Solving the Clever Compound Cartoon Puzzle

[tgm1024]

Clever compound cartoon...Dying to figure out the formula to calculate the ink used.

There is a strange compound nature to this that I can *almost* calculate. This is just something nagging at me.

Take a look at this (IMHO) extremely clever cartoon:

http://sphotos-a.xx.fbcdn.net/hphotos-ash4/s720x720/402912_10150957378703003_949615057_n.jpg

Pretend for the moment that the graphics involved aren't drawn freehand, and that they are clean geometric shapes and the lines are of a uniform width, etc. I don't want the freehand nature of this thing goofing this conversation up.

All each of the 3 panels' drawings are based upon all 3 panels. There are some things that will have to remain as unknown constants of course (the ink used by the words) but I've been trying to figure out (for fun and internal torment):

1. What is the equation for each panel?
2. What is the outcome---how would this cartoon "look"? I'm not sure if this was truly calculated as it stands.
3. Is it possible that there is more than one possible version of this cartoon?

Thanks in advance!

 

[Stephen Tashi]

There is a strange compound nature to this

It's an example of something that is self-referential.

Some such examples are allowed in mathematics (such a recursive definitions - e.g. "An expression is anyone of the following: 1) an identifier 2) a string consisting of an expression followed by the character "+" followed by an expression ...). Some examples are not allowed (e.g. "This statement is false.").

It isn't clear (to me) whether a version of it could be created that gave correct information. The panel on the right would have to contain a small picture of itself. Forgetting the practical difficulty of doing that, it is possible as a mathematical abstraction. ( I think it would be an example of a "fractal"). It might be possible to assign an amount of ink needed to draw the right panel. Perhaps some expert on fractals can comment on this.

Assuming that problem is overcome, the question still remains whether the the picture can be drawn with an amount of ink that informs us correctly of the amount of ink that is used. It's a problem of finding an equilibrium point. For example, for a specific function f(x), there may be a point where f(x) = f( f(x)). But for some functions there is no such point. If we have freedom in picking things like the thickness of the axes, the style of letters, etc, we can think of it as having a whole family of functions f(x,a,b,c,..) where the a,b,c are constants we can also vary. I'd be optimistic about finding at least one function in the family that crossed the line y = x, but we'd need more specific assumptions about the situation to do a mathematical proof that this is possible.

 

[tgm1024]

It's an example of something that is self-referential.

Some such examples are allowed in mathematics (such a recursive definitions - e.g. "An expression is anyone of the following: 1) an identifier 2) a string consisting of an expression followed by the character "+" followed by an expression ...). Some examples are not allowed (e.g. "This statement is false.").

It isn't clear (to me) whether a version of it could be created that gave correct information. The panel on the right would have to contain a small picture of itself. Forgetting the practical difficulty of doing that, it is possible as a mathematical abstraction. ( I think it would be an example of a "fractal"). It might be possible to assign an amount of ink needed to draw the right panel. Perhaps some expert on fractals can comment on this.

Assuming that problem is overcome, the question still remains whether the the picture can be drawn with an amount of ink that informs us correctly of the amount of ink that is used. It's a problem of finding an equilibrium point. For example, for a specific function f(x), there may be a point where f(x) = f( f(x)). But for some functions there is no such point. If we have freedom in picking things like the thickness of the axes, the style of letters, etc, we can think of it as having a whole family of functions f(x,a,b,c,..) where the a,b,c are constants we can also vary. I'd be optimistic about finding at least one function in the family that crossed the line y = x, but we'd need more specific assumptions about the situation to do a mathematical proof that this is possible.

Verrrrrrrrry well stated. The 3rd panel doesn't worry me as it almost certainly approaches a limit, or at least my Calculus classes 30 years ago would lead me to believe that.

I've written a number of recursive algorithms, fractals included, and calculating area usage is a matter of finding the limit of the function as it recurs.

I totally appreciate your phrasing of "equalibrium point". Ironically, it's how I recently explained this to a friend of mine. One of my biggest questions is: "do you suppose that there is only one possible point for that?" (Only one valid way of drawing this cartoon?)

Perhaps a set of 3 large (one per panel) simultaneous equations would sift this out?

In any case, I like the level of cleverness in it. It has a sort of humor I rarely see.

 

[SteveL27]

It has a sort of humor I rarely see.

Credit where credit's due, this is of course the wonderful xkcd. Regular readers see this sort of humor every day :-)

http://xkcd.com/688/

This is one of my favorites ...

http://xkcd.com/1047/

 

[blue_raver22]

I can appreciate your curiosity and determination to solve this clever compound cartoon puzzle. While I am not an expert in graphic design or art, I can offer some scientific insight into this problem.

Firstly, it is important to recognize that this cartoon is a visual representation of a chemical compound, with each panel representing a different aspect or component of the compound. In order to accurately calculate the amount of ink used, we would need to know the dimensions and properties of the panels, such as the thickness and type of ink used, as well as the dimensions of the overall cartoon.

Next, we would need to consider the structure and arrangement of the panels. Are they arranged in a linear sequence or are they overlapping? Are there any hidden or overlapping components that may add to the overall ink usage? These factors would need to be taken into account in order to accurately calculate the ink usage.

In terms of the equation for each panel, it would be helpful to break down the cartoon into its individual components and analyze the geometric shapes and lines used. This would allow us to determine the surface area and volume of each panel, which can then be used in an equation to calculate the ink usage.

As for the outcome, it is possible that there could be more than one version of this cartoon, depending on the variables and assumptions used in the calculations. However, it is also possible that the cartoon was designed with a specific outcome in mind, so it may be difficult to determine an exact version without input from the creator.

In conclusion, solving this clever compound cartoon puzzle will require a combination of scientific analysis and artistic interpretation. It may be helpful to consult with experts in both fields to come up with a comprehensive solution. Best of luck in your calculations!