Monthly Archives: December 2013

Crates of Fruit Puzzle

You work at a fruit packing plant.

There are 3 crates in front of you. One crate contains only apples. One crate contains only oranges. The other crate contains both apples and oranges.

And each crate is labeled. One reads “apples”, one reads “oranges”, and one reads “apples and oranges”.

But the labeling machine has gone crazy and is now labeling all boxes incorrectly.

If you can only take out and look at just one of the pieces of fruit from just one of the crates, how can you label ALL of the crates correctly?

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Hotel room solution

We have to be careful what we are adding together.

Originally, they paid £30, they each received back £1, they now have only paid £27. Of this £27, £25 went to the manager for the room and £2 went to the bellboy.

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The Liar’s Paradox

In philosophy and logic, the liar’s paradox  is the statement “this sentence is false.” Trying to assign to this statement a classical binary truth value leads to a contradiction (see paradoxes).

If “this sentence is false” is true, then the sentence is false, which is a contradiction. Conversely, if “this sentence is false” is false, then the sentence is true, which is also a contradiction.

This paradox is obliquely related to the City of Lies or Truth puzzle.

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City of Lies or Truth Puzzle

You are at an unmarked intersection …one way is the City of Lies and another way is the City of Truth.  Citizens of the City of Lies always lie.  Citizens of the City of Truth always tell the truth.

A citizen of one of those cities (you don’t know which) is at the intersection. What question could you ask to them to find the way to the City of Truth?

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The Hotel Room Puzzle

Three people check into a hotel. They pay £30 to the manager and go to their room. The manager suddenly remembers that the room rate is £25 and gives £5 to the bellboy to return to the people. On the way to the room the bellboy reasons that £5 would be difficult to share among three people so he pockets £2 and gives £1 to each person. Now each person paid £10 and got back £1. So they paid £9 each, totalling £27. The bellboy has £2, totalling £29. Where is the missing £1?

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How many triangles are there?



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Russell’s paradox

In the foundations of mathematics, Russell’s paradox (also known as Russell’s antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naive set theory created by Georg Cantor led to a contradiction. The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl, and other members of the University of Göttingen.

According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell’s paradox.



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Affirming the Consequent

Affirming the consequent, sometimes called converse error or fallacy of the converse, is a logical fallacy of inferring the converse from the original statement.  For instance, consider the following argument:

Premise 1: If Fiona won the lottery last night, she’ll be driving a red Ferrari today.

Premise 2: Fiona is driving a red Ferrari today.

Conclusion: Therefore, Fiona won the lottery last night.

An argument of this form is invalid, that is, the conclusion can be false even when the premises are true.  Since Premise 1 was never asserted as the only sufficient condition for Premise 2, other factors could account for Premise 2, such as:

  • Fiona could have inherited a large amount of money; or
  • She might just be borrowing the car; or
  • Perhaps she even stole it.

The fallacious argument has the general form:

Premise 1: If P, then Q.

Premise 2: Q.

Conclusion: Therefore, P.

Arguments of this form can sometimes seem superficially convincing, as in the following example:

Premise 1: If I have the flu, then I have a sore throat.

Premise 2: I have a sore throat.

Conclusion: Therefore, I have the flu.

But having the flu is not the only cause of a sore throat since many illnesses cause sore throat, such as the common cold or strep throat.

In contrast, a logically valid form of a similar argument would be:

Premise 1: If P, then Q.

Premise 2: P.

Conclusion: Therefore, Q.

This valid form of argument is a classical logical form known as modus ponens, with a history going back to antiquity.

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The free will paradox

The following set of propositions is described by McKenna (2009:1.5)[1] as the classical formulation of the problem of free will:

1)      ‘Some person, at some time, could have acted otherwise than she did.

2)      Actions are events.

3)      Every event has a cause.

4)      If an event is caused, then it is causally determined.

5)      If an event is an act that is causally determined, then the agent of the act could not have acted otherwise than in the way that she did’.

This formulation involves a mutually inconsistent set of propositions, and yet each is consistent with in our contemporary conception of the world, producing an apparent paradox. It is related to another apparent paradox known as Buridan’s Ass.


[1] McKenna, Michael, ‘Compatibilism’, The Stanford Encyclopedia of Philosophy(Winter 2009 Edition), Edward N. Zalta (ed.), URL = <>

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Faulty risk assessment

by Tim Harding B.Sc., B.A.

