The popular squares game seen last week gives rise to a combination -and therefore a strategy- more complex than its apparent simplicity suggests. In a 3×3 grid, the first player can mark one of the sides of the central square or one of the sides of the peripheral squares that do not coincide with those of the central one; the first 4 are interchangeable with each other and so are the other 8, so the first player really only has two options. If the first player marks one of the peripheral sides, as in the figure, the second player can close a small square (the one in the upper left corner), but he may want to follow another strategy. I leave to my astute readers/is the analysis of the situation, less simple than it seems as soon as we move to a 4×4 or 5×5 grid.
Let’s now imagine that the 3×3 grid is the outline of a comic page divided, as usual, into 9 panels. The simplest layout is one vignette per panel, that is, 9 equal vignettes on each page; but a panel can take up 2, 3, 4, 6, or all 9 panels (not 5 or 7 or 8, if the panels are to be rectangular), so a page can be laid out in many different ways. For example, on the comic page Watchmen attached two double vignettes respectively occupy two of the upper panels and two of the lower ones, in one of the different possible combinations. How many exactly?
The deceitful MacGuffin
And speaking of problems that are not what they seem…
Moviegoers know well what a MacGuffin is: a red herring or at least misleading, something that seems important to the development of the plot and is not. The denomination is due to Alfred Hitchcock, and he himself used that resource profusely in his suspense tapes. And there are problems that throw us off in a similar way: they divert our attention to irrelevant details or lead us to think that they are more complicated -or simpler- than they really are.
A couple of weeks ago, Manuel Amorós mentioned a problem that may seem trivial to those who are familiar with numbers and their squares or very difficult to those who are not, but which can be tackled with no tools other than the multiplication table and a little common sense:
In the sequence 1, 11, 111, 1111, 11111, 111111… Is there any term that is a perfect square?
And while we’re at it, let’s look at a few more easy, laid-back but potentially misleading mathematical-logical MacGuffins, like the very vacation activities of these days:
Three children eat three strawberries in three minutes. How many children would it take to eat 100 strawberries in 100 minutes?
The sides of a triangular piece of land measure 24, 48 and 72 meters respectively. What is its surface?
“We were born on the same day of the same year and we have the same mother and the same father, but we are not twins or twins,” say Pedro and Pablo. How is it possible?
And a well-known classic could not be missing here, but it must be mentioned in this context:
Two boys, 20 kilometers apart, are cycling along the same straight road toward each other, and a fly on the handlebars of one of the bikes starts flying directly at the other cyclist. As soon as she reaches the other handlebar, she turns around and flies back to the first one. The fly flies like this, handlebar to handlebar, until the two boys meet. Both pedaled at a constant speed of 10 km per hour, and the fly flew at a constant speed of 15 km per hour, how far did the fly travel in total on its round trips?
And the rigorous meta-question: Is there any inconsistency or inaccuracy in the previous approach?
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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