Guess guess which hand the golden ball is in
Back in the pre-Covid era (oh, when was that?) I was once asked to participate in the "green school". In addition to appropriate rest, the meeting was devoted to mathematics, namely the sphere and its properties. This topic is usually omitted in school because... well, I don't know why.
After that, even students of geology do not know what longitude is (the real thing happened to me - I gave a lecture at the appropriate department of the university). The meeting was extremely successful, applause to the leadership and the three teachers who organized it all. Teaching is not only about transferring knowledge in 45-minute increments from 8 AM to XNUMX PM or XNUMX PM. Well, now everything is different with distance learning. More and more teachers are discussing how to change the traditional classroom system to ... so what? Recall that we are experimenting on a "living organism" - children. Where is the golden ball containing the wisdom of life?
I am reviewing student applications for a National Children's Fund scholarship for exceptionally gifted children. There were always quite a lot of messages from Leszno. They were in that year, but every second child (a student of the lower grades) wrote: "Since my teacher, Mrs. I., left, I have lost interest in mathematics." However, there were many applications from Lublin, which have so far been little submitted. A riddle for readers: what city did Mrs. I. from Leshno move to? To Lublin? Yes, but how did you come up with it, readers?
The surface of a sphere is a sphere (from the Latin sphere 'ball, sky'). This mathematical term has entered the colloquial language: we are talking about the spheres of influence of the great powers, the sphere of one's interests and social spheres. “Oh, she is not of our sphere,” said the countess to this beautiful country girl with whom the young lord had fallen in love. And then everyone imagined society as concentric shells, impenetrable to each other: on the one hand, we are in the best company, of course, on the other, this poor girl and even geometry says “Cinderella: stay where you are!”.
It's not hard to be fascinated by the spherical shape. It is enough to be under the open sky at night, away from cities, preferably on a high mountain in winter. Let's look up: don't we see the celestial sphere clearly? Distant stars join it, against their background wandering celestial bodies move along closer spheres: planets. Ptolemy taught that the Earth is the center of the universe and is surrounded by nine concentric crystal spheres.
In the first seven there are seven known planets: Diana (= Moon), Mercury, Venus, Apollo (= Sun), Mars, Jupiter and Saturn. The eighth sphere contained the fixed stars. Nine o'clock was like a handle that regulates the movement of the clock: the spring equinox moved along them. In the Middle Ages, a tenth sphere was added to this system: the Prime Mover, like a spring, everything moving, a driving force, a hard shell that separates the world from non-existence. The Pythagoreans believed in the harmony of the spheres - that the planets moving through their spheres emit extremely pleasant sounds. After all, the world is a number and music.
In the Mathematics Olympiad for schoolchildren, which we organized in the aforementioned green school, the competition was fierce two teams. Well, one won (33:31), the other lost. Like it is in sports.
The algorithm for dividing starters into two teams is so interesting from a mathematical point of view that I will dwell on it in detail. The problem here, of course, is the equal qualification of strong and weak teams. But what is stable? Apparently, the best choice is random: each player takes a piece of paper with the words 1 or 2 from the box and goes to the appropriate team. But... if you flip a coin 10 times, only 25 percent of the time the result will be 5:5, which is five heads and five heads. So, we see that with a 75 percent probability the teams will be unequal.
There is a blatantly unfair way in which two previously appointed captains choose their team members one after the other: once you, then me. The first captain always has the advantage, he can choose the best of the rest. Similarly, in football, the winner of a cup match is determined from a penalty spot. One team always shoots first. Things are better in tennis where the server is always in the best position. In a tie-break game, after A's first serve, the second serves two times, then A twice, and then two serves, alternating B, A, ... to an advantage of two winning points.
This method is also not very suitable for selecting two student teams. The method that I will describe was created by mathematicians on an idea taken from the so-called Steinhaus algorithm. It is commonly used in math matches, such as pre-Olympic preliminaries. Interestingly, we used a very similar system in my backyard when we wanted to “play football” in the then empty square behind the house. There were many boys (I come from the first wave of the post-war baby boom).
The algorithm is like this. The coin decides which of the captains (A or B) will be chosen first. Let it be A. He points to the player, and now (attention!) Captain B decides whether this player will go to the first or second team. And so alternately. One selects the player, the other appoints him. The second indicates that the first highlights.
