Frédéric Dandurand, P. Eng.

Professional software engineer for enterprise and research applications

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Gizmo Problem Solving - The counterfeit coin problem

Can you find, with three uses of a balance scale, the one gizmo that is either heavier or lighter than the rest of a set of 12 gizmos? Try it out in the applet below!

This task has been investigated in a series of scientific studies and simulations of learning and reasoning.

Have fun with this brain teaser and remember that this difficult (yet solvable) task is not in any way a test of your general learning ability or intelligence. In fact, most participants in experiments involving this task, including smart university students in engineering and psychology, found it very hard to reliably find the target gizmo without ever having to guess! Here is what two of them said during Think Aloud Protocols:

"It's impossible unless you get lucky." (Participant 11)

"I can't even imagine how I'm going to do this!" (Participant 13)

Giving up? Click on the <I am stuck - please help me!> button for instructions and demonstrations. Also, you can consult Figure 2 of this paper, and for an exhaustive list of all solutions, read Appendix 1 of this paper (starting on p. 39).

Note: Java required to view the applet below.

The science of the Gizmo task

As part of my research, I have used the Gizmo task to study complex and planning-intensive problem solving. To get started, you can consult a summary of the work (in slides format) presented during a talk at McGill University. For experimental work, there are also presentation contents from talks on learning by imitation, by instruction, and by reinforcement, the role of memorizing and understanding in imitation, strategies, heuristics and biases, and online experiments in psychology. For computational modeling work, you can consult presentation contents on solving the task with reinforcement learning.

More specifically, my work on problem solving involves:

studying how people learn to solve problems using imitation, instruction or reinforcement learning, and what strategies, heuristics and biases can account for their performance
using computational models (computer simulations) to simulate human learning by imitation, instruction and reinforcement
investigating if participants can generalize what they learn by imitation to new and more difficult tasks
analyzing and comparing performance when participants take part in the experiment in the lab versus online.
using computational models to simulate how humans learn to solve the task using learning based on a distance-reduction heuristic

For a mathematical perspective on the Gizmo task (often called the Coin problem), and a reference to seminal work in cognitive psychology, visit the reference section.

References: peer-reviewed journal articles and conference papers

Experimental studies

Dandurand, F., Shultz, T. R., & Onishi, K. H. (2008). Comparing online and lab methods in a problem-solving experiment. Behavior Research Methods, 40(2), 428-434.
Dandurand, F., Samuel, S., & Shultz, T. R. (2007). Imitation learning in problem solving tasks: Memorizing or understanding? In the Proceedings of Cognitio 2007. Newcastle upon Tyne, United Kingdom: Cambridge Scholars Publishing.
Dandurand, F., Shultz, T. R., & Onishi, K. H. (2007). Strategies, heuristics and biases in complex problem solving. In the Proceedings of the Twenty-Ninth Meeting of the Cognitive Science Society (CogSci 2007) (pp. 917-922). New-York: Lawrence Erlbaum Associates.
Dandurand, F., Bowen, M. & Shultz, T. R. (2004). Learning by imitation, reinforcement and verbal rules in problem-solving tasks. In the Proceedings of the Third International Conference on Development and Learning (ICDL'04) Developing Social Brains. La Jolla, California, USA: The Salk Institute for Biological Studies.

Computational models

Dandurand, F., Shultz, T. R., & Rey, A. (2012). Including cognitive biases and distance-based rewards in a connectionist model of complex problem solving. Neural Networks, 25, 41-56.
Dandurand, F., & Shultz, T. R. (2009). Connectionist models of reinforcement, imitation, and instruction in learning to solve complex problems. IEEE Transactions on Autonomous Mental Development, 1(2), 110 - 121.
Dandurand, F., Shultz, T. R., & Rivest, F. (2007). Complex problem solving with reinforcement learning. In the Proceeding of the Sixth IEEE International Conference on Development and Learning (ICDL-2007) (pp. 157-162): IEEE.

Mathematical analyses and seminal work

Guy, R. K., & Nowakowski, R. J. (1995). Coin-weighing problems. American Mathematical Monthly, 102(2), 164-167.
Halbeisen, L., & Hungerbuhler, N. (1995). The general counterfeit coin problem. Discrete Mathematics, 147(1), 139-150.
Simmel, M. L. (1953). The coin problem: a study in thinking. American Journal of Psychology, 66, 229-241.