Risk’ may be defined as the probability of something bad happening multiplied by the resulting cost/benefit if it does happen.  Risk analysis is the process of discovering what risks are associated with a particular hazard, including the mechanisms that cause the hazard, then estimating the probability that the hazard will occur and its consequences.

Risk assessment is the determination of the acceptability of risk in two dimensions – the likelihood of an adverse event occurring; and the severity of the consequences if it does occur,[1] as illustrated in the diagram below.

risk assessment diagram

By way of illustration, the likelihood of something bad happening could be very low, but the consequences could be unacceptably high – enough to justify preventative action.  Conversely, the likelihood of an event could be higher, but the consequences could low enough to justify ‘taking the risk’.

In assessing the consequences, consideration needs to be given to the size of the population likely to be affected, and the severity of the impact on those affected.  This will provide an indication of the aggregate effect of an adverse event. For example, ‘major’ consequences might include significant harm to a small group of affected individuals, or moderate harm to a large number of individuals.[2]

A fallacy is committed when a person focuses on risks in isolation from benefits, or takes into account one dimension of risk assessment without the other dimension.  To give a practical example, the new desalination plant to augment Melbourne’s water supply has been called a ‘white elephant’ by some people, because it has not been needed since the last drought broke. But this criticism ignores the catastrophic consequences that could have occurred had the drought not broken. In June 2009, Melbourne’s water storages fell to 25.5% of capacity, the lowest level since the huge Thomson Dam began filling in 1984. This downward trend could have continued at that time, and could well be repeated during the next drought.


Melbourne’s desalination plant at Wonthaggi

No responsible government could afford to ‘take the risk’ of a major city of 4 million people running out of water.  People in temperate climates can survive without electricity or gas, but are likely to die of thirst in less than a week without water, not to mention the hygiene crisis that would occur without washing or toilet flushing.  The failure to safeguard the water supply of a major city is one of the most serious derelictions of government responsibility imaginable.

A similar example of fallacious reasoning is in the area of climate change, where the public debate wrongly focusses on whether the science is true or false, rather than on the risks and consequences of it being true or false. This video explains the fallacy quite well.

Other examples of this fallacy are committed by the anti-vaccination and anti-fluoridation movements, often accompanied by conspiracy theories.  They both focus on the very tiny likelihood of adverse side effects without considering the major benefits to public health from the vaccination of children and the fluoridation public water supplies.  Hardly anybody has ever died or become seriously ill in Australia from the side effects of vaccination or fluoridation [3]; yet large numbers of people have died from the lack of vaccination.[4] The allegation of a link between vaccination and autism has been discredited, retracted and found to be fraudulent.  The benefits of fluoridation are well documented. The risks of general anaesthesia for multiple tooth extractions are not to be idly contemplated for children, and far outweigh the virtually nonexistent risk from fluoridation.[5]

[1] This is based on the Australian/New Zealand Standard for Risk Management.

[2] State Government of Victoria (2007) Victorian Guide to Regulation 2nd edition. Department of Treasury and Finance, Melbourne.

[3] In 2010, increased rates of high fever and febrile convulsions were reported in children under 5 years of age after they were vaccinated with the bioCSL Fluvax® vaccine. bioCSL Fluvax® has not been registered for use in this age group since late 2010 and therefore should not be given to children under 5 years of age. The available data indicate that there is a very low risk of fever, which is usually mild and transient, following vaccination with the other vaccine brands: Agrippal®; Fluarix®; Influvac®; and Vaxigrip®.  Any of these vaccines can be used in children aged 6 months and older. This and further information on flu vaccination is available here.

[4] The former Commonwealth Chief Medical Officer, Prof. Jim Bishop has argued that the flu vaccination program “changed dramatically the flu outlook for this country”, with admissions to intensive care from swine flu falling from 681 in 2009 to just 60 in 2010, and hospitalisations dropping from nearly 5000 to 600. Swine flu killed 191 Australians in 2009 and 36 in 2010. In contrast, seasonal flu killed 1796 Australians that year – but, unlike swine flu, the victims were mainly the frail and elderly. Prof. Bishop cautioned that one in every three hospital patients were “perfectly fit and well” before they caught swine flu, which was severe in pregnant women, teenagers who had lost their innate childhood immunity and indigenous people who tend to suffer underlying health problems. Three pregnant women died of swine flu, and 280 ended up in intensive care.  


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