Problem Solving

The Mathematician’s Trap & How Intelligent People Deceive Themselves

This little story about the great mathematician John Von Neumann has always been one of my favorites. I will tell it the way I first heard it. I have since heard a few variations on the story, leading me to think that there may be a component of Urban Legend to it. But I really don’t care, because I think it’s such a great story that it’s worth retelling.

John Von Neumann was considered by many to be one of the most brilliant minds of the twentieth century. He reportedly had an IQ of 180. He was a pioneer of Game Theory, which was very important during the nuclear arms race. (Because GT assumes that all players act in their own best enlightened self-interest, GT turned out to be a much better model for evolutionary biology than for human behavior.) He was also one of the two people (Alan Turing being the other) who is credited with being the father of the modern computer.

The story goes that someone once posed to Von Neumann the following problem:

Two trains are 20 miles apart on the same track heading towards each other at 10 miles per hour, on a collision course. At the same time, a bee takes off from the nose of one train at 20 miles per hour, towards the other train. As soon as the bee reaches the other train, it bangs huwey and heads off at 20 miles per hour back towards the first train. It continues to do this until the trains collide, killing the bee.

Back to our friend the bee. We now have an expression for how far the bee flies after n legs
7. d’n = 2D * 1/2*(3^n – 1)/3^n = D*(3^n – 1)/3^n
and we need to solve it for how far the bee flies before dying. In actuality, the bee will stop flying (okay, “in actuality” this would never happen) when the distance between the trains is less than the body length of the bee. However, since the summation quickly converges on the solution, we can assume that the bee is a point, ignore the famous paradox, and do the summation up to n=infinity.
7. d’n = D*(3^n – 1)/3^n = D*(3^n/3^n – 1/3^n) = D*(1 – 1/3^n)
Quick refresher on infinite limits: as we let n get infinitely large, 3^n approaches infinity, and 1/3^n approaches zero. Therefore the limit as n approaches infinity is
d = limit(n–>infinity)[D*(1 – 1/3^n)] = D*(1-0) = D = 20 miles!

We have just solved this problem by the infinite series method. Infinite series are very important in Mathematical Analysis and Pre-Calculus as they form the basis for derivatives and integration and everything which is Calculus and Differential Equations. That’s why all students of Math, Physics and Engineering get to do a ton of infinite series problems before they graduate. The reason I call this problem the “Mathematician’s Trap,” is because virtually all mathematicians who see this problem will try to solve it the way we just did.

However, if you were to give the problem to someone who’s only had basic Algebra, they might solve it differently. The trains crash at the midway point which is at 10 miles. Since each train is going 10 mph, this takes one hour. During that same hour, the bee is flying at 20 mph, therefore the bee flies20 miles! Wow, that was a lot easier!

The moral of the story is that being smarter or better educated can often times put you at a disadvantage. When someone is trained at doing something a certain way, that action is virtually automatic. It takes great insight to be able to “step outside of the box” and ask if there’s an easier way to do it. (I’m currently working on another post on just this subject which should hopefully be up soon.) Even brilliant mathematicians will fall into the trap, which brings us back to John Von Neumann.

When posed with the above problem (or some variation of it), JVN took all of five to ten seconds to come up with the correct solution. This floored the questioner who said “I’m impressed that you didn’t fall for the Mathematician’s Trap.” After getting a perplexed look from our genius, he asked “How did you solve the problem?”

“By infinite series, of course!”


 Another good one:

Highly intelligent people may turn out to be rather poor thinkers.

They may need as much, or more, training in thinking skills than, other people.  This is an almost complete reversal of the notion that highly intelligent people will automatically be good thinkers.

1)    A highly intelligent person can construct a rational and well-argued case for virtually any point of view.  The more coherent this support for a particular point of view, the less the thinker sees any need actually to explore the situation.  Such a person may then become trapped into a particular view simply because he can support it (see Hypothesis Traps).

