which mentioning the absence of
minor light sources implies that that the major one – the sun – is not
present. The Bus Ride relies on instantiating five dollars as a single
bill, due to cognitive economy. The Potato Bag relies on representing
the bag as full of potatoes, because containers referenced by the
contents they hold are visualized as full of those contents.

These accounts were supported by two means. First, respondents were asked
to imagine target objects or events, devoid of context or narrative
setting, and report their mental imagery directly. Second, respondents
were asked to recall their stumper verbatim, and we searched for
"tells" that revealed what they had imagined.

As noted earlier by Tversky and Kahneman (1982), prototypical
visualization is not driven solely by statistical frequency or base
rates. (1) Although the prototypical accountant is apparently male,
the typical accountant actually is not (as of 2015 in the U.S., nearly
60\% of all U.S. accountants were female –
\url{http://www.bls.gov/cps/cpsaat11.pdf}). (2) Driving and most other
human activities predominantly occur during daytime, yet the speeding
car was visualized at night. (3) Although one \$5 bill is easier to
imagine than five \$1 bills, \$1 bills outnumber \$5 bills by more
than 5 to
1 (\url{http://www.cnbc.com/2015/04/21/100s-closing-in-on-1s-for-most-common-currency.html}).
(4) Since empty potato bags are usually thrown away, one can argue
that base rates do contribute to the dominant visualization here. But
the prototypical image does not reflect statistical incidence alone:
e.g., a wine decanter rarely contains wine, but is typically imagined
with wine in it.

Recall that we began our paper by declining to define
riddles. Stumpers are a particular kind of riddle, to which a common
reaction is puzzlement. We did not aim for, nor do we offer, a general
theory of what stumps people, whether in riddles or in other forms of
problem solving. Our focus was on what determines how people visualize
a scene to fit the stumpers we used, but we could instead have focused
on linguistic stumpers, such as those below
\begin{itemize}
\item[i] What gets wet as it dries?
\item[ii] How can a boat be full of people, without a single person on it?
\end{itemize}
These riddles work because people interpret a polysemic word in a way
that creates semantic conflict. They imagine the \textit{same} object
is getting both wet and dry, and that \textit{full} and
\textit{single} both reference the number of the boat’s occupants. Had
we focused on this class of riddles, we might have sought to
understand the psychological impediments to interpreting words
differently.

Our focal riddles work because the construal evoked by the scenario
dominates attention in a way that prevents other construals that are
required to solve the problem. As shown in Appendix 4, this mechanism
is part of a broader class of phenomena in the JDM literature, in
which a particular mental construction (that need not be a visual
image) displaces or inhibits alternate constructions.

JDM studies typically focus on how this process leads to non-normative
responses, but with stumping riddles, it can inhibit the production of
any response.

That may be primarily a difference of approach. In other words, if
known stumpers can be vehicles for discovering novel psychological
effects, perhaps known effects can be vehicles for inventing novel
stumpers. Below are some preliminary attempts to turn a few known
effects from the JDM literature into stumpers. These are intended more
as illustrations of the concept, with no pretensions to being great
riddles, which (like good jingles, cartoons or logos) require more
than the implementation of a feasible idea.
\begin{itemize}
\item[i] In 1957, in Rhode Island, twice as many pedestrians were
  killed crossing on a green light as on a red light. Explain (based
  on Huff, 1959).
\item[ii] A notebook and pencil cost \$1.10 in total. The notebook
  costs a dollar more than the pencil. Bonnie bought a pencil, handed
  the cashier a dime, and received some change. Explain (based on
  Frederick, 2005).
\item[iii] Four cards show a letter on one side and a digit on the
  other. The cards' upside faces show, respectively, A, Z, 4, 1. Check
  the following claim: when there's a vowel on one side of a card,
  there's an even number on the other. It costs 6 points to turn each
  of the letter cards over, 4 points to turn the 4 card over, and 1
  point to turn the 1 card over. Nellie only had nine points, which
  was enough to turn over all and only the cards needed to check the
  claim. Explain (based on Wason, 1968).
\item[iv] There are three closed boxes. One has \$10,000 in it, the
  other two are empty. Lou chose a box. The game show host promises to
  reveal one of the losing boxes from the others that remain. Lou is
  given the opportunity to switch the box he initially chose to the
  remaining closed box, for \$100. He happily pays. Explain (based on
  Selvin, 1975).
\end{itemize}

Of course, since we are psychologists (rather than comedians or
magicians), we care less about using known psychological effects to
construct new riddles than using existing riddles to discover new
psychological insights.

\section{References}
\begin{hangparas}{1em}{1}

Bar-Hillel, M. (1980). The base-rate fallacy in probability judgments. \textit{Acta Psychologica, 44}(3), 211--233.

