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You have to draw a house in one line.  You must not lift your pencil while drawing. You must not repeat a line. During drawing you have to say: "Das ist das Haus des Nikolaus" (="This is Santa Claus' house.").  You must speak a syllable of this sentence for every straight line.

There is also a saying for two houses: Das ist das Haus vom Ni-ko-laus und ne-ben-an vom Weih-nachts-mann. This is the house of Santa Claus and next door of Father Christmas.

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Wer <N>ach-ge-<DACH>t <Z>uvor am Start, der kennt den Weg von Le-on-hard!

Sent by Gerald Fischböck from Vienna

Number of the Houses 
Three Examples

...	Everybody, who knows the problem, has a solution in his mind and can draw a house. The question is:  How many ways are there for drawing a house?

All sequences start in 1 and end in 2. You can conclude that there are also 44 houses starting in 2 and ending in 1. They are the same because of their symmetry. 

Euler Path   top
The "house of  Santa Claus" is more than a simple children's game. The next statements show it.
For historical reasons I will start with the Königsberger bridge problem (Königsberg is at the river ("old and new") Pregel. It was the capital of Ostpreußen before 1945 and is now called Kalingrad.). 
The Swiss mathematician Leonhard Euler (1707-1783) worked on this problem and solved it.

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The problem is to find a way through Königsberg, so that you cross all seven bridges not more than once. Is it possible? You can simplify the locality by a graph of four points with lines as paths.

Euler found out, that a tour starting in one point and returning to the same point is not possible because of the odd numbers of paths. He generalized to his theorem: A tour this way is only possible, if all the numbers of paths passing a point are even. 

The problem of  "The Seven Bridges of Königsberg" is the origin of an important branch of mathematics, the Graph Theory as a part of the Discrete Mathematic. 
In the terminology of the Graph Theory a point is a vertex and a path is an edge. The whole figure is a network. Every vertex has a valence. This is the number of edges, which meet at a vertex. If the tour begins and ends in the same vertex, you have an Euler circle, otherwise an Euler path. 
So Euler's theorem says: An Euler circle is only possible, if the valences of all vertices are even.

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If you look at the house of Santa Claus again, you realize that there is no Euler circle because of the two lower vertices with the valence 3. But there is an Euler path beginning and ending below.

By the way the network is planar, because you can draw it in such a way that the edges don't cross (drawing). It is not complete, because not all possible edges are drawn. 

Hamilton Path  top
There is a second problem. You don't watch the edges, which you must pass once. You must go through the vertices once. These paths are called Hamilton paths. If the journey is closed, you have Hamilton circles, otherwise Hamilton paths. 
Euler worked on a second popular problem, which needs Hamilton circles, the knight's tour. 

You are to move a knight on the chessboard, so that each field is passed once and it returns to the start.  You look for an Hamilton circle. The vertices are the centres of the squares of the chessboard. The Hamilton paths use two edges, which meet in a vertex. There are thousands of paths for the knight's tour. Actually all tours are handsome. There is one of Euler's solutions on the left.

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After three rounds, the figure is finally created by a closed curve.  You start in 1, passes 2 and end in 3.

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The best way to create the figure is to use a dot pattern. You get a large square by placing small squares in the corners and in between them half a square, a square and a half square in this order. The large square gets four more points by mirroring the centers of the small square.  The points are connected appropriately.

More Figures  top

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There were two houses drawn separated above.  You can leave out a wall. Then the saying is  "Das ist das Haus vom Ni-ko-laus, ne-ben-an vom Weih-nachts-mann" (=This is the house of Santa Claus, next door of Father Christmas.). This figure is insoluble. There are four vertices with odd valences.

If you remove the common wall, it is soluble again.

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Looking for Euler circles is easier, if you colour the figures in grey and white.  This proves the solution of Lewis Carrol's problem with three squares on the left.

Hamilton circle You can illustrate the Hamilton circle with a employee of a parcel post, who must deliver the parcel to one customer (=vertex) after the other.

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Euler circle You can illustrate the Euler circle with a postman, who passes definite streets (=edges) every day to deliver the post. .

Drawings in one (closed) Line top

1st Example: 

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There are Lissajous figures, whose lines have no beginning and no end. They are well known in physics and mathematics.  You find more similar lines on my website Egg curves  or my German page  Spirograph.

It is situated at Bodenheim.

The template of this drawing is on a tile bought in the tourist shop of the Real Alcazar in Sevilla.

Three Old Writing Games top
Houses
This is a competitive drawing game for children called "houses". 

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You spread out ten houses on a sheet of paper. You symbolize them by circles with numbers. One player after the other must connect the houses by lines (=paths). They must not touch the already drawn paths or even cross them. You get a penalty point for every contact. If there is a touch, the following players can use it as a crossing without getting penalty points.  The drawing on the left shows that the players are older, because there are already harassments in form of spirals.

Probably I shouldn't make the following statement because of possible imitating. But I do: You must have one hair torn for every penalty point ;-(. 

Mrs. Meier,  two eggs, one sausage, one comb. Please surround them with two handles to carry. Some pins, one matchbox, one big cake with raisins inside, one broom, one scrubber. - It makes 66 altogether.

A girl looks out of the window. There is the milkman.  She fetches milk. She runs, stumbles and stands up (1st leg). She runs further. She stumbles again, falls down and stands up again (2nd leg). She pours out the milk (spiral tail)  and runs home (back).

References   (German only) top

(1) Hermann Schubert: Mathematische Mußestunden, Berlin 1941

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Als mir dieses Buch in die Hände fiel, versah ich den ersten Satz dieser Seite doch lieber mit einem Fragezeichen. (Das Haus des Nikolaus ist ein uraltes (?) deutsches Zeichenspiel für kleine Kinder.) Im Kapitel Eulersche Wanderungen wird auch ein Figur gezeigt, die wie beim Haus vom Nikolaus keine Eulerkreise, wohl aber Eulerwege zulässt. Die Figur links ermöglicht es, den Satz des Pythagoras zu beweisen.  Wäre Schubert das Haus vom Nikolaus bekannt gewesen, hätte er diese ungewöhnliche Figur wohl durch das Haus ersetzt.  Wenn ich zurückdenke, ich lernte das Nikolaushaus in den 1940er / 1950ger als Kind kennen.

The House of Santa Claus on the Internet     top

English

Behance
One-line-Animal-logos

Eric W. Weisstein (MathWorld)
Eulerian Cycle 

Torsten Sillke
Counting Eulerian Circuits and Tours

Wikipedia
Eulerian path, Seven Bridges of Königsberg,  La Linea ,  Space-filling curve, Glossary of graph theory

Dirk Brundelius
Das ist das Haus vom Nikolaus (alle 44 Häuser als "animated Gifs")

Klaus Huneke 
Dies ist das Haus vom Nikolaus, Acryl auf Nessel, Startseite

Lutz Tautenhahn 
Nikulus

Wikipedia
Haus vom Nikolaus, Eulerkreisproblem, Hamiltonkreisproblem, Königsberger Brückenproblem,  La Linea,  FASS-Kurve, 
Glossar Graphentheorie