Category: InEnglish - Ivan's island

Imagination and intelligence

9/20/2021

Eagles plane on the wings of the wind and, similarly, intelligence uses the wings of imagination. And as the wind is mightier than the eagle, the imagination is mightier than the intelligence. Proof: When your intelligence is not so great, your imagination might persuade you that it is.

The contrary, i.e. the claim that when your imagination is not so great, your intelligence might persuade you that it is, is not true. That is why some very intelligent, but not very imaginative, people prefer to busy themselves with writing 300-page treatises on the philosophy of Thomas Aquinas or with learning the first 22000 digits of Pi (which someone more imaginative has already found how to calculate).

One conjecture a day keeps Alzheimer's away

9/9/2021

There exist numbers, e.g. 24, 30, 42, whose set of divisors can be partitioned into two disjoint subsets with equal sums and cardinalities. Prove or disprove the following conjecture of mine: If k is such a number, then so is 2k.

Hardcore pornography and prime numbers

8/14/2021

What a piece of work is a man! He can't define what hardcore pornography is, but he can recognize it when he sees it (in the words of the world-famous American judge Potter Stewart).

With prime numbers, the situation is quite different - we can easily define them, but we cannot recognize them when we see them. Pierre de Fermat thought that
2^32 + 1 = 4294967297
is a prime number, but much later Leonhard Euler discovered it is not.

Invisible hands

6/20/2021

We worship Adam Smith for "his" invisible hand. Oh, how well he described, with the help of an abstraction, what is happening in the economy, we say to ourselves with admiration.

It is an abstraction, yes, but not quite. On the one hand, it is an abstraction as it is a hand unattached to a body. On the other hand it has many fingers – producers and consumers, and therefore it resembles the human hand. A true abstraction would be the bodiless and fingerless hand. Even geniuses can hardly envisage such a hand. For example, Euclid could see how the invisible hand of prime numbers, with its countless fingers, is spinning the Machine of mathematics, but he was unable to imagine the number zero – the number of fingers of a fingerless hand.

Achilles Last Laugh

6/7/2021

They tell you progress is self-evident and give you justifications: thanks to physics (new ways of javelin-throwing) and chemistry (new javelin materials) a modern record-breaker can throw farther than the ancient Achilles. Even so, today's javelin throwing is "half-throwing". In the time of Achilles, in addition to distance the following things were sought:
> accuracy,
> ability to throw repeatedly,
> ability to throw while keeping your life and health, i.e. ability to throw wearing uncomfortable armor and helmet.

Oh, how I pity the modern record-holder who would face Achilles.

Sounds in the forest

4/22/2021

There is an old philosophical problem that can be compressed into just one sentence: If a tree fals in the forest and no one is around to hear it, does it make a sound? The subjectivists argue that, since sound is subjective experience, no subject means no sound. The objectivists argue that, since sound is vibrations in the air, the falling tree makes a sound. The followers of the middle path, some of them otolaryngologists and psychiatrists, think that in order to have a sound there must be objective reasons leading to a subjective experience; otherwise we have deafness (reasons without experience) or hallucination (experience without reasons).

Many people think they are intelligent and spiritual themselves, because they have a subjective experience of their potential. No, in the "human forest" subjective experience is not enough. To pass as intelligent and spiritual you have to demonstrate objective results. Otherwise people will think your intelligence and spirituality are your hallucinations.

Amicable pairs

4/14/2021

A pair (a, b) such that sigma(a) = sigma(b) = a+b (where a<b and sigma(n) is the sum of the positive integer divisors of n) is called amicable pair. The number a (OEIS A002025) is always abundant and the number b (OEIS A002046) is always deficient. Is a+b (OEIS A180164) always abundant?

1000 words for a picture?

4/8/2021

They say a picture is worth a thousand words. In fact, it was Henrik Ibsen who first said it. I wonder why he preferred to say it instead of paint it.

P.S.
August 27, 2021
Maybe the answer is The grass is always greener on the other side.

Mission impossible?

3/29/2021

Let c(x) be the x-th composite number, c(y) be the least composite number greater than c(x) that is co-prime to c(x), and d = c(y) - c(x). Is there an x, such that the respective d is neither prime, nor perfect square?

