De Gruyter Advent Calendar | Door 6
A MATHEMATICAL PERSPECTIVE ON SANTA CLAUS'S HOUSE
"That's Santa Claus's House" is a popular puzzle in Germany. The challenge is to draw "Santa Claus's House" without taking the pencil off the paper and without following the same line twice. Most German children have at some point drawn one crooked house after another in their attempts to solve the puzzle. Dr. Dr. h.c. Norbert Herrmann, our resident author and mathematician, has a better idea.
For one thing, he has never tried to solve the puzzle with a pencil and paper. To him, the puzzle is not a house, but a geometric figure consisting of points and straight lines. He solves the problem by dint of mathematical logic:
- There is always one point at which one begins and one point at which one ends.
- Each point on the figure has a different number of lines leading to it. The number of these lines is either an odd or an even number.
- If an even number of lines lead to a point, it is a transit point; it is arrived at and then left again.
- If an odd number of lines lead to a point, it can serve only as either a starting point or an end point.
And now for the logic:
In order to draw the house, we need a starting point and an end point. Since only points with an odd number of lines can serve as starting and end points, we must have
=> No more and no fewer than two points in the figure to which an odd number of lines lead.
Is this the case in Santa Claus's house? It is:
And if we know that, we also know where our starting and end points are.
The advantage of Mr Herrmann's method over trying to draw the figure, like the rest of us do, is that it can be used to solve more than just the puzzle of Santa Claus's house.
Any other puzzle in which a geometrical figure has to be drawn in the same way must always have exactly two points to which an odd number of lines lead. If the figure has more than two such points, the puzzle cannot be solved.
The puzzle can also be solved if the figure consists solely of intersecting straight lines. In this case, any point can be taken as the starting and end points.
Try it for yourself!