The theorem of marriage

could a hypothetical mercury enjoy a double sunrise? Yes: as you approach perihelion, Mercury's orbital angular velocity (VAO) would increase to equal its angular rotation alvelocity (VAR), at which point the Sun would stop in the sky; from there, the VAO would briefly overtake the VAR and the Sun would go back a long way, as if taking a wagon before re-following its normal course, and a conveniently placed mercury would see it sink back on the horizon after dawn to return again r to go out soon after,

The fact that Mercury's rotation period (58.7 Earth days) is 2/3 of its translation period (88 Earth days) is not a coincidence. The ratio would normal ly be 1/1 as in the case of the Moon with respect to Earth (i.e. the period of rotation equal to the translation period), and so it was believed to be until 1965; but the considerable eccentricity of Mercury's orbit, with consequent fluctuations in its translation rate, makes an orbital resonance of 2/3 more stable than the typical gravitational coupling

. "weddings and candy"

And speaking of coupling, our feature-leading user Francisco Montesinos raised last week a nice problem which, in turn, refers to an interesting theorem about pairings. Here's a simplified version of the problem:

we have two orange candies, two lemon and two strawberry candies, and we put each pair of candies of the same flavor in a different box. Obviously, if we take a candy out of each box, we'll have one of every flavor. But then we empty the three boxes on the table, stir the six candies as if they were dominoes and put them back in the boxes at random, two on each. Whatever the distribution, can we get all three flavors by taking a candy out of each box? What if instead of three flavors and two candies of each we have four flavors - and four boxes - and three candies of each flavor?

It seems like a simple problem, but it's not so much if we generalize it to m flavors - and boxes - and n candies of each flavor, and one way to address it is through Hall's theorem, also known as a marriage theorem because it is usually illustrated by two groups , one of men and one of women, with the same number of individuals in each group. For each woman, there are one or more men willing to marry her, and any man would marry any woman who would like to marry him (a situation similar to that of the famous musical Seven brides for seven brothers). Is it possible to successfully match all these people?

I suggest to my shrewd readers that they start with the simplest problem of candies (three flavors and two of each) and advance to the never well-weighted conjugal complexities in their possible variants