Manfred Börgens
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How many liars?

Everybody lies.
Our figures are a total mess.
Our statistics are less reliable than the horoscopes in the yellow press.

Authentic quotation from a letter, written by the director of the Italian national statistics authority just before he committed suicide in his office.

A central part of the following problem was suggested by Mr Georg Arends who received his diploma in mathematics from University of Applied Sciences Giessen-Friedberg in 1992.

The Ministry for Truth of the Kingdom of Veritania has got problems with the subjects, too. A certain quarter of the capital attracts special attention: From there come an exorbitant number of wrong tax returns, fraudulent insurance claims, untrue census informations etc. The minister wants to make an example of the 210 inhabitants of the quarter and gives instructions to his officers. They have to choose  N  houses, and  N  floors in each of the chosen houses, and  N  persons living in each of the chosen floors. All these persons have to be brought to the ministry for questioning.

The subjects chosen by the officers gather in the ministry, but their examination is delayed a little. As these people know each other well they use the delay for discussing the imminent examination. There are two groups: One group will give in and speak the truth in the examination, the other group will resist and make untrue statements only.

The interviews reveal to the officers the behaviour of the two groups  -  the truth-tellers and the liars. They decide to ask each of them how many liars are in the sample of people chosen to be examined. Here are the records of the first answers given:

"At least four."
"More than seven."
"Not only one."
"At least six."
"All are liars."


After having finished the interviews the officers realize that all answers are different - no two of them meant the same. A sorted list of the answers (brought into a mathematical form with  L  the number of liars) would read like this:

L ≥ 1
L ≥ 2
L ≥ 3
L = "all"

How many persons were interviewed? How many of them lied?


last update  2005-02-17
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