Cross-figure
Encyclopedia
A cross-figure is a puzzle
similar to a crossword
in structure, but with entries which consist of numbers rather than words, with individual digits being entered in the blank cells. The numbers can be clued in various ways:
Cross-figures which use mostly the first type of clue may be used for educational purposes, but most enthusiasts would agree that this clue type should be used rarely, if at all. Without this type a cross-figure may superficially seem to be impossible to solve, since no answer can apparently be filled in until another has first been found, which without the first type of clue appears impossible. However, if a different approach is adopted where, instead of trying to find complete answers (as would be done for a crossword) one gradually narrows down the possibilities for individual cells (or, in some cases, whole answers) then the problem becomes tractable. For example, if 12 across and 7 down both have three digits and the clue for 12 across is "7 down times 2", one can work out that (i) the last digit of 12 across must be even, (ii) the first digit of 7 down must be 1, 2, 3 or 4, and (iii) the first digit of 12 across must be between 2 and 9 inclusive. (It is an implicit rule of cross-figures that numbers cannot start with 0; however, some puzzles explicitly allow this) By continuing to apply this sort of argument, a solution can eventually be found. Another implicit rule of cross-figures is that no two answers should be the same (in cross-figures allowing numbers to start with 0, 0123 and 123 may be considered different.)
A curious feature of cross-figures is that it makes perfect sense for the setter of a puzzle to try to solve it him or herself. Indeed, the setter should ideally do this (without direct reference to the answer) as it is essentially the only way to find out if the puzzle has a single unique solution. Alternatively, there are computer programs available that can be used for this purpose; however, they may not make it clear how difficult the puzzle is.
Given that some basic mathematical knowledge is needed to solve cross-figures, they are much less popular than crosswords. As a result, very few books of them have ever been published. Dell Magazines
publishes a magazine called Math Puzzles and Logic Problems six times a year which generally contains as many as a dozen of these puzzles, which they name "Figure Logics". A magazine called Figure it Out, which was dedicated to number puzzles, included some, but it was very short-lived. This also explains why cross-figures have fewer established conventions than crosswords (especially cryptic crosswords). One exception is the use of the semicolon to attach two strings of numbers together, for example 1234;5678 becomes 12345678. Some cross-figures voluntarily ignore this option and other "non-mathematical" approaches (e.g. palindromic number
s and repunit
s) where same result can be achieved through algebraic means.
Puzzle
A puzzle is a problem or enigma that tests the ingenuity of the solver. In a basic puzzle, one is intended to put together pieces in a logical way in order to come up with the desired solution...
similar to a crossword
Crossword
A crossword is a word puzzle that normally takes the form of a square or rectangular grid of white and shaded squares. The goal is to fill the white squares with letters, forming words or phrases, by solving clues which lead to the answers. In languages that are written left-to-right, the answer...
in structure, but with entries which consist of numbers rather than words, with individual digits being entered in the blank cells. The numbers can be clued in various ways:
- The clue can make it possible to find the number required directly, by using general knowledge (e.g. "Date of the Battle of Hastings") or arithmetic (e.g. "27 times 79") or other mathematical facts (e.g. "Seventh prime number")
- The clue may require arithmetic to be applied to another answer or answers (e.g. "25 across times 3" or "9 down minus 3 across")
- The clue may indicate possible answers but make it impossible to give the correct one without using crosslights (e.g. "A prime number")
- One answer may be related to another in a non-determinate way (e.g. "A multiple of 24 down" or "5 across with its digits rearranged")
- Some entries may either not be clued at all, or refer to another clue (e.g. 7 down may be clued as "See 13 down" if 13 down reads "7 down plus 5")
- Entries may be grouped together for clueing purposes, e.g. "1 across, 12 across and 17 across together contain all the digits except 0"
- Some cross-figures use an algebraic type of clue, with various letters taking unknown values (e.g. "A - 2B, where neither A nor B is known in advance)
- Another special type of puzzle uses a real-world situation such as a family outing and base most clues on this (e.g. "Time taken to travel from Ayville to Beetown")
Cross-figures which use mostly the first type of clue may be used for educational purposes, but most enthusiasts would agree that this clue type should be used rarely, if at all. Without this type a cross-figure may superficially seem to be impossible to solve, since no answer can apparently be filled in until another has first been found, which without the first type of clue appears impossible. However, if a different approach is adopted where, instead of trying to find complete answers (as would be done for a crossword) one gradually narrows down the possibilities for individual cells (or, in some cases, whole answers) then the problem becomes tractable. For example, if 12 across and 7 down both have three digits and the clue for 12 across is "7 down times 2", one can work out that (i) the last digit of 12 across must be even, (ii) the first digit of 7 down must be 1, 2, 3 or 4, and (iii) the first digit of 12 across must be between 2 and 9 inclusive. (It is an implicit rule of cross-figures that numbers cannot start with 0; however, some puzzles explicitly allow this) By continuing to apply this sort of argument, a solution can eventually be found. Another implicit rule of cross-figures is that no two answers should be the same (in cross-figures allowing numbers to start with 0, 0123 and 123 may be considered different.)
A curious feature of cross-figures is that it makes perfect sense for the setter of a puzzle to try to solve it him or herself. Indeed, the setter should ideally do this (without direct reference to the answer) as it is essentially the only way to find out if the puzzle has a single unique solution. Alternatively, there are computer programs available that can be used for this purpose; however, they may not make it clear how difficult the puzzle is.
Given that some basic mathematical knowledge is needed to solve cross-figures, they are much less popular than crosswords. As a result, very few books of them have ever been published. Dell Magazines
Dell Magazines
Dell Magazines was a company founded by George T. Delacorte Jr. in 1921 as part of his Dell Publishing Co. Dell is today known for its many puzzle magazines, as well as fiction magazines such as Alfred Hitchcock's Mystery Magazine, Ellery Queen's Mystery Magazine, Asimov's Science Fiction, and...
publishes a magazine called Math Puzzles and Logic Problems six times a year which generally contains as many as a dozen of these puzzles, which they name "Figure Logics". A magazine called Figure it Out, which was dedicated to number puzzles, included some, but it was very short-lived. This also explains why cross-figures have fewer established conventions than crosswords (especially cryptic crosswords). One exception is the use of the semicolon to attach two strings of numbers together, for example 1234;5678 becomes 12345678. Some cross-figures voluntarily ignore this option and other "non-mathematical" approaches (e.g. palindromic number
Palindromic number
A palindromic number or numeral palindrome is a 'symmetrical' number like 16461, that remains the same when its digits are reversed. The term palindromic is derived from palindrome, which refers to a word like rotor that remains unchanged under reversal of its letters...
s and repunit
Repunit
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler...
s) where same result can be achieved through algebraic means.