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

\[\begin{array}{|c|c|c|}\hline 8 & 1 & 6 \\\hline 3 & 5 & 7 \\\hline 4 & 9 & 2 \\\hline \end{array}\]
  • 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 Problem 1 and Problem 2 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.

\[\begin{array}{|c|c|c|c|}\hline 2 & 16 & 13 & 3\\\hline 11 & 5 & 8 & 10\\\hline 7 & 9 & 12 & 6\\\hline 14 & 4 & 1 & 15\\\hline \end{array}\]
  • 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.