color math

One reader accused me of making political allusions in my papers on mathematics. Well, I was only talking about training. School has always been a political topic, even when it was supposed to be apolitical in terms of software. In early April, after the introduction of cardinal restrictions in our public life, the demand for distance learning increased dramatically. Part of my article is a reaction to a TV lecture series for elementary school students. They caused a storm in the world of mathematics teachers - they were full of nonsense, like an old barrel of water thrown into a lake. So that no one accuses me of politicization, I will not write which TV channel it was.

The text is fragmentary - I start with a conversation for young children, but move on to reasoning for adults and vice versa. This is not to bore you. First for the kids. This is my voice in the discussion about how (well, how you can) talk to kids about the “Queen of Sciences”.

Exercise 1. Take a look at my first puzzle. What do you see on it?

Where do you live? Mark. Do you think I chose the colors of our borders by chance, or can you find an explanation why the “top” is blue-green, and the “bottom” is a white figure? But why did I write "above" and "below"? After all, these parts of the world are called ... well, what exactly? And the other two? Or maybe you know why the international designations of the four cardinal points are N, E, W, S?

Exercise 2. Look at the road signs (1). Which can we call square? And why are the corners of the first and third rounded? Find out which road signs are triangular, circular (circular), and octagonal in shape. Why is one triangular sign different from the others? Why only one octagonal sign?

1. Which of these symbols are square?

Exercise 3. Go online. Raise any browser. Type "square", then select "pictures" and... look at the pictures that are there. Not all, but only a dozen. Choose the one you like best. You choosed? Now try convince mewhy this one. Maybe you don't know yourself? Or maybe you know?

Exercise 4. Now look at my puzzle number 2. Do you see squares in it? Exactly - it's red inside. They get bigger. The first, tiny, on the left has one eye, one "button".

I will answer right away. A magic square is a square in which the sum of the numbers horizontally, vertically and diagonally is the same. Let's check: you would probably say that the second one is twice as big because it has two buttons on each side…. Oh, is it twice as big? Count how many buttons he has Four! Let's see what will happen next. The third wide and three loops in height. Count the seams. How many are there? 25. The fourth four is a long and wide (or high) four. Four times four is sixteen. Yes, it has sixteen stitches. And the fifth? There are five stitches on each side, so how many are there in total? Bravo, 25. We say that this square has an area of ​​XNUMX. But you probably knew it. So, as shown in the table on the right.

4+9+2=3+5+7=8+1+6=4+3+8=9+5+1=2+7+6= 4+5+6=8+5+2=15.

Wikipedia rightly writes that magic squares are useless in science. They are only interesting. But the ways they are constructed are more interesting than the squares themselves. It's like in tourism: very often the goal is secondary, the path to it is important. Let's look at how to build a square of twenty-five square meters. We put the one in the middle and remember the already forgotten “royal game”, that is, chess. We will jump straight to the NNE (North-North-East). Already the "troika" falls out of the square. We take it to its place (the last one in the second row from the bottom). Reminds me of the musical "reduction to the first octave". We apply this principle consistently... as long as possible. He gets stuck at six. It doesn't matter, we put the six under the red five, which is already within our square.

2. Why is this square "magic"?

Back to math for kids. Now look at the top of my puzzle #2. Are there any squares there? Not! What are these figures called? Beata, how are you? You are right, rectangles. Why are they called that? Because they have right angles? We'll talk about this a little later, but for now let's remember what a right angle is. Bartek, how would you explain this to someone who doesn't know? Maybe it's such an even angle. Well, let it be. If we are driving a car and turning at a right angle, then neither too far forward nor too far back, but exactly exactly to the side. Selina, get up and turn around at a right angle. Left or right? whichever way you want.

Let's also talk about the shapes above, that is, the rectangles. Which one is fat, thin, slender, tall, short, less oblong, more oblong? You will surely agree that the yellow on the right is long, thin and tall. But be careful. If it lies on its side, it will also be long, but short. Would you call him "fat"?

