The Magic of Logic (7)
It’s all about scales and weights. Why are so many young kids currently weighing potatoes? Perhaps you can help them?!
Recently (in January 2020) I travelled to Chennai, India, where a former World Chess Champion and a World Champion challenger, both good friends of mine, were doing a training camp with the most talented young chess players in the world. It is a remarkable experience, and I have reported on it here. The kids are twelve to sixteen years old, basically children, playful and noisy. But at the same time most are fully qualified chess grandmasters, who are making life difficult for some of the best players in the world. It is not easy to convey to non-chess readers how mind-blowing this is: it’s like pointing to a rowdy child on a playground and saying: “That’s Stephen. He’s twelve. He has just completed his PhD in quantum physics and is working at the cyclotron in Zürich.” Whaaa?
The kids in Chennai spent six to eight hours of each day studying chess. Then in the evening they would unwind with football on the lawn, running and kicking — until someone saw me watching. He would inevitably run over to me and say: “Come on, please give us some logical puzzles!” And then they would spend an hour or more solving them.
I should mention that as part of a research project I am currently trying to determine whether exceptional chess talents are better at working out non-chess logical problems than normal kids — or equal, or perhaps worse. In this endeavour psychologist friends have urged me to use one standard problem (and not an assortment) so I can reach a valid conclusion. I selected the problem suite I had conducted with grandson Enders when he was six years old. I tried it on the super-talents in August last year (2019) and reported on that experience. I will report on the overall results when I have a larger sample.
New weighing problem
Since my visits many of the chess kids have been in touch, asking for more puzzles, and challenging me with some of their own. Currently they are pondering over a new weighing problem I gave them. I got it from Fabian, a young next-gen programmer in our company:
A marketeer sells potatoes. Unfortunately he has forgotten to bring along the weights for his scale. So he borrows weights from a neighbouring vendor, who cannot relinquish all his weights. So our marketeer takes as few as possible. How many must he take, and which ones, to be able to weigh anything between one and forty pounds, in a single weighing?
This is not an easy puzzle (if your name isn’t Gauss). My best solver, Tommy, found the answer in two minutes. We spent much more time testing the solution, figuring out how to weigh out random numbers of pounds: 4, 13, 29, 34. It worked every time — but not without careful thought. You can sell anything between one and forty pounds with one weighing.
Today, almost a week after my proposing it, none of the chess kids have solved the problem. Yesterday I tried it on my prime tester, eight-year-old Enders, who is a very proficient logical problem solver. I gave him a simpler version — which weights do you need to weigh any amount between one and fifteen pounds — in the car driving home from school. He came up with crazy answers, like 120, until wife Ingrid discovered he did not know what market scales are and how they work. At home I showed him an example.
But the problem was still much too hard, and his younger brother Hennes, six years old, wanted to participate. So I started from scratch (and this section is meant as a tutorial for working with young children):
“Let’s say you want to weigh just one pound,” I said, “how many weights will you need to borrow? Ha-ha, easy peasy: just one, a one-pound weight. Two pounds? Two weights, naturally: one and two. Anything up to three pounds? Two weights! Clever boy, Enders figured out immediately that 2 + 1 can be used to weigh out three pounds. Okay, what do you need to weigh anything between one and four pounds? Three weights: one, two and four, both lads agreed. But they could see in my face and my demeanour: there is a better solution. And suddenly Enders had it: two weights! You have one and three pound weights, which you can use together to measure four pounds. One, and three are also easy, using 1, and 3. But what about two? Well, Enders had found the trick: put the three pound weight on one side and the one pound on the other, together with the potatoes. That will weigh out two pounds!
Now with the trick of putting weights on both sides of the scales, Enders was ready for more. We discovered (together!) that for one to five pounds we needed three weights. With 1, 3, 5 we could weigh anything up to six pounds, and could do more. Eight and nine were easy, but seven required some effort. When someone in the family asked Enders something else, about his day in school, he refused to answer: “I’m thinking!” So his involvement in the logical problem was high.
With 1, 3, and 5 pound weights you can weigh seven pounds by weighing 5+3 against 1. Eight and nine are trivial, and with that we had reached the maximum — which is always the sum of all the weights you have. But what if we borrow different weights? We discovered, by trial and error, that 1,3,6 allows you to weigh one to ten pounds. We tried 1,4,7, but could not weigh out nine or ten pounds. Then we had a nice idea: 2,3,7. That works for one to ten and twelve — but eleven is impossible. And then we tried 1,3,7. We had to make sure we could weigh everything from one to eleven with them, and struggled for a while on five and nine pounds, which seemed impossible — until we thought of 7+1–3 and 7+3–1. We had worked out all combinations.
It was a great evening, with lots of discussion and sharp thinking. Do try it with your kids, it is lots of fun and very bonding. And if one of them has a natural aptitude for mathematics, you may be laying the foundation for a great career. If you yourself are mathematically inclined, or have masochistic tendencies, try and figure out which weights you need to weigh out anything from one to 121.