Problems

Let \(A\), \(B\), \(C\) and \(D\) be four points labelled clockwise on the circumference of a circle. The diagonals \(AC\) and \(BD\) intersect at the centre \(O\) of the circle. What can be deduced about the quadrilateral \(ABCD\)?

Consider the 7 different tetrominoes. Is it possible to cover a \(4\times7\) rectangle with exactly one copy of each of the tetrominoes? If it is possible, provide an example layout. If it is not possible, prove that it’s impossible.

We allow rotation of the tetrominoes, but not reflection. This means that we consider \(S\) and \(Z\) as different, as well as \(L\) and \(J\).

In the following grid, how many different ways are there of getting from the bottom left triangle to the bottom right triangle? You must only go from between triangles that share an edge and you can visit each triangle at most once. (You don’t have to visit all of the triangles.)

Explain why a position \(g\) is a winning position if there is a move that turns \(g\) into a losing position. On the other hand, explain why a position is a losing position if all moves turns it into a winning position.

A technique that can be used to completely solve certain games is drawing game graphs. Given a game \(G\), we draw an arrow pointing from a position \(g\) to a position \(h\) if there is a move from \(g\) to \(h\).

As a simple example, the game graph of \(\text{Nim}(2)\) is shown below.

Draw the game graph of \(\text{Nim}(2,2)\). Is \(\text{Nim}(2,2)\) a winning position or losing position?

Is \(\text{Nim}(2,5)\) a winning position or a losing position?

Let \(x,y\) be nonnegative integers. Determine when \(\text{Nim}(x,y)\) is a losing position and when it is a winning position.

Is \(\text{Nim}(1,2,3)\) a winning position or a losing position?

Is \(\text{Nim}(1,2,4,5,5)\) a winning position or a losing position?

When we write 137 in decimal, we mean \(1 \cdot 10^2 + 3 \cdot 10 + 7 \cdot 1\). If we write it instead using powers of 2, we have \(137 = 1 \cdot 2^7 + 0 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0\). To tell apart binary representation from decimals, we can use the following notation: \(137 = (10001001)_2\).

What is the number 273 in binary? Note that using binary is useful for finding whether a particular Nim game is a winning position or a losing position.