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Connect each red circle with each black circle by drawing a line and the lines should not touch.

From each red circle, 3 lines must be drawn which connect red circles with black circles, but the lines must not touch.

I am 32 years old now. When I was in 4th class, a friend of mine challenged me to solve this problem and I still can't solve it.

This is a famous problem called the Three utilities problem, which is part of the mathematical field of graph theory. The problem basically asks for a planar embedding of the utility graph, which does not exist.

While it's not exactly know when the puzzle was invented, it was published at least as far back as 1913 and it took mathematicians until 1930 to solve it. That means it's rather difficult to prove why the puzzle is impossible to solve; it requires some advanced mathematics to do so, beyond the scope of almost all readers here. This is again a case of "it's hard to prove a negative".

Simply put, there are just too many lines which need to be drawn and too few points. This is just a 'fact' of two-dimensional mathematics, just like the icosahedron being the largest (three dimensional) polyhedron.

While many other answers have covered the fact that this is provably impossible as typically understood. I remember as a kid being presented with this as a riddle and the way it was phrased was similar to how you phrased it. In that case it is often acceptable to pass a line through one of the dots, resulting in:

Of course this is a bit of a cheeky answer, but the way it is often worded, especially if intended to actually be solved, this is a solution.

This is a problem I've thought about quite a bit too and here is my "proof" for why it cannot be done

For convenience, let us label the black circles $B_1, B_2, B_3$ and the red circles $R_1, R_2, R_3$.
Suppose there exists a way to connect all the circles as specified. Then the circles $B_1, R_1, B_2, R_2$ form a closed loop in the plane.
Since the circle $B_3$ and $R_3$ must be connected to each other, they must be both inside the loop or both outside the loop.

Both inside
The interiors of the loops formed by sequences of circles $B_1, R_1, B_3, R_2$ and $B_2, R_2, B_3, R_1$ are disjoint regions whose union is the interior of the original loop. The circle $R_3$ must lie inside one of these loops. If it is inside the first loop, there is no way to connect it to $B_2$ which is outside the first loop. If it is inside the second loop, there is no way to connect it to $B_1$ which is outside the second loop. This is a contradiction.

Both outside
In this case, either the sequence of circles $B_1, R_1, B_3, R_2$ or $B_2, R_2, B_3, R_1$ forms a loop whose finite interior is disjoint from that of the original. Without loss of generality, suppose it is the former. Then $R_3$ must be within this loop or outside of it. If it is within the loop, then there is not way to connect it to $B_2$, which is outside the loop. If it is outside this loop, there is not way to connect it to $B_1$ which is within the larger loop formed by $B_3, R_1, B_2, R_2$. Hence another contradiction.

Further reading

This is a corollary of Kuratowksi's Theorem and is an example of what they call a $K_3,3$ graph.

It has to do with graph theory and non-planar graphs.

If (and only if) a graph is planar, you can draw it on a flat piece of paper without the lines ever crossing.

On the other hand, Kuratowski's theorem states that a finite graph is planar if (and only if) it doesn't contain the K5 graph or the K3,3 graph in any form.

The K5 graph is the completely connected graph with 5 nodes. The K3,3 graph is the complete bipartite graph with 3 nodes on either side, which is also known as "the utility graph" because of this very puzzle. (It's usually given in the form of connecting utilities to houses.)

So this puzzle asks you to draw one of the two "archetypal" non-planar graphs, and therefore it's impossible to solve it on a flat piece of paper. Which is why some silly guys "fixed" the puzzle by printing it on a mug. :-)

I am 32 years old now. When I was in 4th class, a friend of mine challenged me to solve this problem and I still can't solve it.

If your friend ask you to do this on paper page, there is a solution with this small "legit hack":

just make a hole in the paper, and draw a line through it, then continue the line on the other side and comme back through another hole !

The answer is not so nice but there is nothing to do because this puzzle is unsolvable in 2D.look here ,It's the same principle
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