This blog is about Simon, a young gifted mathematician and programmer, who had to move from Amsterdam to Antwerp to be able to study at the level that fits his talent, i.e. homeschool. Visit https://simontiger.com
Simon learned this game on Brilliant.org at https://brilliant.org/practice/winning-moves/?chapter=competitive-games (Warning: this link will only work if you have a Premium Subscription to Brilliant). Brilliant describes the game as follows: “Luk tsut K’i is a board game from China in the time of Confucius. In medieval Europe, it went by the title Three Men’s Morris. This game is very similar to tic-tac-toe; the objective is for one player to get their three pieces all on the same line. If this occurs, that player wins”.
After a whole night working on my writing and not feeling very fresh in the morning, I told Simon about the three ages of life: the young age is when one can party all night long and the next morning feel like one has been sleeping like a rose, the middle age is when one parties all night long and the next morning feels like one had been partying all night long, and the old age is when has been sleeping all night long and the next morning feels like one has been partying all night long. He immediately drew these pictures, telling me it’s just like 1-input 1-output logic gates, but the only one that makes sense is the OR.
Three boxes with fruit, all the three labels are misplaced. What is the minimum number of times one will have to sample a random piece of fruit from one of the boxes to know how to label all the three boxes correctly? From Mind Your Decisions.
Connect A and A’, B and B’, C and C’, D and D’ so that no lines intersect. (Neva added colors).
Dividing 11 coins among three people: “How many ways can you divide 11 coins to 3 people? How many ways are there if each person has to get at least 1 coin?” From Mind Your Decisions.
Solving a simple quadratic equation geometrically: the geometric interpretation of “completing the square”, a notion from deriving the quadratic formula. From Mind Your Decisions.
Which way do the arrows point? (Simon made this drawing in Microsoft Paint):
How many times, on average, do you have to roll a dice until you get a repeated value? I saw this probability challenge on the Mind Your Decisions channel. I decided to test it experimentally. First, I repeated the experiment myself in two sets of 50. Then I created a diagram in the Wolfram Language to visualize the distribution. Finally, I made a p5.js sketch to roll the dice thousands of times.
In a complete binary tree, every node has two children (except for the bottom nodes that don’t have any children at all). This means one mind-blowing thing: that the bottom row always has more nodes than the number of nodes in the entire rest of the tree! Example: if there’s one node at the top of the tree, two nodes in the second row, four nodes in the third row and eight nodes in the bottom row, the bottom row has more nodes (8) than the remaining part of tree (7). I’ve been thinking about this, and I applied this to the real world:
The average number of children a parent has in the world is 2.23 (I’ve used an arithmetic mean, which is oversimplistic, should have probably used the harmonic mean). Does this mean that currently, the number of children exceeds the number of parents? The definition of “children” I’m using are people who don’t have children, so the last row of nodes so to speak. By “parents” I’m counting all generations. If you just want to talk about now, the parents living now, then you have to trim the top rows (the already dead generations). If the average number of children is 2 or more, are there going to be more children in the world than parents?
Well, in this model, I’m ignoring crossover. This means we should consider every node in our tree for 2 people*. So, now, if the average number of children is 4 or more, there’re going to be more children than parents. So, what I said earlier was wrong. The average number of people doesn’t exceed 4, so there aren’t more children than parents. But the number of children today may still exceed the number of parent generations still alive.