In Robert Frost’s famous poem,

__The Road Not Taken__(sometimes mistakenly referred to as “The Road Less Travelled”), a traveler is faced with the difficult choice of which road he should follow.

This brings
to mind a classic logic puzzle, with mathematical implications.

Here is my
poetic interpretation of the puzzle.

__THE ROAD NOT TAKEN – Mr. Wagneezy Version__

**Two roads diverge in a yellow wood**

**One leads to certain death**

**The other leads to riches untold**

**I stop to catch my breath**

**I soon discover I have no clue**

**Exactly which road is which**

**I look to the right, and then to the left**

**My eyes begin to twitch**

**Suddenly two gnomes appear**

**From out of nearby briars**

**One of them is a truthful gnome**

**The other one is a liar**

**In looking at these gnomes, alas**

**I cannot tell the difference**

**Which one speaks truth? Which one speaks lies?**

**I fight the urge to wince**

**These seemingly identical gnomes**

**Both know which road to take**

**But instead of making it clear to me**

**They make it nearly opaque**

**The gnomes agree that one of them**

**Will answer a single question**

**Once I get the answer**

**They will end the conversation**

**I still can’t tell which one speaks truth**

**And which one is the liar**

**They smirk at me, these pesky gnomes**

**That came out of the briars**

**I must determine what to say**

**It is a daunting task**

**To get the gnomes to reveal the way**

**What question should I ask?**

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Successfully
getting these gnomes to show us which road to choose will take careful
planning. Since we have no way of
knowing which gnome tells the truth and which gnome lies, we must be able to
come up with a question that both gnomes would answer the same way.

Obviously,
the direct approach (e.g. “Which road leads to untold riches?”) will not work,
because the truth-teller would point to one road while the liar would point to
the other road.

Therefore,
we must ask an indirect question – one that incorporates both the truth and the
lie. We can accomplish this by asking
one gnome to tell us which road

*the other gnome*would point us towards. (E.g. “If I asked the other gnome which road leads to untold riches, which road would he point to?”)
The logic
here is that the truth about a lie gives the same result as a lie about the
truth.

More
specifically, if we happen to talk to the truth-teller, he will point to the
wrong road because that is the road the liar would have pointed to.

On the other
hand, if we happen to talk to the liar, he will also point to the wrong road
because that is

*not*the road the truth-teller would have pointed to.
In either
case, the wrong road will be indicated, and we can choose the other road to
travel on.

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For movie
buffs – a version of this puzzle appeared in the movie Labyrinth
:

Earlier, I
mentioned that this puzzle has mathematical implications.

Let’s
deconstruct this puzzle into a similar mathematical question.

I hope you
agree with me that telling the truth is

*positive*, and lying is*negative*.
Consider
mathematical operations that can be performed on two numbers. Suppose one of the numbers is positive, and
the other number is negative, but we DON’T KNOW WHICH IS WHICH.

Which
operations are guaranteed to give us results with the same sign, regardless of
which number is positive?

If we
arbitrarily choose ±2 and ±3 for our numbers and use them to explore each
operation, we get the following:

As we can
see,

*multiplication*and*division*are the only operations that fit the bill.
(Not
coincidentally, multiplication and division have the same priority in the Order of Operations.)

Therefore,
the logical argument

“The truth about a lie is equivalent to a lie about the
truth”

seems to
match up with the mathematical concepts

“A negative times a positive is equivalent to a positive
times a negative”

and

“A negative divided by a positive is equivalent to a
positive divided by a negative”.

By the way –
multiplication and division both exhibit the desired property because

1)
Multiplying is the same thing as dividing by the
reciprocal.

2)
Dividing is the same thing as multiplying by the
reciprocal.

and

3) A
number’s sign (positive or negative) is the same as the sign of the number’s
reciprocal.

May all your
roads be wisely chosen.

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