Conversational hat trick

If there is one thing that the pandemic has indubitably brought along is more time at home: not only we are saving the commuting time, but even when we are technically in our home offices we cannot always abstract ourselves completely from our surroundings, so we are in the end more up-to-speed with anything that is happening in the house. In my case, the time that would otherwise go into travel time is now regularly spent having a second coffee at the breakfast table, reading the news or checking my social networks before my actual working day starts. Now that the kids are on vacation and old enough to help themselves to their own breakfast, this long stay at the table occasionally provides golden opportunities of conversation, that I dearly cherish. Today I have had, three conversations, three, with Jason that proved to me how he is maturing and getting ready to set sail any time.

The first exchange came about when he saw on my table the cover page of this video, which explains (now I know) the Collatz conjecture. He asked me if I was interested in "the 3x+1 problem" as well, but I did not know what he was referring to, so I clicked the video to take a look. He immediately protested my gesture and asked for an opportunity to explain the problem to me, because normally it is me doing the explaining. So he grabbed paper and pencil and demonstrated the problem without hesitation: starting with any number, you create new numbers in two possible ways. If the number is even, you divide it by two. If it is odd, you multiply by three and add one. For instance, starting with 7, which is odd, you apply the second rule to obtain 22, which is even. Then you apply the first rule to obtain 11, and so on over 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 4, 2, 1 and the next step will take you back to 4 for an endless loop. The conjecture in itself is that, for every positive integer that you use as a start, the sequence does eventually end in 1.

 

Photo: Bill Gracey

Once the statement of the problem was clear, we went on to discuss how you can work the procedure backwards, demonstrating all the values that can be reached in one step, then in two and so forth, until you close all holes. For instance, he quickly reached the conclusion that the conjecture was true for all the powers of two, because then you only apply the first rule and reach one. He also reasoned that, if the conjecture is true for a number (7 in our example), it is also true for all the powers of two multiplied by that number (14, 28, 56...) since the latter will unavoidably land in the former by application of the first rule and once you reach the initial number there is a known series. Of course, the best part of the conversation was the end, when he admitted that it had been great to discuss the math problem with me.

The second conversation took place during my mid-morning break, while I was watering the plants. I ran into him as I was coming back from the bathroom with a full watering can and I could not help showing him the progress of the avocado saplings that I planted about a month ago. I explained to him that, contrary to the common knowledge the best way to make them sprout is not the one with the toothpicks in a jar of water, but just peeling the pit, wrapping it in a damp paper towel and putting it inside a plastic bag for two to four weeks.

The advantage of this method is that it does not require constant checks of the water level and is also immune to the variations in air humidity levels but, if you are not careful, the shoot might come on the side of the pit instead of the usual growth just through the middle. Jason correctly pointed out this fact in one of the seeds and how the root was coming out precisely on the opposite side, hinting to which direction the pit was lying during the initial sprouting. He also realized that the shoot is leaning towards the window attracted by the light, so I explained that I regularly turned the pot around to prevent it from getting too twisted. Then we went on to examine two additional plants that were less developed and he also made a couple on interesting remarks. In the end, he confessed that the conversation had been very interesting and that he was less scared of having picked biology at school for next year.

The third conversation was directly initiated by him after lunch when he asked if we were somehow exempt from turning in our taxes as he has seen many times in the movies. I explained that we did have to turn in our taxes, but we used an accountant for that, which is the reason why he never saw me doing the work. Then he asked if we could keep doing his taxes as well when he went off to university and I explained that he did not need to do it until he had a reasonable level of income, like a job. This triggered a bit of anxiety in him, because he would not know how to do it or even how to find an accountant who can do it for him, so I assured him that, if that was a problem, we could ask our own to do it in the beginning until he got settled. He admitted that it would lift a weight off his shoulders because he was always scared of hiring people and getting into contracts. He was afraid that he would do something wrong and end in jail or hurt people by making the wrong choices, so that is why he was looking forward to sharing a flat with his schoolmates, where he would gladly take care of many house chores as long as someone else dealt with the utilities, the landlord, etc. Once again, I gave him reassurances that this fear is normal, that I had it myself when I left my home and that he would be able to do things well enough. His own worry was a sign that he would do OK.

It is hard to believe that that baby grew to a toddler, then to a teenager and now is nearing adulthood, having developed many amazing skills and, like everybody else, lacking a bit in some others. Above all, the most amazing part of his personality is how deeply concerned he is about the well-being of everyone around him. He is still occasionally nagging, but it is only with the intention to entertain and amuse. My only hope is that he continues to be lucky in life and does not end up running into someone that might destroy his kind heart, because I want to have many more conversation as interesting as those today, no matter whether it is just one or three at time. Have a nice evening.