A cross-figure is a puzzle similar to a crossword in structure, but with entries that 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 or arithmetic or other mathematical facts
The clue may require arithmetic to be applied to another answer or answers
The clue may indicate possible answers but make it impossible to give the correct one without using crosslights
One answer may be related to another in a non-determinate way
Some entries may either not be clued at all, or refer to another clue
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
Another special type of puzzle uses a real-world situation such as a family outing and base most clues on this
Cross-figures that 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 one gradually narrows down the possibilities for individual cells 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 the last digit of 12 across must be even, the first digit of 7 down must be 1, 2, 3 or 4, and the first digit of 12 across must be between 2 and 9 inclusive. 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 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 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. 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 where same result can be achieved through algebraic means.