3. Start building a 5 by 5 magic square.

4. How to build a 5x5 magic square?

Now again two inserts for older readers. The first is 100. I think 100 is a hundred in any Slavic language. This is important for linguists. The name of this number distinguishes two groups of Indo-European languages, which include all languages ​​on our continent, except for Finnish, Hungarian, Estonian Basque and the little-known Breton.

In the languages ​​that developed in the first wave of migrations, the word 100 developed into (Greek) and (Latin), from which both French and German (and, of course, English) originated. That is why we call these languages ​​centums.

Our language belongs to the group of central or satemic languages, because after palatalization (softening) the proto-language took this beautiful and short form of a hundred. One hundred years, one hundred years, long live...

5. For connoisseurs. Magic square made up of prime numbers.

The second insert is longer, but perfectly on point.

Mathematician and

Index BMI I inquired out of necessity. Let me remind you that this is an indicator that compares and evaluates the compliance of the weight of an adult patient with a theoretically established norm. The math formula is simple: Divide your weight (in kilograms) by the square of your height (in meters). The limit for overweight is assumed to be a quotient of 25. On this scale, renowned Spanish tennis player Rafael Nadal is almost overweight (185 cm, 85 kg), giving a BMI of 24,85. Skinny as a chip, his Serb rival Novak Djokovic is 21,79 and fits easily into normal weight limits. The author of these words ... I will not say how high this figure is. However, as the lower limit of the correct weight for me (180 cm), this is ... 61 kg. A 180-kilogram guy with a weight of 61 kg would surely fall with any gust of wind. I believe that although the principle of the indicator itself is correct, this setting of parameters was probably imposed by pharmaceutical companies (diet pills).

Doctors themselves are aware that this indicator does not take into account the personal characteristics of the patient. I'll also add a math fact. Older people are losing weight. Their spine collapses. In my youth, I was 184 cm tall, now 180 cm. If I weighed 100 kg, then “then”, that is, with a height of 184 cm, this would give an indicator of 29,5 (I degree overweight), and now that with a height of 180 cm, it will be 30,9 (overweight of the second degree). And yet "I" did not shrink, only the spine twisted.

Let's check the BMI index for "constancy of indicators." The point is that it shouldn't matter if the data is given in the metric system (kilograms and meters) or, for example, in English pounds and feet. Of course, the numbers will be different, as will the numbers expressing the speed of movement in miles and kilometers. But one can easily turn one into the other without contradiction. Here is a digression. Miles can easily be converted to kilometers. But when asked how big the refrigerator is, my Canadian friend replied, "27 cubic feet." And be smart here. The situation is even worse when determining the fuel consumption of a car. In the US and Canada they rate it as "How many miles per gallon will I drive?" Reader, maybe you can judge (calculate) whether 60 mpg is too much or too little? The other US gallon is different from the Canadian (also called imperial) gallon. True, metric measures have been in effect in Canada for many years, but changing habits is not so easy.

But with BMI everything is in order. Since an English foot is 30,48 cm and a pound is 0,454 kg, the result of the English BMI (expressed in pounds of weight per square foot of height) must be multiplied by 0,454 and 0,30482, which equals 4,88. A 180 cm person weighs 220,26 pounds and 5,9 feet. Both methods of calculating BMI are the same, 30,9.

Now the most interesting (from the point of view of mathematics). In one of my books, I described the "roundness index" - how much rounded shapes look like a circle. How much - that is, mathematically "how many percent." The wheel is, of course, 100 percent round. And other numbers? How to measure it?

Let's apply this idea to measuring how much a rectangle "looks" like a square. Let's call it "destruction measure". The square should be 100% cracked, right? The mathematician prefers to say that the crack of a square is 1, and the crack of narrow rectangles is correspondingly smaller.

Let's apply something like body mass index to the rectangles. Divide the area by the square of the perimeter. How much is a square with side a? It's just 1/16 of the accounts. To get an index of 1, let's multiply by 16. So the body mass index for rectangles is

Now imagine that the rectangles go to the doctor. “I'm going to calculate your BMI,” the doctor says. Please, one by one. Here are your results. Which one to lose weight?

