On last week’s first problem, Raúl Toral comments: “This is an example of an ‘oboe’ (displaced by an error). When counting the duration of the 6 chimes, we count the time from the first to the last. In this time 5 intervals have passed (from the first to the second chime, from the second to the third, from the third to the fourth, from the fourth to the fifth and from the fifth to the sixth). Therefore, each interval lasted 6/5 = 1.2 seconds. In the 12 bells there are 11 intervals lasting 11×1.2=13.2 seconds.
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As for Kant’s clock problem, it is quite simplified without changing the basic argument, considering that the philosopher returns home immediately after seeing the time on his friend’s clock. Suppose the stopped clock reads 2. Kant winds them up, goes to his friend’s house whose clock reads 5, returns home and sees that his clock reads 3 now, meaning it took him an hour to go there and back, that is, one half an hour has passed since he saw that it was 5 o’clock on his friend’s watch; hence it is 5.30.
arithmetic puzzle
The Nine Digits and the Changing Zero is a verse by Borges who says in the first quatrain of his sonnet El enamorado:
moons, ivories, instruments, roses,
lamps and Dürer’s line,
the nine digits and the alternating zero,
I have to pretend things like that exist.
And it is also the title of a delicious book by the Colombian mathematician Bernardo Recamán, in which, as the title suggests, there are many arithmetic puzzles, some easy and some not so easy, many of them invented by the author himself and all interesting and ingenious. Let’s look at three of them:
1. I asked my neighbor how many grandchildren he had, and he replied, “Each is a different age and the sum of their ages is 73, which is my own age.” And has no other set of different whole numbers whose sum is 73 a larger product than the age of my grandchildren.
How many grandchildren does the neighbor have and how old are they?
2. Some of the 5,000 members of the World Arithmetic Society (each with a different membership number between 1 and 5,000) got together to discuss a problem and as they queued for a drink, they found that their membership numbers were a series of consecutive Numbers made numbers, and that none of the members stood next to anyone whose number was coprime to their own. (Remember that coprime or coprime numbers are numbers that have no more common divisors than 1.)
How many members met and what were their membership numbers?
3. The square of a number and its cube usually have at least one digit in common, though not always; For example, the square of 14, 196, and its cube, 2,744, have neither in common.
What is the largest number you can find whose square and cube have no digits in common?
Recaman’s sequence
Mathematics according to Bernardo Racamán, in an image from YouTube.
In addition to his recreational math books, Bernardo Recamán is best known in the field of computer science for the sequence named after him. There are actually two sequences attributed to Recamán, but this is the best known:
0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62…
Which numbers follow?
The main image of this article shows a visualization of the Recamán sequence.
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