## Balancing Scales

#### Problem

In order for the scales to balance the block on the right side must be 10 kg.

It is possible to use the same weights to weigh, for example, 4 kg.

In fact using 2 kg, 3 kg and 5 kg weights it is possible to weigh all but one value from 1 kg to 10 kg. What is that value?

(Note: You don't have to use all three weights.)

#### Solution

The following weights can be found, with the desired weight indicated in brackets.

Left Side | Right Side | |

3 | 2 (1) | |

2 | (2) | |

3 | (3) | |

2 5 | 3 (4) | |

5 | (5) | |

3 5 | 2 (6) | |

2 5 | (7) | |

3 5 | (8) | |

2 3 5 | (10) |

And so it is not possible to weigh 9 kg.

What if you used 1 kg, 3 kg and 6 kg to weigh all the weights from 1 kg to 10 kg?

What about using 1 kg, 2 kg and 7 kg?

If `x` + `y` + `z` = `T`, what must be special about `x`, `y` and `z` so that all the weights from 1 kg to `T` kg can be weighed?

(Hint: Try `x` + `y` = `T` to begin with.)

Problem ID: 91 (Dec 2002) Difficulty: 1 Star