Note the psychological difficulties. If captain A chooses the best one, B will immediately add him to his team. If he points to the weakest one, B will say: “very good, take him” ... but after a while the roles will be reversed. So how do you play?
A certain amount of social engineering and social psychology helps here. Especially when young people are involved in the selection, likes and dislikes are very important, and not just an indefinite “level”. In particular, after selecting any of the players as captain A, captain B evaluates whether said teammate (indicated friend) is subjectively in the top or bottom half of the participants. Do I want him/her or let him go to A? I have a choice. If I see the specified member in the upper half, I take him in, and if I see the specified member in the lower half, I leave him. The role is changing, now B and A scores qualify. He does the same. If he evaluates the indicated person well, then he takes him personally, if he is mistaken, he leaves. This way, every captain gets better (or at least equal!) every time. Everyone thinks that he made the best choice, that his team is better. Mathematics can please people even in such delicate matters.
It was with such questions that the contestants in my green school fought. As you can see, there are some questions. non-mathematical, challenging and fun.
- What is loxodrome?
- You have 20 balls. What is the height of the tetrahedron that can be made up of them? How many balls do you need for a 10-layer tetrahedron?
- I left the tent. I walked a kilometer west, then a kilometer north, then a kilometer south. This is how I ended up in my tent. A bear sat in front of him. What color was it?
- How many balls of diameter 1 will fit in a ball of diameter 2?
- Sort from smallest to largest ball used in the following sports: tennis, table tennis, football, volleyball, basketball, water polo.
- Which ball is neither spherical nor oval (as in rugby or American football)?
- List proverbs and sayings related to the ball.
- Come up with a joke that begins with the words "A bullet flies to the doctor."
- A sphere is inscribed in a cube with a side of 1 meter. Is there enough room for a 20 cm ball in the corner?
- Can a cube with a side of 1 inch fit in a sphere with a radius of 1 centimeter?
- As you know, in the past, cannonballs were indeed spherical. Today they are not. What made you change the shape of rockets?
- The volume of the sphere is p2 cubic centimeters. Calculate its surface area.
- This is a circle with a radius
it may lie on a sphere of radius
- Container B contains 100 white balls, container C contains 100 black balls. We randomly select 10 balls from container B and drop them into C. Out of the 110 balls currently in C, we randomly select 10 and drop them into B. Are there more black balls in B or white balls in C?
- What shape can be the shadow of the ball?
- Which parallel on Earth is half the length of the equator?
- Planet T is evenly covered in grass. At some point on the planet, a goat is tied. How long should the chain be so that the goat can reach exactly half the grass on the planet?
- In the poem, Pan Tadeusz Stolnik was shot. Whose rifle was hit by a bullet?
- How many four-letter words (meaningful or not) can be formed by rearranging the letters in the word KULA?
- Is there a ball touching all the edges of the cube? If so, calculate its radius. If not, justify.
Comments. I propose to find out (from which Internet?) What is Loxodrome.
Task 2 is quite difficult. Twenty identical balls can be made into a tetrahedron 10 + 6 + 3 + 1 (ten balls at the bottom, then six, three and one). Such a block has four layers, but it is less than four times the diameter of each sphere - the balls fall into the recesses of the lower floor.
I will discuss this challenge though... I won't decide. I'll leave that to the willing reader. I mean, among others, my friend Kazimierz from Szczecin. Kaziu - you will definitely like it. After all, we associate the task with the school. This is the bunch we see in the photo. These oranges were very good... Any salesman knows that it is best to put apples, oranges, lemons and other hard fruits like this (tomatoes could crumble). Well, it was only at the end of the last century that the problem posed by Johannes Kepler in 1610 was solved, namely, how to mathematically show that this is really the best way. More precisely, equal spheres occupy the smallest place in space with this arrangement. This is just under 75 percent. This is an exciting mathematical problem because it occurs in large spaces, but that is again a topic for another article.
The school I went to, well, quite a while ago, was still eleven years old. In the penultimate, tenth grade, the whole year was mostly geometry and trigonometry. I remember Henryk Paśniewski's set of problems - what wasn't there? Tetrahedra, prisms and pyramids are cut in all possible ways. Oh yes, crutches were few. Because it is difficult, even drawing is not easy.
