Bookish problems

The key to Sam Loyd’s chess problem raised last week is that, since Black’s pawns have not moved, on the previous move Black had to move either the king or the rook, which is why he cannot castle; in this way, White mates in two by moving the queen to a1, to bring it to h8 on the next move: checkmate.

Regarding the unusual position of Geza Schweig, it is interesting to note that there is one by Ernest Clement Mortimer, from 1991, suspiciously similar, as Rafael Granero has commented (see comment 1 from last week, with a link to an article on Mortimer’s position) . As for Schweig’s position, it seems impossible, until we realize (or not) that the surviving black knight, which appears to be situated in its starting square, might not be the queen’s knight, but the king’s knight after take a walk and capture the missing white horse. (You can see a detailed analysis of the game on the excellent page “divulgators.com“: https://divulgadores.com/ajedrez-el-analisis-retrospectivo/).

The number 2021 is not prime, but its prime factorization is not obvious: 2021 = 43 x 47; and its relationship with 90 has to do precisely with this decomposition, since 43 + 47 = 90.

On the other hand, 2021 is two thousand twenty-one in base 10, but in base 3 it is sixty-one, in base 5 it is two hundred sixty-one, in base 7 it is seven hundred one … Therefore, the next number in the sequence 61, 261, 701… is 2021 in base 9: 1477.

The divisors of 2021 are 1, 43, 47 and 2021, whose sum is 2112, next year capicúa.

Problematic books

And speaking of factoring, our regular commentator Salva Fuster has proposed an interesting problem related to the topic (and by saying this I give a hint):

Several consecutive pages have been torn from a book, the numbers totaling 9131. What pages have been torn?

The solution is in the comments section of last week, for those who are not in a position to think after the Christmas excesses.

And let’s continue with the bookish riddles, among which a classic cannot be missing:

The five volumes of an encyclopedia, of 300 pages each, are neatly arranged on a shelf. A meticulous moth begins to eat paper on page 1 of the first volume and ends, piercing sheet after sheet, on page 300 of the last volume. How many sheets have you punched in all? (The covers of the books are not taken into account).

And a few more of various kinds:

My memory works in a strange way: I do not remember on which page I interrupted reading a mystery novel, but I do remember that the figures on that page add up to five times those on the next page. What page did I stay on?

One typographer claims to have used exactly 3,000 characters to fold (number, in the jargon of the trade) the pages of a book, and another replies: “I don’t believe it.” Is your unbelief justified?

A child entertains himself by stacking the volumes of an encyclopedia, all the same, so that each one stands out as much as possible from the one underneath it without actually falling. If the child had an unlimited supply of books, how far could the book at the top of the stack move away from the vertical from the base?

And, speaking of unlimited supply, is it the number of writable books?

Carlo Frabetti is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including ‘Damn physics’, ‘Damn mathematics’ or ‘The great game’. He was a screenwriter for ‘La bola de cristal’.

You can follow Matter on Facebook, Twitter, Instagram or subscribe here to our newsletter