Now that I have your attention, let’s talk about math. It was a rainy Sunday evening in Puchberg am Schneeberg, the fifth day of the Club Alpbach Lower Austria Hub. May it be due to the rain, the combination of ambition and dizziness of the scholars or the lack of motivation for typical evening activities, such as drinking, after an intense Saturday night, however, at some point almost out of nowhere everyone was talking about a certain riddle. That is:
You are given twelve (billiard) balls. They all have the same weight but one – one is either heavier or lighter. You are also given a balance scale, with which you can measure the weight of a single ball or a set of them against another set of balls. Your task is to find the ball that has a different weight and determine if it is heavier or lighter than the others. The difficulty is, you may only measure three times.
In the following hours a scenery evolved, that reminded me back to the mornings before my math exercise classes at TUW. People crowded in small circles, drawing sketches of possible solutions on napkins, throwing arguments for why their particular approach is better than that of the others. As Alpbach-unusual as this may sound, it was a lot of fun! Around 2 am, the first working solution was found – a reason to cheer, for all that were still awake.
However, we did not let go of the problem right away. Could we find an algorithmic approach to solve the riddle? Could we generalize it to a higher number of balls and measurements? What if two out of twelve balls had a different weight – or if we used a balance scale with three arms instead of two, and does that kind of scale even exist? Finally we set up a formula to calculate in many (not all) of those cases, how many non-equivalent measurements one could do with the scale and the balls, and how many non-equivalent combinations of measurements there are – in our case 73788 and 66955816363436. An algorithm for the calculation of all possible solutions to the riddle was already in the making, when we decided to stop escalation the situation further and went to bed to be at least somewhat fit for the talks in the next morning.
Dear Reader, I hope I didn’t scare you too much with these paragraphs. If you are not into balls-riddles, that’s fine, you don’t have to be. However, this common riddle-solving strengthened my impression, that scholars of the Forum Alpbach are interested in and fascinated by a vast variety of things – not only politics, economy and society, but apparently also math, natural
sciences and informatics. So after all, this was a fun and bonding evening and I’m looking forward to any further riddle-solving (or other, more usual evening activities) with my co-scholars in the future.
P.S.: Have you puzzled over the riddle yourself and want an exemplary solution? I’m sorry, I won’t provide one here. But you can always get in touch with the Club Alpbach Niederösterreich and ask for one! 😉
von Felix Wagner