Bar-Hillel, M.,\& Falk, R. (1982). Some teasers concerning conditional probabilities, \textit{Cognition, 11}(2), 109--122.

Benartzi, S., \& Thaler, R. H. (1995). Myopic loss aversion and the equity premium puzzle. \textit{The Quarterly Journal of Economics, 110}(1), 73--92.

Camerer, C., Loewenstein, G., \& Weber, M. (1989). The curse of knowledge in economic settings: An experimental analysis. \textit{Journal of Political Economy, 97}, 1232--1254.

Duncker, K. (1945). On problem-solving. \textit{Psychological Monographs}, \textit{58}(5), 270.

Fagerlin, A., Wang, C. \& Ubel, P.A. (2005). Reducing the influence of anecdotal reasoning on people’s health care decisions: Is a picture worth a thousand statistics? \textit{Medical Decision Making. 25}(4), 398--405.

Fischhoff, B. (1975). Hindsight is not equal to foresight: The effect of outcome knowledge on judgment under uncertainty. \textit{Journal of Experimental Psychology: Human Perception and Performance, 1}(3), 288--299.

Frederick, S. (2005). Cognitive reflection and decision making. \textit{The Journal of Economic Perspectives, 19}(4), 25--42.

Garth, T. R. (1920). The psychology of riddle solution: An experiment in purposive thinking. \textit{Journal of Educational Psychology, 11}(1), 16--32.

Grice, H. P. (1975). Logic and Conversation. In P. Cole \& J.L. Morgan
(eds.) \textit{Syntax and Semantics, Vol. 3, Speech Acts,} 41--58, New
York: Academic Press.


Herrnstein, R.J., \& Prelec, D. (1992a).  Melioration.  In G. Lowenstein and J. Elster (eds.)  \textit{Choice Over Time }(ch. 10, 235-263).  New York:  Russell Sage Foundation.


Herrnstein, R.J., \& Prelec, D. (1992b).  A theory of addiction.  In G. Lowenstein and J. Elster (eds.)  \textit{Choice Over Time} (ch. 13, 331-359).  New York:  Russell Sage Foundation.


Heyman, G. M. (1996). Resolving the contradictions of addiction. \textit{Behavioral and Brain Sciences, 19}(4), 561--574.

Huff, D. (1959). \textit{How to Take a Chance.} Harmondsworth: Pelican Books.

Kahneman, D. (2011). \textit{Thinking, Fast and Slow.} Macmillan.

Kahneman, D., \& Lovallo, D. (1993). Timid choices and bold forecasts: A cognitive perspective on risk taking. \textit{Management Science, 39}(1), 17--31.

Kosslyn, S. M. (1980). \textit{Image and Mind.} Harvard University Press.

Legrenzi, P., Girotto, V., \& Johnson-Laird, P. N. (1993). Focusing in reasoning and decision making. \textit{Cognition, 49}(1), 37--66.

Luchins, A. S. (1942). Mechanization in problem solving: The effect of Einstellung. \textit{Psychological Monographs, 54} (whole no.\ 248).

Maier, N. R. (1930). Reasoning in humans. I. On direction. \textit{Journal of Comparative Psychology, 10}(2), 15--43.

Margolis, H. (2008). A note on neglect defaulting. \textit{Judgment and Decision Making, 3}(4), 355--363.

Martyna, W. (1978). What does ‘he’ mean?. \textit{Journal of Communication, 28}(1), 131--138.

McCaffery, E. J., \& Baron, J. (2006). Isolation effects and the neglect of indirect effects of fiscal policies. \textit{Journal of Behavioral Decision Making, 19}(4), 289--302.

Read, D., Loewenstein, G., \& Rabin, M. (1999). Choice bracketing. \textit{Journal of Risk and Uncertainty, 19}(1--3), 171--197.

Rosch, E. (1978). Principles of categorization. In: E. Rosch \& B.B. Floyd (eds.) \textit{Cognition and Categorization}, 27--48. Hillsdale, NJ: Lawrence Erlbaum Associates.

Schkade, D. A., \& Kahneman, D. (1998). Does living in California make people happy? A focusing illusion in judgments of life satisfaction. \textit{Psychological Science, 9}(5), 340--346.

Selvin, S. (1975). On the Monty Hall problem.\textit{ American Statistician, 29}(3), 134--134.

Simonson, I. (1990). The effect of purchase quantity and timing on variety-seeking behavior. \textit{Journal of Marketing Research, 28,} 150--162.

Tversky, A., \& Kahneman, D. (1982). Judgments of and by representativeness. In D. Kahneman, P. Slovic \& A. Tversky (eds.). \textit{Judgment under Uncertainty: Heuristics and Biases} (ch.\ 6, 84--98). Cambridge: Cambridge University Press. 

