Clumsy wagon

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Clumsy wagon
Clumsy wagon.jpg
On display at Matemateca
Type Hands-on, puzzle
Topics Plane curves

Clumsy wagon is an exhibit on display at Matemateca.


The exhibit presents a wooden board on a table. A groove (the ‘rail’) traced along a winding pathway goes from one side to the opposite side of the board. A piece of wood (the ‘wagon’) is placed over the groove with two pins inside it. The pins have ‘heads’ under the groove that prevent the wagon from being removed off the rail. As the wagon has always two of its points necessarily on the rail, its movement is particularly difficult if the path is very winding. The goal is to carry the wagon from one endpoint to the other.

Activities and user interaction

Users usually have no difficulty into figuring out what they have to do, after reading a hint or listening to the mediator instruction. Then they are surprised by how painful it is to handle the wagon in order to achieve the task. There are moreover some ‘traps’ in the way, where the user may wander in circles if they do not stop to think at some special positions. The exhibit poses a very unusual problem, but in some sense also very real. It is defying to see a unidimensional path without bifurcations making people lost. The path does not indicate clearly where are the difficulties and what will be the necessary movement to cross it from one side to the other. This is so because the ‘right’ way of seeing the problem is through the map of level curves of a two-dimensional function. In the end, they get there, but a mysterious atmosphere remains.

Mathematical background

This exhibit is inspired by the “Moving needle problem”, appeared in the blog Area 777, authored by Conan Wu. There is a conjecture saying that wagons of arbitrary size can cross arbitrarily chosen paths if they start and end at two points of the same line. The conjecture is stated for Cr smooth paths, with r greater than zero, since there are C0 counter-examples. The interesting point here is that the problem can be reformulated in terms of the connectivity of level curves of a suitable two-variable function.

History and museology

The exhibit originated in Matemateca, inspired by the ‘moving needle problem’ in the mentioned blog, Area 777. We are not aware that any other museum constructed some- thing similar so far.