Mathematics: Ethical Problems

The simplest and most elegant solution to the first problem from last week is that there are 7 candles and they last for 7 days: since it takes 4 hours for a candle to burn completely, we have 28 “candle hours”, and we spend 1 on the first day, 2 on the second, 3 on the third.. And the seventh 7, i.e. 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. But…, is this the only possible solution? (See the latest comments from last week).

As for the old man saddened by the transience of time, he celebrates his birthday on the 31st of December: on the 30th he had the 77-candles (not “in”, small mistake in the wording) his previous birthday cake, on the 31st he is 78 and on On January 1, it is believed that the next year he will be 80 years old.

“In the third problem – as our distinguished user Francisco Montesinos says – it is convenient to take as a unit of measure x cm which remains on the thinnest sail once the required time has elapsed. When calling d1 and d2 for the dips experienced by each candle, it is immediately that d1 = 15x and d2 = 12x. Therefore, t = (d1/v1) = (d2/v2) = (15x/4x) hours = (15/4) hours = 3h 45m”.

After 3 hours, the long candle in the fourth problem, since it takes 7 hours to burn completely, will have consumed 3/7 of its height and thus will measure 16 cm, while the short candle will consume 3/11 of its height. its initial height, h; Thus (since there is an 8/11 sail on the left), 8h/11 = 16, whence p = 22 cm.

Strange relatives and descriptive problems

The puzzle became a classic (used in its day to pick out creative people) and consisted of connecting 9 dots arranged in a 3 by 3 grid with only four straight strokes, without lifting the pencil off the paper or going over it twice. The same hit. But the known solution in the figure is not unique. Can you find another solution? And the strict meta-problem: What is moral?

Or in other words, why do so many people fail to solve the problem and even come to the conclusion that it is impossible to connect the dots in less than 5 strokes? What mental block makes it difficult to find a simple solution once you know it?

A second descriptive problem relates to another classic problem, which we dealt with two years ago: dividing an obtuse triangle into acute triangles. If you don’t already know that, spend a few minutes experimenting with division before moving on to the definition problem, which is:

The union of 9 points with 4 strikes and dividing the obtuse triangle into an acute triangle do not seem to have anything to do with each other, yet they are two closely related problems. What is your relationship?

As a climax, there are two other moral classics and they are also related to each other depending on the kind of difficulty involved in their solution: A peasant who loves geometry wants to plant four trees on his land so that each of them gives the same amount as the other three. Can you do it or is it impossible?

With 6 matches, it’s easy to make a regular hexagon. And by moving just two matches and adding one more, you can make two equal rhombuses of hexagons, without a match at the top and using them along their entire length. Even more difficult, with the 6 matches, you can make 4 equilateral triangles. how?

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