Tversky, A., \& Kahneman, D. (1983). Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment. \textit{Psychological Review, 90}(4), 293--315.

Wason, P. C. (1968). Reasoning about a rule. \textit{The Quarterly Journal of Experimental Psychology, 20}(3), 273--281.

Wittgenstein, L. (1953). \textit{Philosophical Investigations}. London, Basic Blackwell.

Zwaan, R. A., \& Radvansky, G. A. (1998). Situation models in language comprehension and memory. \textit{Psychological Bulletin, 123}(2), 162--185. 

%\vfill
%\break
\end{hangparas}

\bigskip

\subsection*{Appendix 1: The stumpers (Study 1 versions)}

\begin{tabular}{@{}l p{6in}@{}}\toprule
Accountant &“That attorney is my brother”, stated the accountant. Later, that attorney stated that he has no brothers. Both were telling the truth.\\ & How is that possible?\\
Speeding Car &  A black man all dressed in black was walking alone down the middle of a black asphalt road. The man had no flashlight, and no streetlights were on. A car was travelling towards him at full speed, its headlights off, yet managed to see him in time to brake and avoid running him over. \\ & How was the driver able to see the man? \\
Bus Ride & Bus tickets cost one dollar each, unless you buy a card, which is good for six rides. The card costs five dollars. A passenger boards the bus, and hands the driver five dollars, without saying a word. The bus driver takes her money, and hands her a card, with one of the six holes punched out. The passenger thanks the driver and takes a seat. \\ & How did the driver know that this was what the passenger wanted?\\ 
Potato Bags & Tom broke his arm badly, and it was in a cast for weeks. When the cast was removed, he trained as follows: He extended his arm to the side, straight, and while holding a small potato bag, maintained this position for as long as he could. Once he could keep it that way for a whole minute, the small bag was replaced by a medium bag, and the exercise repeated. Once he could hold the medium bag for a full minute, it was replaced by a large bag. As soon as Tom could hold a large potato bag that way for an entire minute, one potato was added to the bag. Tom’s arm collapsed almost immediately. \\ & How come? \\
Farmer Joe & Farmer Joe eats two fresh eggs from his own farm for breakfast every day. Yet there are no chickens on his farm. \\ & Where does Farmer Joe get his eggs?\\\bottomrule
\end{tabular}

\newpage
\rule{0in}{0in}
\newpage

\subsection*{Appendix 2: Instructions}

\paragraph{Opening Instructions for Study 1:}

On some of the following pages you will find short vignettes. Read them carefully, as you will be asked one or two questions about them. On other pages you will be asked to visualize some objects in your imagination, and to report what you see in your "mind's eye". We hope you enjoy the task. Thank you for your cooperation. 


\paragraph{Opening Instructions for Study 2:}

This study is comprised of several different tasks, and you will be instructed about them as you proceed. On a couple of the following screens you will find short vignettes. Read them carefully, as you will be asked a couple of questions about them. On several other pages you will be asked to visualize some things in your imagination, and to report what it is that you see in your “mind’s eye”. Dwell on them for a while, and try to flesh them out with some detail.

A few other little problems will follow, which you will be asked to
solve. The survey will conclude with a series of moral
judgments.\footnote{These data do not pertain to the present study.}
It is separate from the earlier questions, and payment does not depend
on your answers to it. Throughout the study, when you wish to advance
to the following screen, click on the NEXT sign at the bottom of the
screen. It will not be possible to return to earlier screens. We hope
you enjoy the task.  Thank you for your cooperation!


\paragraph{Instructions for the lead stumper in Study 2.}

If you can think of a good answer, please enter it in the text box. If
you can’t come up with what seems like a satisfying answer, then
please just enter the words “I am stumped.” Note: We are not trying to
test your ability to come up with some original, creative, atypical,
exotic “solution.” In other words, you don't need to struggle to say
something, if you know that it couldn’t be the expected answer. We are
just trying to understand when people do or do not feel stumped. Your
payment does not depend on whether you are stumped, nor on any answers
you may provide.

\subsection*{Appendix 3: Examples of respondents' answers.}

Some respondents who failed to solve the stumper nonetheless gave answers that were consistent with the stumper's premises, and we deemed them acceptable. Popular examples follow.

\textbf{Accountant:} In Study 1, 28\%\footnote{These percents are smaller than those shown in the "acceptable" column, because they are examples of such answers, but not a comprehensive list of them.} of the respondents pointed out that the salutation "brother" has more than one meaning – e.g., friend, fellowman, etc. In Study 2, we explicitly stated that the two protagonists shared parents, leaving no room for acceptable non-canonical answers. 

