Problems

The left figure is formed by two interlocking loops joined to a solid ball. The right figure is formed by two unlinked loops joined to a solid ball. Describe how to transform the left into the right without cutting, tearing or passing the loops through each other.

A pair of points on a circle are said to be antipodal if they are on two opposite ends of a common diameter. P and Q in the picture are antipodal points. If we glue every pair of antipodal points on a circle, then what is the resulting shape?

A surface P is created by gluing every pair of antipodal points of a disc (a circle with inside filled in). We represent P on the plane by a disc in the following picture and bear in mind that the antipodal points are glued.

Explain why the two diameters in the pictures are in fact two circles on P and how to stretch it so that it becomes a single loop not touching any of the glued points.

It is possible to play tic-tac-toe on a torus: gluing the sides means that the bottom row is above the top row and the right most column is also to the left of the left most column. Is one of the players guaranteed to win if they play all the right moves?

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

Show that $\sqrt[3]{3}$ is irrational.

What is logically the opposite of the statement “every $n$ is odd or $p<q$"?

Let $a$, $b$ and $c$ be the three side lengths of a triangle. Does there exist a triangle with side lengths $a+1$, $b+1$ and $c+1$? Does it depend on what $a$, $b$ and $c$ are?

There is a triangle with side lengths $a$, $b$ and $c$. Can you form a triangle with side lengths $\frac{a}{b}$, $\frac{b}{c}$ and $\frac{c}{a}$? Does it depend on what $a$, $b$ and $c$ are? Give a proof if it is always possible or never possible. Otherwise, construct examples to show the dependence on $a$, $b$ and $c$.
Recall that a triangle can be drawn with side lengths $x$, $y$ and $z$ if and only if $x+y>z$, $y+z>x$ and $z+x>y$.

There is a triangle with side lengths $a$, $b$ and $c$. Does there exist a triangle with side lengths $|a-b|$, $|b-c|$ and $|c-a|$? Does it depend on what $a$, $b$ and $c$ are?
Recall that a triangle can be formed with side lengths $x$, $y$ and $z$ if and only if all the inequalities $x+y>z$, $y+z>x$ and $z+x>y$ hold.