Langman's Paradox:
How Is It Possible?

Martin Gardner attributes the following variant of Hooper's paradox Dr. Harry Langman of New York city. As before, Langman's paradox is based on some properties of the Fibonacci numbers. More accurately, its explanation lies in the identity that also justifies Curry's paradox.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

What if applet does not run?

To make the paradox work for small values (3, 4, 5) of the parameter, check the "Cheat" box. For greater values, the effect is virtually imperceptible. To see how the paradox works, drag the pieces from one rectangle into their designated locations in the other.


  1. M. Gardner, Mathematics Magic and Mystery, Dover, 1956, pp. 137-138

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