6. Which rectangle is for weight loss, and which is anorexic? Calculate them

Statement. BMI treats people as flat creatures! This indicator works well (without taking into account the settings of the limit levels). However, mathematicians are skeptical. It's too simple to be generic. Too simple mathematical formulas for describing biological and social phenomena should be treated with great caution.

We are back to chat for younger children. Let's take another look at puzzle number 2. We agreed, dear children, that it is true that a rectangle has only right angles. It would be strange if it were otherwise. But the figures below (the blue pyramid), the purple "twist" and the blue pinwheel also have only right angles. Maybe they are rectangular? No, people agreed that rectangles are only those that have four right angles, no more.

Learn to think right. Look:

If something is a rectangle, then it only has right angles. This is not the same as:

If something has only right angles, it's a rectangle.

Why? Instead of a rectangle, take a cat and a dog, instead of right angles, take paws. Do you understand now? Definitely!

Commentary for adults (and not only). In my youth there was a slogan: Thinking has a colossal future! I wish it was so long ago.

Understand. Important question. Is a square a rectangle? There is! It has four right angles! We can say that a square is the most even rectangle. Each side is the same length.

We will continue to make beautiful puzzles. You know exactly what an even number is. If the class is set in pairs, then either someone will be left without a pair, or ... not left. Is 12 an even number? Yes. When twelve people want to play volleyball, it is easy for them to form two teams. Twice six is ​​twelve. And if the same people want to play ping-pong, they can form six pairs. Six times two is also twelve.

What do they have in common: a match, a wedding, a duel, a mirror and a coin? Number two. In a match, two teams, a man and a woman get married (yes, a man and a woman - he gets married, she gets married). Two opponents are fighting in a duel, in the mirror we see a slightly different "" me. The medal has two sides. What are their names? Heads or tails. We have an eagle on Polish coins. Do you know anyone who has a twin brother or sister? A long time ago, “twins” were used in the villages - two connected vessels, one for soup, the other for ... a second course.

Or maybe you understand the words: double, symmetry, inversion, duality, opposite, twins, duet, tandem, alternative, negative, denial?

If a room has two exits (or entrance and exit, whichever you prefer), shall we say it has "two doors"? No, it's somehow not right. How is it right? Why do we say so? And if we add another entrance to a two-door room and put a door in there, how many doors will there be? Three? Oh no….

The "front" goes hand in hand with the "rear". Where there is “left”, there is also “right”, if something is not “above”, then it can be “below”. If there were no plus, the minus would not be needed. Number two is great.

They sing: “Two dogs…” Do you know the melody? If not, learn.

How many blocks are in the next puzzle? I don't know, we won't even count. I mean without counting, I know there is an even number. Why? Kasper, how do I know this? Oh, you already know? As you say? That everyone is equal? For the same!

Smoothly. To a couple. Doesn't it bother you that the pink on the left is darker than the one on the right?

Which is not even there. I remember that as a child I played football, there was always a problem if there were seven, nine, eleven, thirteen of us ... It was impossible to divide into two equal teams. The solution was that we played for one goal. The goalkeeper did not belong to any of the teams. He had to defend himself from every blow.

A challenge… not just for adults. Give examples of vehicles that have an odd number of wheels (we do not count the spare wheel in the car). One day I noticed that it could be... a cable car to Kasprowy Wierch - a car rolled along the cable on seven wheels. But now I don't know how it is.

How many blocks are in the fourth puzzle? Is there an even or odd number? Petrek, this is for you! How will you solve it? Do you want to count and then you will know? Well, are you wrong in this calculation? See if it doesn't matter.

In ancient times, odd numbers were considered the best. Today we prefer parity. Did you know that if we give someone flowers, then there must be an odd number of them? Of course, this does not apply to giant bouquets.

A conceivable challenge... maybe not just for adults. Who is worthy of words of gratitude, flowers and respect from all of us (and let's not be afraid of this - a solid reward!) For selfless, exhausting, long, hard and risky work so that we do not get sick, and if we get sick, recover as soon as possible?