# HEX | Science | THE COUNTRY

We were wondering last week in how many different ways you can express the number 210 as a product of natural numbers (whole and positive). The first thing is to break it down into prime factors: 210 = 2 x 3 x 5 x 7; the different factorizations will be the different ways of grouping these cousins, with which the problem is equivalent to that of traveling friends (if we exclude the option of going all together in the same car, which would correspond to the trivial factorization 1 x 210):

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Advertise Here210 = 2 x 3 x 5 x 7 = 2 x 3 x 35 = 2 x 5 x 21 = 2 x 7 x 15 = 3 x 5 x 14 = 3 x 7 x 10 = 5 x 7 x 6 = 2 x 105 = 3 x 70 = 5 x 42 = 7 x 30 = 6 x 35 = 10 x 21 = 14 x 15

Regarding the number 2310, in its prime factor decomposition there is none repeated (that is, with exponent): 2310 = 2 x 3 x 5 x 7 x 11, so, as in the previous case, the number of possible decompositions in whole and positive factors will coincide with the Bell B number_{5} = 52 (with the exception of excluding 1 as a factor).

### The inventions of Hein

Our regular commentator Carlos Gaceo has sent a link to an interesting game called MatHex (see comment 72 of last week), which is a digitized update of the Hex, a board game invented in the 1940s by the Danish engineer and writer Piet Hein.

The Hex is played on a board of 11 x 11 hexagonal squares, where two players alternately place their chips, white one and black the other, trying to join the two opposite sides of the board that correspond to each player with a chain continuous of your chips. It's a static chip set, like the Go: once placed on the board, they remain in the same square until the end of the game. The four squares of the corners belong to both concurrent sides and, therefore, to both players.

Although the most common board is the 11 x 11, boards of other sizes can be used. In the figure we see the end of a game won by white on a 5 x 5 board. But the position is unlikely. Why?

The reduced board facilitates game analysis. Is there a winning strategy? Can you prove that one of the players has an advantage? Can the game finish in a draw?

Piet Hein also invented the cube soma, a three-dimensional puzzle that we have dealt with on occasion, and although he did not discover the superelipse, he studied its properties and applied them to design and sculpture. But that is another article.

**Carlo Frabetti** He is a writer and mathematician, a member of the New York Academy of Sciences. He has published more than 50 scientific dissemination works for adults, children and young people, among them *Damn physics*, *Damn mathematics* or *The big game*. He was a screenwriter *The Cristal ball.*