Le Monde puzzle [#845]

Yet another one of those Le Monde mathematical puzzles which wording is confusing to me:

Take the set of integers between 1 and 1000. endow all of them randomly with red or blue tags. group them by subsets of three or more (grapes). and also group them by pairs so that a switch can change the colour of both integers.  Is it always possible to activate the switches so that one ends up with all grapes being multicoloured?  Unicoloured? 

I find it (again!) ultimately puzzling since there are configurations where it cannot work. In the first case, take a grape made of four integers of the same colour, reunited two by two by a switch: activating the switch simply invert the colours but the grape remains uni-coloured. Conversely, take two integers with opposite colours within the same grape. No mater how long one operates the switch, they will remain of an opposite colour, won’t they?!

This issue of Le Monde Science&Médecine leaflet actually had several interesting entries, from one on “the thirst of the sociologist for statistical irregularities“—meaning that regression should account for confounding factors like social class versus school performances—to the above picture about weighting the mass of a neutrino—mostly because it strongly reminds of Escher, as I cannot understand the 3D structure of the picture—, to  another tribune of Marco Zito informing me that “quark” is a word invented by James Joyce—and not by Carroll as I believed—, to an interview of Stanislas Dehaene, a neuroscientist professor at Collège de France and a (fairly young) member of the Académie des Sciences—where he mentions statistical learning patterns that reminded me of the Bayesian constructs Pierre Bessière discussed on France Culture—.

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