Archimedes' box

In the simplest variant of nim, mentioned last week, which consists of removing 1 to 3 units from a row of 20 ―let's say they are matches―, to win we have to leave 5 to our opponent in our penultimate move, since then, take the ones you take, in the next move we will leave you the last match.

And to leave him 5, first we have to leave him 9, and before 13... That is, whoever leaves a multiple of 4 plus 1 matches on the table wins.

And since at the beginning there are 20, if the first player removes 3 and leaves 17 (16 + 1) he is assured of victory.

As for the chessboard with 8 pieces per side that advance through the columns, it is equivalent to a variant of the nim with 8 rows of 6 matches in each one and being able to remove all the matches you want from the same row in each turn.

In this case, the second to play would win -let's say it's black-, whose strategy would consist of making the same move as white, but on the column symmetrical with respect to the middle of the board.

And regarding the "perfect chess game", in which both players play in the best possible way, there is a very broad consensus that it would end in a draw, a few -very few- experts believe that white would win and nobody thinks that it could have a winning strategy for Black;

but there are no rigorous proofs, nor are there likely to be soon, given the enormous complexity of the game, whose possible positions are in the order of septillions.

the

ostomy

And from the familiar square board divided into 64 equal squares to another lesser-known -but also fascinating- one divided into 14 different and apparently capricious parts: the

, an ancient puzzle of the same type as the tangram (although it seems that both games developed differently).

independently), also known as

(Archimedes' box), as the great Greek mathematician studied it in the 3rd century BC.

C., as stated in a fragmentary Byzantine copy known as the Archimedean Palimpsest.

As in tangram, a first challenge is to mix up the pieces and put them back in their square box, which can be done in different ways (17,152, to be exact), and the pieces can also be rearranged to form a rectangle ( of what dimensions, if we take as a unit the twelfth part of the side of the square?), as well as numerous more or less recognizable silhouettes, such as those shown in the figure.

The name of the puzzle (literally “battle of bones”) suggests that it was originally a game with movable and reversible pieces for several players, who competed to see who could compose certain silhouettes with greater fidelity.

To form squares and rectangles, a sheet of graph paper can be used;

but to compose less simple silhouettes it is convenient to cut the pieces out of a sheet of cardboard or buy a plastic or wooden version in a specialized store.

Long and instructive hours of fun -or despair- guaranteed.

Although, as a first step, I invite my astute readers to calculate the areas of each and every one of the pieces (with respect to the largest unit that gives whole areas) and to draw the pertinent conclusions.

And as a last question, one wonders if it is a coincidence that the

has exactly twice as many pieces as the tangram or…

Carlo Frabetti

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