Problems

How would you describe the surface obtained by glueing the sides of the octagon as on the picture? Sides of the same colour are glued together in the same direction as shown.

Describe the surface which we can get if we start with a rectangular sheet of paper, make a cylinder by glueing the opposite sides in the same direction and then glue the other opposite sides of the paper band in the opposite direction as on the picture.

Scrooge McDuck has \(100\) golden coins on his office table. He wants to distribute them into \(10\) piles so that no two piles contain the same amount of coins. Moreover, no matter how you divide any of the piles into two smaller piles, among the resulting \(11\) piles there will be two with the same amount of coins. Find an example of how he could do that.

A parliament has 650 members. In this parliament there is only one house and every member has at most three enemies. We wish to split this parliament into two separate houses in such a way that each member will have at most one enemy in the same house as them. We assume that hard feelings among members of parliament are mutual, namely if \(A\) recognises \(B\) as their enemy, then \(B\) also recognises \(A\) as their enemy.

Is this splitting possible?

Suppose you have a coffee mug made of stretchy and expandable material. How do you mold it into a donut that has a hole inside?

Take a straight string of finite length. Put the right half on top of the left half to form a hoop, and then pass it (the piece now on top) through the hoop from underneath. Glue the two ends together to form the trefoil knot.

Draw a sequence of pictures to illustrate the formation of the trefoil knot.

In the picture below, each side is glued in the same direction as its opposite side. Draw a sequence of pictures to show the gluing process and the final shape.

Take a straight string of finite length. Wrap the right half over and then under the left half so that it is pointing to the right again and there is a hoop. Pass the right pointing piece through the hoop from above. Glue the two ends together to form the figure-8 knot.

Draw a sequence of pictures to illustrate the formation of the figure-8 knot, seen below.

Explain why it is possible to have a picture of a figure-8 knot with 100 crossings.

The unknot is the simplest knot, formed by taking a piece of straight string and gluing its two ends together. In reality, there is no knot! Now, which of the following two knots is the unknot?

Show that a knot that can be drawn with only one crossing must be the unknot.