P.S.
March 30 2021
Wow! Fausto Morales found that
(75083 * 250480717) - (2^2 * 3^2 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 547) = 35

666 and 777

3/22/2021

Find a link (not numerological) between the Number of the beast (666) and the Number of God (777). Palindromes? Multiples of a repunit? Think harder!

P.S.
My answer was simple (as I am)

EulerPhi(777) = 2 * EulerPhi(666)

but Fausto Morales surprised me with the following rock-poetry (italics mine):

I got thinkin' Rock-hard...

Multiplying the beast's interpretation of God's number (3 sevens) by God’s interpretation of the beast’s, we get

37*666 = 24642

A two-sided, even-spaced, ladder to a perfect number, i.e. a Stairway to Heaven.

On the other hand, multiplying the beast's interpretation of its number (3 sixes) by God's interpretation of His number, gives

36*777 = 27972

A two-sided, unevenly spaced, higher path to some odd number, i.e. a Highway to Hell

Who says great minds think alike?

Ivan's chessboard problem 2

3/9/2021

Ivan has a chessboard of n^2 squares. A pawn is standing on the lower left corner of the chessboard, i.e. on the lower left corner of square 1,1. The pawn’s goal is to reach the upper right corner of the chessboard, i.e. the upper right corner of square n,n. The only moves allowed are diagonal shortcuts through squares that do not result in passing through a square that has already been passed through (within the same route).

The routes are considered distinct if they are represented by different sequences, e.g. the sequences
1,1/2,2/3,3/4,4/4,5/3,5/3,4/4,3/5,3/5,4/5,5/6,6 and
1,1/2,2/3,3/4,3/5,3/5,4/4,4/3,4/3,5/4,5/5,5/6,6
define two distinct routes, regardless of the fact that the pawn traces (see below) might look the same from above.

How many distinct routes f(n) are there?

Ivan's chessboard problem

2/27/2021

Ivan has a white chessboard with a side of N unit squares, whose diagonal (a1, b2, c3, ...) is painted black. How many are the composite squares (i.e. the squares with a side of C unit squares, where 1<C<N), whose vertex unit squares are all white?

P.S.
Fausto Morales found the following formula
(2N^3 - 9N^2 + 10N)/6 - [1 - (-1)^N)]/4
and I'll write something similar to the inscription on Carnegie's tombstone, namely
Here blogs a man who knows how to attract readers cleverer than himself.

Milionnaires

2/17/2021

The say the first million is the hardest. I'd say something different:
Most people will find it easier to earn a million than to find the sum of the digits of the first million numbers.

Smarts and luck

2/12/2021

"Power is nothing without control," was Pirelli's motto. "Smarts is nothing without luck", is how John Atanasoff's life can be summarized.

According to Jane Smiley, author of Atanasoff's biography, the United States is the only country where patent applications are approved on item creation date basis (instead of patent application date basis). In any other country, Atanasoff would not have been recognized by the court as the creator of the first electronic digital computer because neither he nor Iowa State University (with whose funding Atanasoff created his computer) had ever applied for a patent.

Atanasoff was doubly lucky: if Sperry had offered Honeywell a licensing agreement similar to the one signed with IBM (to use their patents for 10 million dollars), Honeywell would have agreed. But the people of Sperry were greedy and demanded 250 million. Honeywell refused and filed the famous lawsuit, which some people say was the most important patent case of the 20th century.

Religion

2/9/2021

"There is no God but Allah and Muhammad is his messenger," cried out the Muslims, waved the green flag and succeeded in capturing the lands from India to France. Even if we have doubts about their faith, we cannot fail to admire its power, i.e. their readiness to die in the name of Allah.

"There is no greater God but Evolution and Darwin is its messenger," cry out today's believers. Even if we doubt the truth of their faith, we cannot fail to admire their pragmatism: as they wave the flags of Evolution, they head en masse to the vaccination points. Faith is dear, but life is dearer.

Futurologists

2/7/2021

The futurologists are strange creatures. They tell you that 20 years ago it was impossible to predict you'd carry in your pocket a computer more powerful than those of NASA were when you were born. Next minute they tell you that by 2040 the renewable energy will be at least 80% of all energy produced, if the unforeseen developments are ruled out.

