I am a mathematician with an interest in the history of mathematics, and I've been researching the history of recreational mathematics for about 25 years. I first came across the Three Rabbits as a puzzle in English and American puzzle books of the 19th century.

The oldest known example of the type is in The Girl's Own Book, which first appeared in about 1833, with the following text: "Can you draw three rabbits, so that they will have but three ears between them; yet each will appear to have the two that belongs to it?"

The puzzle aspect of the image goes back to about the 16th century in Europe. At Paderborn, Germany, the image is from about 1500 and there is a folk rhyme, but its date is not known: 

"Der Hasen und der Löffel drei,
und doch hat jeder Hase zwei."
[The hares and ears are three,
and yet each hare has two[, you see].]

More specifically, Jurgis Baltrusaitis's Le Moyen Age Fantastique shows a 1576 Dutch engraving with the puzzle given in Dutch and French around the image.

The secret is not great when one knows it.
But it is something to one who does it.
Turn and turn again and we will also turn,
So that we give pleasure to each of you.
And when we have turned, count our ears,
It is there, without any disguise, you will find a marvel.

These are the oldest known dated examples of the Three Rabbits as a puzzle. I feel it is likely that the popularity of the image stems from its being a puzzle.

The puzzle occurs in the present day. A recent book of philosophy problems gives a three rabbits picture as a problem in perception. Each rabbit can be individually seen as correct—it is only when you try to see all three at once that you see the problem. This is similar to Roger Penrose's Impossible Tribar.

There are several other types of puzzling images based on anatomical rearrangement. We have recently learned of early 13th century examples of faces arranged in a circle and sharing eyes. These may  derive from 'tricephalic sculptures' where a head has three faces sharing eyes, so that the three faces have only four eyes all together, though only two eyes may be shown.  Such images date back to the Gallo-Roman period in the first through third centuries. These may derive from Roman images of the double-faced god Janus, which may derive from Hittite or Babylonian ideas. In China, we noted that Avalokitesvara is often shown with multiple heads, with some sharing of features. 

Another related and ancient variation of the idea is to have several bodies with one head or one body with several heads. The oldest, and perhaps the most beautiful, example has three fish sharing a head on a blue faience bowl, from 18th dynasty Egypt, about 1500 BC. This image has been popular ever since all around the world—there is a fine example in the Shaanxi History Museum, Xi'an, on an eared jug from the Yuan dynasty (1271-1368).

Another famous example is from the Ajanta Caves in India, showing four deer sharing a head, done by Buddhist artists sometime from the 2nd century BC to the 4th century AD.

A different type of puzzle based on anatomical rearrangement is Sam Loyd's Trick Mules of 1871. You have to cut the printed image into three pieces and put each rider on a mule. P.T. Barnum bought millions of copies of this for advertising. Here is a nice version showing the solution.

In looking at 19th century puzzle books, it is clear that the Trick Mules derives from the 'Dead Dogs Puzzle'. The earliest known example is in The Family Friend of 1849. The caption on the problem reads:

These dogs are dead you well may say:
Add four lines more, they'll run away!

And the solution's caption reads:

See now the four lines. "Tally-ho!"
We've touch'd the dogs, and away they go!

In the early 1980s, I learned that a similar image was popular throughout the world and long before the 19th century. In China it is known as Sixi or Four Happinesses. It is a general image of good luck and is often used on plates, toggles, scrolls, scrollweights and even on matchboxes, with the inscription: Lian sheng gui zi tu (Many treasured sons diagram).

The image is not restricted to China and Japan. It appears in India, Persia, Europe and Tibet. Perhaps the finest example is the following, with four heads but twelve horses.

There are some old European examples. Baltrusaitis shows a nice Dutch engraving of 1576. A version frm south Germany or Switzerland, late 15th century, had a rotating piece to allow the heads and bodies to be connected either way, and is one of the earliest known examples of a printed design with a movable part.

But the image goes back before printing. A version appears in the Peterborough Psalter of about 1310, now in Brussels. The earliest known example of the pattern in Europe is a sculpture on the Booksellers' Doorway of the Cathedral at Rouen, France, from 1275?-1300.

There are several beautiful occurrences in Tibet, from the Kingdom of Guge, based at Tsaparong (= Zhaburang) in the very southwest of modern Tibet. However, the dating of these is unclear—the Kingdom was active from the mid 9th century until about 1300 and somewhat later.

The description of the picture says this was a generally adopted composition in this time and place. The book also says that they made "auspicious revolving poems" which are a design of squares, each with a syllable so one can read it "in a crosswise, diagonal or circular way."

Other examples occur at other Guge sites. Since the dates of the Guge images are not known to any degree of precision, it is difficult to decide if they are earlier than the earliest European examples. So the question of the origin of this pattern remains open until we learn of more examples. In both Europe and Tibet, the earliest known examples are well developed, so one suspects that earlier examples must exist in or near both areas. It would seem natural for the image to have come from the Arabic world to both Europe and Guge, but no Arabic examples are yet known.

The Sixi image is an example of what visual psychologists call a 'gestalt' phenomenon, where there are alternative views of the same image. The two interpretations of the image, like the two interpretations of a Sixi image, compete in our mind. We can see one interpretation and then the other, but it is hard to see both at once.

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Adapted from a presentation at the International Conference on Grottoes Research, Dunhuang, China, August 2004.