Thursday, February 28, 2013

The Road Not Taken (Logic Puzzle)

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
(Line 1 by Robert Frost)

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?


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.


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”


“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.
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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Friday, February 8, 2013

What’s the difference between 90% and 100%? (Traits of HIGHLY successful math students)

The differences between the students who fail in math and the students who get decent grades – C and above – are generally clear cut, and easily identifiable (unless there is a learning disability involved).  Most of these differences aren’t even math specific… failing students in any subject can generally improve their grades by improving one or more of the following areas:

1)      Get organized
2)      Do (and turn in) assignments on time
3)      Be willing/able to seek help when needed
4)      Be willing to put in the time/effort necessary to be successful
5)      Practice/Prepare for tests
6)      If the people you hang out with aren’t committed to success, hang out with different people
7)      Don’t accept defeat easily (“If at first you don’t succeed, try, try again.”)

However, I recently found myself wondering about what sets the truly excellent math students apart from the merely-great math students.  I have a number of students earning a grade of A or A- (at or above 90%).

Why do a select few of these students consistently earn grades near 100% (or above 100%, if extra credit is offered) in math class?

In analyzing my students, I have noticed a few traits that differentiate merely-great math students from highly successful math students.

1.       Merely-great math students tend to believe that knowing a lot and being good at math is the most important component of good test taking.

Highly successful math students understand that knowing a lot and being good at math is actually the second most important component of test taking.  The most important component is being able to successfully communicate your knowledge and math excellence to your teacher/professor.

There are many merely-great math students who turn in tests containing problems with ambiguity in the solutions.  Their brains might have been doing all the right steps, but their work is suspect.  Highly successful math students provide complete solutions that are clearly mathematically sound.  As a result, they tend to receive higher scores on their tests.

Laziness also plays a role in this.  Many merely-great math students seem to follow the philosophy, “When in doubt, show less work.”  Most highly successful math students follow the philosophy, “When in doubt, show more work.”

2.       Merely-great math students don’t tend to place a high value on neatness.

Highly successful math students tend to make neatness a priority.

I can’t count how many times I’ve been grading work and discovered that a student got the wrong answer due to a mistake caused by poor handwriting.  I’ve seen “4” turn into “9”, “z” turn into “2”, “7” turn into “1” and a myriad of other sloppy mistakes.  These mistakes often prove to be the difference between a 95% test and a 100% test.

3.       Merely-great math students use lectures and class time to learn what they need to know in order to succeed.

Highly successful math students also use lectures and class time to learn what they need to know, but they tend to use this as a starting point in their learning process.  They are adept at looking in textbooks for additional examples in order to enhance the material covered in a lecture.  They are also eager to explore alternative methods and they have a desire to know WHY a particular method works or doesn’t work.

4.       Merely-great math students do their best to find out what material will be covered on a test.  (Some are even quite assertive in trying to get specific review topics from their instructors.)  They then review this material diligently to make sure they can complete the test successfully.

Highly successful math students emphasize material they know will be covered on a test when studying.  However, they also attempt to review/practice all of the other concepts from a chapter.  This is helpful in (at least) two ways:

1.       Since concepts are often interrelated, highly successful students gain a deeper understanding of the “key concepts” by broadening their review.

2.       Teachers who put extra credit items on their tests will often draw from these “other” concepts.  When these extra credit test items show up, highly successful students are ready for them.

5.        Merely-great math students sometimes put too much emphasis on being right at the expense of focusing on what went wrong.

Highly successful math students attempt to learn as much as possible from their mistakes so that they can avoid making the same kind of mistakes in the future.

 I can think of a number of students who are experts at “nickel-and-diming” teachers out of extra points.  They submit work that is good (but not great) and then have to verbally defend their work and try to convince the teacher that they deserve 100% credit. They rejoice when their arguments are fruitful and they receive a small increase in score… their mission has been accomplished.  Sadly, these students tend to get in the habit of submitting less-than-stellar work and find themselves arguing with their teachers a lot.

Highly successful math students, on the other hand, are willing to admit when their work is less than ideal.  They may argue the merits of their work with their teacher in an attempt to get more points, but their chief concern is learning how to produce stellar work in the future that will be above reproach.

The above list of traits is not exhaustive – there are probably more traits of highly successful math students that I’ve missed.  (If you’d like to add to my list, leave a comment below.)

Also, the above list does not come from a scientific study.  It is simply an anecdotal summary of my own personal observations.

I hope it will help you as you strive for true excellence.


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