Since then, trigonometry at school has been very truncated, degenerated. Like any old man, I tend to remember that everything “then” was better. This, of course, is not true. Not all. Trigonometry is no longer so necessary in the daily work of an engineer and surveyor. Wooden triangulation towers on mountain peaks are no longer needed. One of the largest buildings was located in Lyuban on Kroshchenko Street. He even had a mountain name "patria". Okay, enough of these digressions. Let's look at the picture. First, we determine the length of the segment s. It's a 60 degree angle. From AC we find BC, then the height BH. But this is the height of the side wall of our orange pyramid. From here, the height of the pyramid is obtained by multiplying the HV by the sine of the angle of inclination of the wall to the base, which ... also needs to be calculated, but this is easy and standard.
Can I say that "task solved". Unfortunately, I associate this with the increasingly widespread distance learning. I sit in front of the screen and "talk to the picture", and the students - because I teach them - have to work according to my instructions. Anyway, that's how I learned Microsoft Teams, Inspery and other gadgets for conducting such classes. The instructor was at home, I was at home, everyone was drinking their coffee, he was "talking to the picture" and I was trying to imitate.
→ At the school where I teach, they already know that even when “normality returns”, the lectures will continue in the same spirit. This form has many advantages. This is a topic for another article. This makes it even better for those who…want to learn.
→ Unfortunately, there are not as many of them as it seems to us, teachers. I was late to submit this article to the editor, but if you are reading this, then it ended well. Namely, I was frustrated by the behavior of the freshmen, whom I gave too much freedom during the exam. I will draw conclusions and “police” methods will return. And my mood meant that instead of finishing the text, I went for a long walk through the snowy fields near Warsaw. It was cold, I was cold...
→ Let's return to the competition with the ball. Question 6 can be answered with a hacksaw or a crushed ping-pong ball. The best joke (task 8) was the one where the bullet complains to the doctor: "I don't know what's wrong with me, but I'm completely confused." There was also a good one, in which the bullet complained of radiant pains and that it was bursting in diameter. The proverb associated with the ball (task 7) is placed in the title of the article, we also know what the ball at the feet is (this is the Earth, right?). The bear-colored task has a long beard (of course, the bear was white, because such a route is only possible in the polar regions).
Cannonballs (problem 11) are no longer spherical because we can make threaded barrels, which gives the projectile rotational motion.
Question 13 turned out to be interesting. Later I even gave them to students. They tried to make calculations that were not very sensible, they were horrified by the element at 17. Meanwhile, the task is trivial. A sphere with a given radius is tiny, while a circle is large. It won't fit. To question 18, the correct answer was: a Muscovite whose rifle Jacek Soplica snatched out. The students answered incorrectly: Jacek.
I will return to the sphere, because it attracts me very much. And the proof is the following “ode to the spherical”.
Kulo! You are my! Big or tiny
You are the same around the end.
I want to sing all your beauty today.
I can do equations. But I will write an ode!
How much to appreciate, he will only know
Who saw you. But your beauty in all its decoration
Those who studied at the green school in Slavkov will appreciate it!
Your beauty extends from the center to the edges!
Everyone is equally drawn to your geometry.
I'm from a small child, from a preschooler,
I've always wanted to understand why you are like this.
How painful these paternal words sounded:
“You will know when you grow up. Today you are too small!
When I was a boy and an innocent child
Four thirds wanted to know your secret
And how deep the heart was deep
When I wanted your secret four-story plaza.
I swallowed up a lot of fleeting youth
Integrating your basic rounds into thought,
And when the youth's face was covered with stubble,
I saw a triangle with three right angles
She has. And immediately I could decipher how beautiful she is.
The surface geometry is elliptical.
And how the young man was influenced by Oscom,
When she finally showed me her rhumb line
What is each meridian?
At the same cutting angle ... and it follows that
That the length of the curve grows to infinity,
And that both poles are intertwined with love.
Later, looking at your Riemann tensor
I calculated the areas of fields until the morning
And on your surface gracefully called a sphere,
The main curves never went to zero.
So stay true to me, unshakable, homogeneous,
Just as cute and funny in every way.
Always leave only round sections!
More than once my heart will yearn for you.
Let young people now, not knowing gray hair,
They study your curves, unknown to Gauss.
