Homework

Problem 1

The Game of Fifteen turns out to be identical to Tic-tac-toe, though it is not immediately apparent. To see why this is, consider the following $3\times 3$ grid of numbers

$$\begin{array}{|ccc|}\hline 8& 1& 6\\ 3& 5& 7\\ 4& 9& 2\\ \hline\end{array}$$

• Part A Show that each row, each column and each diagonal of the above grid adds up to $15$. Squares with this property are called magic squares.
• Part B List (up to reordering) all the possible triples $(a,b,c)$ of integers between $1$ and $9$ which don’t repeat digits and which sum up to $15$. Show that each occurs as either a row, column, or diagonal in the matrix above.
• Part C Use the results of Part A and Part B to explain why the Game of Fifteen is the same as Tic-tac-toe.

Problem 2

Consider the table below, which is an example of a $4\times 4$ magic square.

$$\begin{array}{|cccc|}\hline 2& 16& 13& 3\\ 11& 5& 8& 10\\ 7& 9& 12& 6\\ 14& 4& 1& 15\\ \hline\end{array}$$

• Part A Show that each row, each column and each diagonal of the above grid adds up to $34$.
• Part B Find some examples of four entries in the matrix above which sum to $34$ but are not rows, columns, or diagonals.
• Part C Explain why even though we have a similar situation as in Problem 1, it is not true that the Game of Thirty-four is the same as the $4,4,4$-game.