## Hint for Problem 4

("Know/Don't Know Game" Math. Mag. 1983, p. 177) After B's first announcement,
the sum of squares must be attainable in at least two ways, and the
possibilities are reduced to a table that begins as follows.

Sum | Sum of Squares |
---|

8 | 50 |

9 | 65 |

10 | 50 |

11 | 65, 85 |

13 | 85, 125, 145 |

14 | 130, 170 |

15 | 125 |

16 | 130, 200 |

17 | 145, 185, 205 |

. . . | . . . |

The only appearances of 50, 65, 85, 125, and 145 as a sum of squares in this
table are already displayed. Keep going and you will find one possible answer.
But how do you know it's the only possible one?

B's first announcement gives A a lot of information. But when A makes his
first announcement, B already knows that A cannot possibly tell what the two
numbers are. What good does it do B to listen to A?