There are always unforeseen developments, why should we rule them out? If there were no unforeseen developments, then 20 years ago I and the futurologists would have known that today we'd carry in our pockets computers more powerful than those of NASA were at the time of our birth.

Watches and people

2/1/2021

Which watch is smarter: the Swiss one (dozens of springs) or the Korean one (thousands of transistors and lines of code)? Which watch owner is smarter: the one with the mechanical watch (which costs thousands of Swiss francs) or the one with the Korean watch (which costs few hundred euros)?

The Swiss watch is simple, you put it on your hand and forget about it. A Korean watch needs time and effort to set up, update, and connect to another smart and much more expensive device that also requires setup and updates. Therefore, a Korean watch wastes a lot of your time. On the one hand, the people with a lot of time seem smarter. On the other hand, if they dedicate their time to their watches, they don't seem so smart. On the one hand, the people with mechanical Swiss watches seem to earn more than enough and are therefore smarter. On the other hand, if they dedicate their money to Swiss watches and similar knick-knacks, they don't seem so smart.

I think the smartest watch is the watch that does not say what time it is, but what to do with your time. The same goes for people: the smartest are not those who know what time it is, but those who: a) know what they spend their time on, and b) are happy about it. They definitely don't need watches for that.

Of numbers and men

1/28/2021

Perfect numbers like perfect men are very rare. René Descartes
Zumkeller numbers like good men are at least one in a dozen. Yours truly

Albrecht Dürer's Melancholia

1/23/2021

They say that the magic square in Albrecht Dürer's Melancholia is interesting because the painter's goal was to show the year of the sad event (the death of his mother in 1514 AD) that provoked his melancholia. This worthy goal had several unintended consequences. The couples of symmetric quadruplets (first+ultimate and second+penultimate) form identical trapezoids, the second of which is rotated by 180 degrees. There is one more interesting thing: the sums of numbers on the numbered vertices of the two hexagons - blue (3+2+14+15) and red (10+11+7+6) - equal the magic constant of the square.

Blankets and commutativity

1/19/2021

Oddly enough, combining blankets is not commutative. Blankets A + B can make you warmer than B + A.

Check your divergent thinking!

1/10/2021

Take 7 distinct numbers out of the numbers {1,2,...,9} and use them to fill in every circle so that the products of the numbers on every horizontal and vertical line are equal.

This was the easy part. The hard thing to do is to find several different approaches to the right solution (embodied in the different first questions you may ask when you start thinking about the problem).

Man's best friend

12/30/2020

"People say that dog is man's best friend," said the dog, "but are silent about whether man is dog's best friend."

Needless

12/29/2020

Once upon a time ... I wrote some program but later found it did lots of needless calculations. I removed the respective part of the program, but instead of running faster it slowed down. When I complained to a friend of mine, who had been programming for decades, he looked down on me the way the Great Incas had been looking down on their subjects and said, "This is what usually happens."

Come to think of it, this is what usually happens not only to our programs but to ourselves. Oh, how often our efforts to remove the needless things from our lives lead us to needless situations. Wishing to remove the needless weight we damage our muscles and tendons and are getting hernias. Wishing to spend the needless money we meet people we do not need to know, do things we do not need to do and waste our time. At the end we learn the hard way that out of all imaginable things time is never needless.

Apples and oranges

12/23/2020

I have a friend who does not know I am a synesthete and is amazed by my ability to compare music to movies or Irish whiskey to Bulgarian white wine. He told me more than once that comparing apples to oranges is a forbidden operation.

Not only am I synesthete, but I am also interested in math. That is why I know one can compare objects of different type. One can even add and multiply them. It’s elementary, my dear Watson, 5 apples plus 3 oranges equals 8 pieces of fruit. 5 jackets times 5 trousers equals 25 different suits.

Thinking is like breathing

12/21/2020

The fact that we think does not necessarily mean we don't have to learn how to think. Same with breathing, once we start practicing sports, yoga, or singing, we learn our "old" breathing doesn't work and we need to learn to breath correctly.