Problems

There are \(n\) balls labelled 1 to \(n\). If there are \(m\) boxes labelled 1 to \(m\) containing the \(n\) balls, a legal position is one in which the box containing the ball \(i\) has number at most the number on the box containing the ball \(i+1\), for every \(i\).

There are two types of legal moves: 1. Add a new empty box labelled \(m+1\) and pick a box from box 1 to \(m+1\), say the box \(j\). Move the balls in each box with (box) number at least \(j\) up by one box. 2. Pick a box \(j\), shift the balls in the boxes with (box) number strictly greater than \(j\) down by one box. Then remove the now empty box \(m\).

Prove it is possible to go from an initial position with \(n\) boxes with the ball \(i\) in the box \(i\) to any legal position with \(m\) boxes within \(n+m\) legal moves.

A paper band of constant width is tied into a simple knot and tightened. Prove that the knot has the shape of a regular polygon.

On the picture below you can see graphs \(K_5\) a complete graph on \(5\) vertices and \(K_{3,3}\) a complete bipartite graph on \(3\) and \(3\) vertices. A theorem states that these graphs cannot be embedded into plane, namely one cannot draw graphs \(K_5\) and \(K_{3,3}\) on a plane in such a way that there would be no intersecting edges.
A converse statement is also true: if a graph \(G\) cannot be embedded into a plane (drawn on a plane without intersecting edges), then this graph contains either \(K_5\) or \(K_{3,3}\) as a subgraph.
The question is: can you draw the graph \(K_5\) without intersecting edges on a torus?

Is it possible to draw a graph \(K_{3,3}\) without intersecting edges on Moebius band?

Is it possible to link three rings together in such a way that they cannot be separate from each other, but if you remove any ring, then the other two will fall apart?

We start with a rectangular sheet of paper, preferably with proportions more than \(6:1\), so it looks more like a band, for now assume that one can stretch or shrink the paper band as needed. If we glue the opposite sides of the paper band in the same direction as on the picture, we will get a cylinder. What surface do we get, if we glue the circles of the cylinder in the same direction as well?

Describe the surface which we can get if we start with a rectangular sheet of paper and then glue the opposite sides of the paper band in the opposite direction as on the picture.