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Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. Two solutions are considered to be the same if, as in the example shown, they contain the same six triples. How many different solutions can you find?

2 | 8 | 3 |

6 | 4 | 9 |

5 | 7 | 1 |

2 | 3 | 8 |

5 | 1 | 7 |

6 | 9 | 4 |

Show that it is impossible to place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that the diagonals, as well as all the rows and columns, add up to prime numbers.