This puzzle is often worded like this: Pipes from a gas, electricity and water supply need to be laid in three houses. Is it possible to do this without having to lay one line over another?
The puzzle is often presented to both children and adults, as it is a splendid example of a task of the kind that mathematicians seek to solve. In fact, the answer is well known among mathematicians, but it does not in any way diminish the value of people spraying themselves on it.
If readers have not already taken the time to look at the example, they are strongly encouraged to bring out a pen and paper or any other tool that comes to mind and try to get a feel for it themselves before reading further.
The puzzle concerns an extremely interesting branch of mathematics called graph theory. A graph is a mathematical phenomenon that consists of a number of vertices and edges that connect one of the nodes, two and two. The nets can be drawn on a piece of paper and then it does not matter how the net turns or how the knots lie in relation to each other, but only which knots are connected together with legs.
Here are four pictures from the same network. This net contains four knots and four legs.
A planar graph is a net that can be engraved in the plain, the equivalent of drawing it on a piece of paper without two legs intersecting. Therefore, the question here is equivalent to whether the network in the picture above (so-called fully-fledged 3.3-part network, often representedK3 , 3) is a pipe net – whether there is any way to draw it so that the legs do not intersect.
Those who have tried the puzzle have, of course, run into various difficulties. When everything seems to be going well, it suddenly becomes clear that the last leg can only cut one of the others.
No matter how hard you try to draw the legs, they always seem to cut.
The truth is, this is impossible! It can be seen as follows:
If we mark the nodes on the left and right with A, B and C on the one hand and X, Y and Z on the other, we must at least somehow get the tubes that connect the nodes AYCXBZA and mark it as a kind of circle – a so-called channel . It can therefore be assumed that they have already been drawn. Without losing information, the knots can be spread out so that the image is clearer, as is done in the video below.
Here the knots are rearranged so that you can see how two of the last three legs preclude the very last one from being drawn.
When you get here, it is clear that the legs AX, BY and CZ are left. For each of them, you have to choose whether to draw inside or outside the road that has already been built, but only one leg can be drawn on each side! Therefore, it is inevitable that the network’s cables will intersect somewhere.
Fully equipped net with 5 knots.
In 1930, the Polish mathematician Kazimierz Kuratowski demonstrated in his article that all non-wiring networks can be scaled down and simplified so that they are either published. K3 , 3 or another network, the complete network with 5 nodes, K5, which is shown here. This conclusion is called Kuratowski’s statement.
It is not so difficult to see that it is possible to draw all the grids on a flat sphere instead of a flat surface. If a hole is made in the ball, a so-called torus is obtained , a kind of donut ring shape . On such surfaces, these nets can in fact be drawn well, without two legs intersecting. If any given network called clan (e. Genus) network, the number of street to be taken in the sphere until the drawing can network on the plane without legs intersect. The lineage is therefore 0, lineageK5 and K3 , 3 is 1 and so on.
Before the author wanders too far from the starting point, it is worth summarizing: The answer to the question is no . This drawing cannot be performed on normal surfaces.