A 9-panel grid, like the one used on the pages of Watchmen and many other comics, can be divided into panels in different ways, as one panel can span multiple panels, as we saw last week. Since there are 12 frame spacings, and each frame can be kept or removed to create larger frames, there are theoretically 2¹² = 4096 different possibilities; but in this way we also include vignettes in L (3, 4 or 5 panels), in C, in U and in other forms that are not very suitable for sequential narrative; if only rectangular bullets are allowed, the number decreases significantly (by how much exactly?).
And then, a review of the “MacGuffin problems” of the previous installment:
A number whose square ends in 1 can only end in 1 or 9; the tens digit can be any, but it is easy to verify that in the square of a number ending in 01, 11, 21… or 09, 19, 29… the tens digit is always an even number and, therefore, Therefore, there is no number in the sequence 1, 11, 111, 1111, 11111… that is a perfect square.
If three children eat three strawberries in three minutes, between the three of them they eat at a rate of one strawberry per minute, then in 100 minutes they will eat 100 strawberries. But this solution is open to discussion (see the first comments from last week).
As for the triangular terrain, it is the typical problem-joke: the largest side is equal to the sum of the other two (24 + 48 = 72), so the triangle is actually a rectilinear segment and its surface is 0 .
That of non-twins born on the same day and with the same parents is also a problem-joke: Pedro and Pablo are part of a trio of triplets.
The paradox of the still fly
With regard to the well-known classic of the fly that flies from one bicycle to another (there is another bloodier version, with a pigeon that flies between two trains that end up colliding), the imprecision of the approach is that the fly cannot fly at constant speed, because it continuously decelerates when reaching a handlebar and accelerates in the opposite direction when leaving it. A situation that, by the way, gives rise to a curious paradox: since the fly flies in the opposite direction after landing on the handlebar, there is a moment when its speed is 0; but since it is in contact with the bicycle, at that instant the speed of the vehicle will also be 0… Can a fly stop a bicycle? Where is the fallacy?
Regarding this puzzle, there is an amusing anecdote (probably apocryphal) attributed to John Von Neumann. The problem is easily solved by seeing that the fly has been flying for one hour at a speed of 15 kilometers per hour, and has therefore traveled 15 kilometers; but it can also be solved “the hard way” by adding the series of decreasing routes that the fly makes between handlebar and handlebar. It is said that on a certain occasion they presented the problem to Von Neumann, who, how could it be otherwise, gave the correct solution, but taking a few seconds longer than expected for what should be an instant response, and that when he asked why he seemed hesitant, he replied that it was not so easy to add the series mentally.
Carlo Frabetti is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including ‘Damn Physics’, ‘Damn Mathematics’ or ‘The Great Game’. He was a screenwriter for ‘The Crystal Ball’.
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