### The Puzzles Page of a newspaper

Once upon a time, daily newspapers were very serious publications. Then, in the early decades of the 20th century, they gradually started to introduce crosswords, and these became very popular. Crosswords have evolved in several different ways since then, with cryptic and simple, enigmatic and convoluted. There are cross-number problems and general knowledge crosswords.

The last few years have seen many British newspapers offering a page (or more) of puzzles. Sudoku logic is common, as are assorted crosswords. Also extremely popular are word problems. Enthusiasts will tell you that it is proven that regularly tackling such puzzles is good for one's mental health.

The "Independent" newspaper which Tina and I read, normally has the following:

a quick crossword, where the first two/three/four across solutions run together to make a pun

a cryptic crossword

three sudoku puzzles

a codeword, which is a crossword without clues, but each letter of the alphabet is represented in the grid by a number from 1 to 26, with two or three given; the object is to fill in the grid by logically deducing the code for each letter

two word ladders - convert MOON into STAR using intermediate words which differ by one letter from the previous one

a word wheel with nine letters, one highlighted, with the object of finding as many words which use the highlighted letter and some of the others

two maths puzzles, described below

In the paper today, there is a letter about the word ladders, asking if a slang word is allowed. The leeter goes on to criticise the text that accompanies the word wheel. "We have found XX words, including one nine-letter word" The writer points out that the setters clearly start with a nine-letter word, check that it has no anagrams, and make the word wheel.

The maths puzzles are interesting. The digits 1 to 9 are arranged in three rows of three columns. In each row and column, the digits are separated by one of the basic arithmetic operations, +, -, *, /. Ignoring the priority rules of mathematics, the result at the right hand side of the rows and at the base of the columns is given. The aim is to find the arrangement of the digits. In the elementary puzzle, two digits are given. In the advanced puzzle, only one is given. For example:

To a mathematician/O.R. person, the puzzle is to solve six simultaneous equations for the unknowns, with the side condition that the unknowns are all different single digit integers. However, it is not necessary to use combinatorial optimisation to solve them.

My interest in the puzzle page is to consider how the puzzles on a particular page might be created. There is an extensive literature about crosswords and sudokus, and computer programmes to help create a crossword with selected words included, and sudokus which range from easy to fiendish. Word ladders have been studied by graph theorists, representing each word as a node and connecting a pair of nodes if they are "adjacent". Donald Knuth's Stanford GrephBase does this, and Donald Knuth is also a fan of puzzles, as he writes:

Word wheels -- well, anyone with a word list and elementary UNIX commands can find the words in them.

Codewords are more intriguing. They are crossword grids with extra constraints. First, every letter must occur at least once). Second, there must be at least one word which can be deduced from its pattern, possibly with the help of one or other of the given letters. Thus, yesterday we had the pattern 1 2 3 3 4 3 4 with no clues. (we also had Z 5 6 7 A 1 with two clues) Third, rule 2 must continue to apply as the solver progresses. Fourth, rules 2 and 3 may possibly lead to a choice of letters, which further study will reveal. Finally, the words must be everyday ones. A friend observed that the vocabulary of the setters includes a large range of words using the infrequently used letters of the alphabet, J, K, Q, Z

But those maths puzzles. I have a theory about the ones that appear in the daily paper. It is that they do not need any given digits to be solvable. I have looked at many of them, ignoring the given one or two digits, and by logic have been able to solve each one. So I reckon that they are created without human intervention. All that happens is that a random grid is created using the digits, with added arithmetic operation and the six results are recorded. Then the digits are removed, and a computer program runs through the 9! (nine factorial = 362880) arrangements to see if there is a unique solution. Then one or two digits are added at random. As a result, the puzzles are generally very simple. I am sure that there is scope for developing a set of construction rules which would be more challenging for the puzzler. Any takers? Did you spot cheeses and zodiac?

The last few years have seen many British newspapers offering a page (or more) of puzzles. Sudoku logic is common, as are assorted crosswords. Also extremely popular are word problems. Enthusiasts will tell you that it is proven that regularly tackling such puzzles is good for one's mental health.

The "Independent" newspaper which Tina and I read, normally has the following:

a quick crossword, where the first two/three/four across solutions run together to make a pun

a cryptic crossword

three sudoku puzzles

a codeword, which is a crossword without clues, but each letter of the alphabet is represented in the grid by a number from 1 to 26, with two or three given; the object is to fill in the grid by logically deducing the code for each letter

two word ladders - convert MOON into STAR using intermediate words which differ by one letter from the previous one

a word wheel with nine letters, one highlighted, with the object of finding as many words which use the highlighted letter and some of the others

two maths puzzles, described below

In the paper today, there is a letter about the word ladders, asking if a slang word is allowed. The leeter goes on to criticise the text that accompanies the word wheel. "We have found XX words, including one nine-letter word" The writer points out that the setters clearly start with a nine-letter word, check that it has no anagrams, and make the word wheel.

The maths puzzles are interesting. The digits 1 to 9 are arranged in three rows of three columns. In each row and column, the digits are separated by one of the basic arithmetic operations, +, -, *, /. Ignoring the priority rules of mathematics, the result at the right hand side of the rows and at the base of the columns is given. The aim is to find the arrangement of the digits. In the elementary puzzle, two digits are given. In the advanced puzzle, only one is given. For example:

A- B* C=-21

+ - -

8+ D/ E=3

+ - +

9- F* G=12

= = =

19 -2 8

My interest in the puzzle page is to consider how the puzzles on a particular page might be created. There is an extensive literature about crosswords and sudokus, and computer programmes to help create a crossword with selected words included, and sudokus which range from easy to fiendish. Word ladders have been studied by graph theorists, representing each word as a node and connecting a pair of nodes if they are "adjacent". Donald Knuth's Stanford GrephBase does this, and Donald Knuth is also a fan of puzzles, as he writes:

*I believe that the creation of a great puzzle or a great pattern is a scholarly achievement of great merit, an important contribution to world culture, even though the author of such a breakthrough is often an amateur who has no academic credentials. Therefore I'm proud to follow in the footsteps of the pioneers who have come up with significant new “mind-benders” as civilization developed.*Word wheels -- well, anyone with a word list and elementary UNIX commands can find the words in them.

Codewords are more intriguing. They are crossword grids with extra constraints. First, every letter must occur at least once). Second, there must be at least one word which can be deduced from its pattern, possibly with the help of one or other of the given letters. Thus, yesterday we had the pattern 1 2 3 3 4 3 4 with no clues. (we also had Z 5 6 7 A 1 with two clues) Third, rule 2 must continue to apply as the solver progresses. Fourth, rules 2 and 3 may possibly lead to a choice of letters, which further study will reveal. Finally, the words must be everyday ones. A friend observed that the vocabulary of the setters includes a large range of words using the infrequently used letters of the alphabet, J, K, Q, Z

But those maths puzzles. I have a theory about the ones that appear in the daily paper. It is that they do not need any given digits to be solvable. I have looked at many of them, ignoring the given one or two digits, and by logic have been able to solve each one. So I reckon that they are created without human intervention. All that happens is that a random grid is created using the digits, with added arithmetic operation and the six results are recorded. Then the digits are removed, and a computer program runs through the 9! (nine factorial = 362880) arrangements to see if there is a unique solution. Then one or two digits are added at random. As a result, the puzzles are generally very simple. I am sure that there is scope for developing a set of construction rules which would be more challenging for the puzzler. Any takers? Did you spot cheeses and zodiac?

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