Some paradoxes resolved


The 'coin rotation' paradox

The 'Achilles and the tortoise' paradox

The 'two envelope exchange' paradox

The Bentley paradox

The Fermi paradox


Explanation of the Coin rotation paradox:

The moving coin completes one full revolution after only going half the way around the stationary coin.

An introduction to this 'paradox' is at:

coin rotation paradox

This is not a paradox, but a useful example for explaining simple relativity.

Movement (in our case rotation) has meaning only when defined with respect to a system of reference.

We spontaneously tend to choose ourselves (the room, the table, our eyes, the eagle coin) as the system of reference, which is case a), below.

However, this is only one of the possible choices:

a) If the system of reference chosen is the "eagle" coin, then the "face" coin rotates a complete turn, from facing the eagle to facing away from the eagle.

b) If the system of reference chosen is the "face" coin, then the "eagle" coin rotates a complete turn, from the left - in front of the face, to the right - the back of the head.

This could be better seen if you place the coins in a small table and ask a friend to rotate the face coin, while you are moving around the table one complete turn, synchronously with the face coin.

c) Finally, if the system of reference chosen is the contact point, then both coins rotate half a turn, both ending upside-down. In this case, if you keep rotating in synch with the contact point, then you will end on the other side of the table.

What remains constant, in all cases, is the apparent movement of the contact point: it "touches" both coins along the top half of the edge of both coins (i.e. half the circumference of each coin).

If you continued the rotation for an additional equal amount, the contact point would reach the original position, having traveled one full circumference in both coins. At that point both coins would be in the original relative position.


Explanation of the 'Achilles and the tortoise' paradox, attributed to Zeno:

When Achilles reaches the point where the tortoise was, the tortoise has advanced further ahead. When Achilles reaches this new point, the tortoise has again advanced, thus Achilles will never reach the tortoise.

An introduction to this 'paradox' is at:


We need to note that Zeno's proposition invites the solver to do a series of steps each time changing his system of reference: The starting system of reference, then the new system of reference where Achilles has reached the point where the tortoise started, then again a new point where the tortoise was, etc., every time essentially re-creating the original problem at a smaller scale in a convergent, infinite series.

After Zeno's proposed first step, or first change of system of reference, the problem is exactly the same as the original, the only change being a difference in "scale". Hence the "solver" has no hope of ever progressing towards a solution by following Zeno's suggested steps.

Changing system of reference essentially "restarts" the problem-solving procedure, the proposed "algorithm". That is Zeno's trick.

In order to actually solve the problem, without changing system of reference, one might think in these terms:

In the observer's system of reference, the speeds are:

Speed of Achilles: va = x*m/s;  (e.g.: 8.06m/s)
Speed of the tortoise: vt = y*m/s; (e.g. 0.06m/s)

An attentive listener would have exposed Zeno's trick by 'eliminating' the changes of system of reference. He would note that, since Achilles is faster than the tortoise, the two are moving towards each other at a certain speed.

In order to 'see' that, it may help to think of Achilles and the tortoise in an empty space.

In a stable system of reference, the two are moving towards each other at the speed of:

 vta = |(x-y)m/s|; (e.g. 8m/s).

The original problem is simply solved by calculating the time necessary for the two to meet if the original distance is 100m. That is:

t = 100/vta   OR  t = 100/|(x-y)m/s|;

(e.g.: t = 100/8 = 12.5s)


Explanation of the "two envelope exchange" paradox

One envelope contains twice as much money as the other.

Having chosen an envelope, before inspecting it, you are given a chance to take the other envelope instead. It would appear that it is to your advantage to switch envelopes.


We do not know the amount on the selected envelope. However, we know that the total amount in both envelopes is fixed and pre-established.

We can 'suppose' that the selected envelope contains any amount x (e.g. $20), but this supposition also implies the total amount for the two envelopes T to be either 1.5x or 3x (e.g: $30 or $60).

The two totals cannot be true at the same time. Thus each scenario must be analyzed separately.

This is the fault of the reasoning in the paradox.

If the two scenarios are analyzed separately, then there is no paradox.

In the first scenario (e.g.: T = $30), our supposition that the selected envelope contains $20 would imply that the other envelope contains $10. However, it is equally probable that the opposite is true: the selected envelope could contain only $10. In both cases, switching the envelope would make a $10 difference.

In the second scenario (e.g. T = $60), our supposition that the selected envelope contains $20 would imply that the other envelope contains $40. However, it is equally probable that the opposite is true: the selected envelope could contain $40. In both cases, switching the envelope would make a $20 difference.

Thus, in both scenarios, there is no probabilistic advantage in switching envelopes.


Explanation of the Bentley paradox:

An introduction to this 'paradox' is at:


This would be true only if the (observable) universe, as a whole, would be without inherent motion, with respect to a 'bigger' universe.

However, if the observable universe, just like a solar system or a galaxy, is also rotating, then Bentleys assumption does not apply.

The explanation of a rotating observable universe is consistent with the "black hole" or "multiverse" cosmological models.


Explanation of the Fermi paradox:

Enrico Fermi realized that due to the sheer number of stars and planets in our galaxy, another civilization in the universe would have had enough time to colonize our entire galaxy and leave an imprint that we could detect. This seems to be logical. Why then we have not seen any such evidence?

An introduction to this 'paradox' is at: and


A basic assumption of the paradox is unfounded: The Fermi paradox formulation, and the Drake equations, assume that finding a habitable planet, as far as environmental conditions, implies finding life on those planets with a certain probability.

However, the Fermi paradox formulation considers space, matter and time for colonization, without considering the time necessary for life to randomly form anywhere.

With simple probability math, we can calculate the minimum time necessary for the simplest bacterium to develop by random mutations. If such random mutations happen once a second, that time would be more than 1020 seconds. This is assuming the best environmental conditions in a particular “spot” in spacetime. See:

The problem is that, according to today’s best scientific knowledge, the observable universe is less than 1018 seconds old.

Random mutations are the only mechanism proposed so far (if we exclude intelligent design) for the evolution of life.

Thus, given the age of the universe and our current scientific knowledge, we conclude that even the simplest bacterium could not have developed anywhere in the observable universe; far less higher forms of life traveling around in spaceships.



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