There are 10 bags full of coins. All coins look the same and all wight 10 grams, except the coins from one bag that are fake and all weight 9 grams. Given a scale, how could you tell which bag has the wrong coins in just one measurement? Explain your answer!
Take one coin from the first bag, 2 from the second, 3 from the third, . . , 10 from the 10th and place them on the scale. The total mass shown by the scale should be $(1+2+\cdots+10) \cdot 10-x$ grams, where $x$ is the number of fake coins (each coin weighted 10 grams, except for the fake ones that are one gram lighter). Since the scale will show $550-x$ grams we can see how many coins are fake and that is enough to deduce from which bag those coines were taken.