In our infinite forest from last week, the probability that a randomly fired bullet will embed itself head-on into a log, counter-intuitively (which suggests that with infinitely many trees, it will certainly occur), is 0. Well, “head-on ” means that the line of the trajectory of the bullet (and therefore its tip) must pass through the axis of the trunk.
This is equivalent to saying that if we randomly draw a straight line from a vertex in an infinite grid, the probability that it passes through another vertex is 0, since if it did, the segment joining the first vertex with the second would be the hypotenuse of a triangle. Rectangle whose legs would each measure integer numbers of units (taking as unit the side of the square of the grid). Therefore, the tangent of the angle formed by the impact trajectory with the horizontal would be a rational number, and if we draw a line at random the tangent of its angle can have any real value, and the infinity of real numbers is of order superior to that of rationals.
If we settle for the bullet reaching a trunk at any point of it, things change. How? (Suppose that in our forest from last week the logs are cylinders with a radius of 10 cm.)
very human chess problems
Our “featured user” Manuel Amorós sent an interesting chess meta problem.
In the figure position, they play white and mate in 3 moves, according to a chess engine; but a human can checkmate in 2. What is checkmate in 2 and why doesn’t the machine see it?
Another famous problem on which, to the delight of technophobes and human supremacists, machines seem less clever than we are, is the one posed in the seventeenth century by the Italian chess player Gioachino Greconicknamed the Calabrian, considered the first professional player in history and author of many interesting games and positions.
He is credited, among other things, with the first version of the “mate de la coz”, so named because the knight chokes the opposing king, immobilized by his own pieces:
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.OO Nf6 5.Re1 OO 6.c3 Qe7 7.d4 exd4 8.e5 Ng4 9.cxd4 Nxd4 10.Nxd4 Qh4 11.Nf3 Qxf2+ 12.Kh1 Qg1+ 13. Nxg1 Nf2#
In the position we see on the adjoining board, the most famous of Greco’s problems, Black plays and seems to have little chance of preventing White from promoting one of his two pawns, as understood by Stockfish, one of the most powerful chess analysis engines. However, Black has the chance to draw.
Can you outplay the machine and find Black’s equalizing strategy?
And here’s another problem that machines seem to have a hard time solving as well: In the board position, they play White and win.
In view of the impenetrable chain formed by Black’s seven pawns and bishop attached to their White counterparts, it doesn’t seem like White, with the rook and two knights blocked, can do much. And yet…
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