### Disclaimer

First of all, I am neither a lawyer nor a trained ethicist. The
following are a list of thought experiments related to “hacking”
(*i.e.*, testing the limits of) the law. Unless otherwise noted,
I have not done any research to confirm whether or not the questions
posted herein are either novel or have already been answered.
Although the following contains some material related to computers, I
have tried my best to write it in such a way as to be accessible to
the widest audience.

### Copyrighting a Number

Is it legal?

It is obviously legal to copyright an artistic work, like a digital photo. A digital photo, however, is really stored on a computer’s hard drive as a sequence of numbers, each representing the color of a dot in the picture. This sequence of numbers could be summed such that it amounts to a single, unique number. Would it be legal for one to give that number—which uniquely represents the copyrighted image—to a friend? The friend could then divide that number back into its sequence on the hard drive, thus reconstructing the original copyrighted picture. If copyrighting numbers is not legal, then I do not see why what I just described would not be legal.

The issue is actually a bit more complicated than it seems.

It is entirely possible that the method used to convert the digital
picture to a single number could be slightly modified (*e.g.*, by
adding 1 to the resulting number). If the recipient of
the number does not know that this was done then the resulting
reconstructed picture will look like noise. If the recipient knows to
subtract 1 from the number before reconstructing the picture, however,
the picture will be exactly the same as the copyrighted picture.

To add even more complication, it is entirely possible that, by adding 1 to the number, the improperly decoded picture might in fact become a completely different copyrighted picture.

#### Example

- Person
**X**has a copyrighted picture, called picture**A**, that he/she legally owns. **X**converts the picture to a number, $n$.**X**sends the number $n+1$ to person**Y**.

Case 1:

**Y**converts the number $n-1$ back to a picture, resulting in picture**A**.

Case 2:

**Y**converts the number $n$ to a picture, resulting in a completely different picture**B**.- Picture
**B**turns out to be copyrighted by person**Z**. - Neither person
**X**nor person**Y**have ever even seen picture**B**before.

### At what point is copyright lost?

Related to copyrighting a number is the following.

When the picture is represented as a sequence of numbers
(representing the colors of the individual dots in the picture), it is
possible to increment each of the colors of the individual dots. For
example, let’s say the dot in the upper left corner of picture
**A** is currently black. We could iteratively increment
the color of that dot so that it eventually becomes white (going
through a sequence of lightening grays in the process). We could even
increment all of the dots in the picture at the same time.

Now, let’s say picture **A** is a photo of the Mona Lisa of which
we do not own the copyright. Picture **B** is a photo of the Empire
State Building that you took and of which therefore own the copyright.
Both of the pictures have the same dimensions; therefore each dot in
picture **A** has a corresponding dot in picture **B**.

Now, we iteratively increment the dots in **A** such that they
all move toward the color of their corresponding dot in picture
**B**. Let’s call the result of this picture **C**. At the
beginning, **C** will look exactly like picture **A**. At the
end, **C** will look exactly like picture **B**. In the middle of
the process, **C** will look like a linear combination of **A**
and **B**.

### Question 1

At what point during the “morph” fromAtoBwill the “copyright” of pictureCtransition from that of pictureAto pictureB?

### Question 2

Is there any point during the process that pictureCmight not be protected byeitherpictureAor pictureB’s copyrights?