Problems

For any triangle, prove you can tile the plane with that triangle.

Prove that one cannot tile the whole plane with regular pentagons.

Draw the plane tiling using convex hexagons with parallel and equal opposite sides.

Draw how to tile the whole plane with figures, consisting of squares \(1\times 1\), \(2\times 2\), \(3\times 3\), \(4\times 4\), \(5\times 5\), and \(6\times 6\), where each square appears an equal number of times in the design of the figure. Can you think of two essentially different ways to do this?

Find a non-regular octagon which you can use to tile the whole plane and show how to do that.

Mark one card with a \(1\), two cards with a \(2\), ..., fifty cards with a \(50\). Put these \(1+2+...+50=1275\) cards into a box and shuffle them. How many cards do you need to take from the box to be certain that you will have taken at least \(10\) cards with the same mark?

\(6\) friends get together for a game of three versus three basketball. In how many ways can they be split into two teams? The order of the two teams doesn’t matter, and the order within the teams doesn’t matter.

That is, we count A,B,C vs. D,E,F as the same splitting as F,D,E vs A,C,B.

How many subsets are there of \(\{1,2,...,10\}\) (the integers from \(1\) to \(10\) inclusive) containing no consecutive digits? That is, we do count \(\{1,3,6,8\}\) but do not count \(\{1,3,6,7\}\).
For example, when \(n=3\), we have \(8\) subsets overall but only \(5\) contain no consecutive integers. The \(8\) subsets are \(\varnothing\) (the empty set), \(\{1\}\), \(\{2\}\), \(\{3\}\), \(\{1,3\}\), \(\{1,2\}\), \(\{2,3\}\) and \(\{1,2,3\}\), but we exclude the final three of these.

Dario is making a pizza. He has the option to choose from \(3\) different types of flatbread, \(4\) different types of cheese and \(2\) different sauces. How many different pizzas can he make?

Determine the number of \(4\)-digit numbers that are composed entirely of distinct even digits.