\textbf{Speeding Car:} In Study 1, 22\% of the respondents enlisted the moon as the source of light by which the driver saw the man on the road. So in Study 2's version, the moon's presence was explicitly ruled out. However, some respondents sought to explain how the driver brakes in time, and assumed the road had 2 lanes, with cow and truck occupying different sides of the road. 

\textbf{Bus Ride:} In Study 1, 26\% of the respondents surmised, in one form or other, that the passenger was a repeat passenger, recognized by the driver, who therefore knew just what transaction was desired. So in Study 2 it was explicitly stated that the passenger was a first time rider. 

\textbf{Potato Bags:} In Study 1, 53\% of the respondents pointed out, in one way or another, the truism that everyone has their physical limits, so the added potato was the proverbial straw that broke the camel's back. In Study 2 the training story was replaced by a market story; 25\% assumed that the "expensive" potato was a very special one – very big, or of superior quality.

Some "solutions" called for more creativity, and were rarer. Examples:

\textbf{Accountant:} Five Study 1 respondents answered that the accountant must have died between the two utterances.

\textbf{Speeding Car:} One respondent conjured a firefly as the source of light. 

\textbf{Bus Ride:} A few subjects suggested that the silent passenger used some form of non-verbal communication (e.g., a written note requesting a card, or holding up 5 fingers).

\textbf{ Potato Bags:} One respondent suggested that “There is one potato left in the market and the price has been raised very high.”

We deemed many answers "unacceptable". Examples:
Some were inane (e.g., the 30\% of all Bus Rides who denied, directly or indirectly, that there could be any reason for handing the driver \$5 except for wanting a card, when of course a reason could be that the passenger had nothing smaller than a fiver). Some were clutching at straws (e.g., the 12\% of all Speeding Car respondents who thought that even in the absence of any light, the whites of one's eye or of one's toothy grin, or mere movement, would suffice to make the man visible). Some were in outright defiance of the vignette's premises (e.g., in Study 2, those who guessed that the attorney is adopted, although told explicitly that the protagonists have the same parents). And some were just plain fantastical (e.g., the respondent who wrote: “He (Tom) was hypnotized, and the "magic" phrase or action to make him drop his arms was executed"). 

\onecolumn

\begin{table}[t]
\subsection*{Appendix 4: Examples of robust JDM effects where having one thing in mind can block or hinder other considerations which are normatively relevant. (Table courtesy of Jon Baron.)}
\centering
%\caption*{Appendix 4: Examples of robust JDM effects where having one thing in mind can block or hinder other considerations which are normatively relevant. (Table courtesy of Jon Baron.)}
\begin{tabular}{L{11cm}L{5cm}}\toprule
\textbf{JDM Effect} & \textbf{Sample Reference} \\
\textbf{Base-Rate Fallacy} \newline Individuating information blocks the judged relevance of background base-rates. &Bar-Hillel (1980)\\ 
\textbf{Non-regressive prediction (along with \textit{representativeness} and \textit{availability}, the major judgment heuristics)} \newline WYSIATI (What You See Is All There Is).& Kahneman (2011)\\
\textbf{The Volvo Effect} \newline A single vivid concrete anecdote overrides plentiful statistical evidence. & Fagerlin, Wang \& Ubel (2005)\\
\textbf{Hindsight Bias} \newline Outcome knowledge hinders access to pre-knowledge prediction uncertainty.& Fischhoff (1975)\\ 
\textbf{Curse of Knowledge} \newline A known fact prevents full imagination of a state of belief without that knowledge.& Camerer, Loewenstein \& Weber (1989)\\
\textbf{Isolation Effect} \newline People focusing on immediate consequences of an option fail to think of side effects or indirect effects. & McCaffery and Baron (2006)\\
\textbf{Focusing Illusion} & \\
1. A salient attribute relevant to a comparison blocks consideration of other attributes relevant to the same comparison.& Schkade \& Kahneman (1998)\\
2. An initial mental model blocks consideration of alternative models. & Legrenzi, Girotto \& Johnson-Laird (1993) \\
\textbf{Choice Bracketing} \newline Choice considered in isolation prevents seeing the broader picture determined by the entire choice set. Similar to other dichotomies:& Read, Loewenstein \& Rabin (1999)\\
Sequential vs.\ simultaneous choice & Simonson (1990)\\
Narrow vs.\ broad decision frames& Kahneman \& Lovallo (1993)\\
Isolated vs.\ distributed choice& Herrnstein \& Prelec (1992a,b)\\
Local vs.\ overall value functions& Heyman (1996)\\ 
Myopic loss aversion & Benartzi \& Thaler (1995)\\
\textbf{Neglect Defaulting}& \\
People who neglect to continue reasoning after the first step may fail to see that further steps can overturn the default.  & Margolis (2008)\\\bottomrule
\end{tabular}
\vspace{2in}
\end{table}

\end{document}
