Page 438 - Deep Learning

Notes to Pages 124–127 421

obtain a certain amount of water (e.g., 4) through a series of pouring actions. See
Atwood and Polson (1979) and Atwood, Masson and Polson (1980) for a process
model of how people solve this task.
58. See, e.g., McNeil and Alibali (2005), Smith (1995) and Wiley (1998).
59. Chronicle, MacGregor and Ormerod (2004), Chronicle, Ormerod and MacGregor
(2001) and MacGregor, Ormerod and Chronicle (2001).
60. The N-Balls Problem is a generalization of the Coin Problem studied by Schooler,
Ohlsson and Brooks (1993). In an N-Balls problem, the task is to determine
which of N similar-looking balls is heavier than the rest, using no other tool or
resource than a balance beam that can only be used a maximum of X times. For
example, to find the heavier ball among 7 balls with 2 uses of the balance beam,
first weigh any 3 balls against any of the remaining 3 balls. If the two groups
weigh the same, the heavier ball is the one not weighed and no second step is
needed. If one group is heavier, it contains the heavier ball. Next compare any
two balls in that group. The possible outcomes are obvious, either one of the
compared balls is heavier or else the one not weighed is the heavier. The class of
N-Balls Problems has been investigated by James MacGregor, Thomas Ormerod
and the late Edward Chronicle (MacGregor, personal electronic communication,
September 13, 2007).
61. Wallas (1926, pp. 79–82). Although Wallas was first to label it, the incubation con-
cept goes back to the personal testimony of Poincaré, von Helmholz and others
(Ghiselin, 1952; Hadamard, 1949/1954).
62. See, e.g., Lubart (2000–2001).
63. Kaplan (1989) and Sio and Ormerod (2009). An unpublished manuscript by
R. Dodds, T. B. Ward and S. M. Smith entitled “Incubation in Problem Solving
and Creativity” supports the same conclusions as the latter two reviews (personal
electronic communication, S. M. Smith, October 3, 2007).
64. Weisberg (1986, p. 30).
65. Simon (1966).
66. The oldest extant version of the Archimedes bathtub story appears in Marcus
Vitruvius Pollio, De Architectura Libri Decem (The ten books on architecture), Book
IX, paragraphs 9–12. The paragraphs are available in English translation on line at
http://www.math.nyu.edu/~crorres/Archimedes/Crown/Vitruvius.
The full English translation of Vitruvius’ text is also available online at: http://
penelope.uchicago.edu/Thayer/E/Roman/Texts/Vitruvius/home/html.
Classicists universally agree that there is little reason to believe the story.
Vitrivius was not an eyewitness or even a contemporary. He was a Roman
who lived in the first century b.c., approximately 200 years after Archimedes.
Other authors in antiquity also tell the bathtub story, but they lived even later so
there is no way of knowing whether they had independent sources or repeated
Vitruvius’s story. One strong reason for believing that the story is not true is that
the difference between the amount of water displaced by a pure gold crown and
a crown of the same weight but made of gold mixed with some lighter metal is
so small that it could not have been measured with any accuracy with the tools
available to Archimedes; see the calculations on display at http://www.math.nyu.
edu/~crorres/Archimedes/Crown/CrownIntro.html.