Problems
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.
What is logically the opposite of the statement “every \(n\) is odd or \(p<q\)"?
Four different digits are given. We use each of them exactly once to construct the largest possible four-digit number. We also use each of them exactly once to construct the smallest possible four-digit number which does not start with 0. If the sum of these two numbers is 10477, what are the given digits?
The picture below shows a closed disc, which is just a circle with the inside filled. The grey interior represents the interior of the disc. Describe the resulting shape when you glue the circular boundary to one point.
Label the vertices of a cube with the numbers \(1,2,3,\dots,8\) so that the sum of the labels of the four vertices of each of the six faces is the same.
Is it possible to construct a 485 × 6 table with the integers from 1 to 2910 such that the sum of the 6 numbers in each row is constant, and the sum of the 485 numbers in each column is also constant?