I think you're suggesting take four medians: the median of the X coordinates of the reds, the median of the Y coordinates of the reds, and the same for the greens.

No. The problem is defining (or understanding) what the median is for a 2D dataset (R2).

Think of 3 points in the form of an equilateral triangle with the lower edge parallel to the X axis. 

+
|        x
|       . .
|      .   .
|     .     .
|   .         .
|  .           .
| x.............x
+-------------------

[download]

Whilst the top point is the median in the X axis (looking up). The bottom right point is the median if you are looking in from the top left. Equally it's the bottom left point, if you look in from top right. Which would be the "correct median" depends upon the relative positioning of the other set of three points; or more correctly, their median. And the above three points can be rotated through 0->120°, giving an infinite number of directions to view the dataset, (or transformations you could apply), in order to access the median.

Which I think means that the warm-up problem is an almost complete red herring!

As you cannot work out which direction to look in (or which transformation of the coordinate system to apply), to determine the median for this dataset, until you know the median of the other. And vice versa. You cannot use a 'sort and take the middle' or K'th ordered element approach to determining the median as you would use for an R1 dataset; for an R2 dataset. Nor for the higher dimensions.

That leads you, (led me?), to think about how to determine the median of a set of points in R2, without reference to the other dataset. And that's when I found the Euclidian distance method.

The premise is that the median of a R2 dataset is that point at which the sum of the Euclidian distances between that point and the points in the datast is minimised.

There are other methods, including the point that minimises the sum of the areas of the sets of triangles formed between that point and pairs of points of the dataset, but that seems much harder to calculate.

Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
Lingua non convalesco, consenesco et abolesco. -- Rule 1 has a caveat! -- Who broke the cabal?
"Science is about questioning the status quo. Questioning authority".
In the absence of evidence, opinion is indistinguishable from prejudice.

In reply to Re^3: Puzzle: The Ham Cheese Sandwich cut. by BrowserUk
in thread Puzzle: The Ham Cheese Sandwich cut. by Perl Mouse

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