2)    Verbal fluency is often mistaken for thinking.  An intelligent person learns this and is tempted to substitute one for the other.

3)    The ego, self-image and peer status of a highly intelligent person are too often based on that intelligence.  From this arises the need to be always right and clever.

4)    The critical use of intelligence is always more immediately satisfying than the constructive use.  To prove someone else wrong gives you instant achievement and superiority.  To agree makes you seem superfluous and a sycophant.  To put forward an idea puts you at the mercy of those on whom you depend for evaluation of the idea.  Therefore, too many brilliant minds are trapped into this negative mode (because it is so alluring).

5)    Highly intelligent minds often seem to prefer the certainty of reactive thinking (solving puzzles, sorting data) where a mass of material is placed before them and they are asked to react to it.  This is called the “Everest effect” since the existence of a tough mountain is sufficient reason for the best climbers to react to it.  In projective thinking, the thinker has to create the context, the concepts, and the objectives.  The thinking has to be expansive and speculative.  Through natural inclination or perhaps early training, the highly intelligent mind seems to prefer the reactive type of thinking.  Real life more usually demands the projective type.

6)    The sheer physical quickness of the highly intelligent mind leads it to jump to conclusions from only a few signals.  The slower mind has to wait longer and take in more signals and may reach a more appropriate conclusion.

7)    The highly intelligent mind seems to prefer – or is encouraged – to place a higher value on cleverness than on wisdom.  This may be because cleverness is more demonstrable.  It is also less dependent on experience (which is why physicists and mathematicians often make their “genius” contributions at an early age).

See also:

By Eric Patton

Well look down Yonder Gabriel, put your feet on the
land and see

But Gabriel don't you blow your trumpet till you hear
from me

There ain't no grave can hold my body down

7 replies on “The Mathematician’s Trap & How Intelligent People Deceive Themselves”

Very interesting food for thought.

Intelligent people can also be quite lazy, if they never learned to work for anything. Such people often cling to sinecures where they can safely squander their intelligence without actually producing or creating anything..

The academic bubble has sucked in large numbers of bright people who might otherwise have been forced to make useful contributions to society. The government bubble and the government lobby bubbles likewise lead to large scale waste of human capital.

Finally, logic is no longer universal in the post-modern age. What passes for logic in many intellectual and pseudo-intellectual communities is is little more than rite, dogma, and lifelong indoctrination — even among the very bright. When particular inconvenient viewpoints and logical systems are driven out of academia and politically correct realms of thought, what remains will often deform itself in very perverse ways..

The academic system is as much a daycare as anything else at this point.

I was just talking to Giovanni about how certain ideas that go against established orthodoxy like egalitarianism are almost impossible to get published in peer reviewed journals, no matter how well researched it is.

(this might change over time: )

There is a bloat of bright people in fields like cancer research and physics that could go to other fields that aren’t as populated and make breakthroughs much easier, instead of trying to stay afloat in cutthroat competition.

The frontiers have always been populated by people with eccentric personalities but I agree that a lot of people without technical training tend to lack the rigor and methodology that frontier work requires.

Thoughtful people tend to do things explicitly and consciously that other people do automatically.

This can be wonderful…or disastrous.
Imagine the burden of responsibility borne by someone with voluntary control over every heartbeat?

Thus, good management skills truly are more important for ‘smart’ people.

Being ‘smart’ is like being a 4 wheel drive vehicle. You have abilities that allow you get over rough terrain that others cannot handle.
But as a consequence, you’re far slower and less efficient on an open road.

No trait is necessarily bad or good except as it relates to the circumstances.

If we had deadly claws on our hands, we’d be better in a fight, but we’d sacrifice some of our fine motor abilities by always having claws in the way.

If we have bigger brains, childbirth becomes more difficult. Bigger brains require huge amounts of energy to sustain.
So unless there’s a huge payoff for being ‘smart’ we’re better off being ‘stupid.’

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