Help for Contradictions

When one unleashes the full power of a 'forcing net', then contradictions are found to occur very commonly. The following contradictions are discovered:

  • Two same numbers in the one unit are both inferred to be true;
  • Two different numbers in one cell are both inferred to be true;
  • All numbers in one cell are inferred to be false;
  • All instances of one number in a row, column or box are false

Basically, if a network of strong and weak links is created based on the false assuption that a number is true, then contradictions may or may not occur. If the network of chains doesn't get off the ground because of few strong links being present, then no contradiction may be revealed. Alternatively, when the network is generated based on the correct assumption that a number is true, then there are no contradictions. Thus, the lack of contradiction does not allow one to make any conclusions (unless every number is assigned true or false without contradiction, delivering the solution to the puzzle).

Finding a contradiction based on the assumption that a number is false allows one to asign this number true. This can be very rewarding if the cell being assigned a single number (ie being solved) had three or more candidates to begin. However, contradictions based on a number being true only yield a single elimination of that number.

In order to clarify the process, a utility is provided which lets you enter the number and its address, and then press the 'Show True/False' button, and the end result of all the strong and weak chains that can be generated is displayed. Red numbers are 'true', blue are 'false', and orange are ones that have been assigned both true and false. These latter are the result of a number at an address which was first assigned true and added to an array of true number+address, was later assigned false, and added to the array of false number+address.

Complex contradictions

When you try your luck with hard puzzles like the Easter Monster, even the above techniques are not enough. Postulating two different numbers are true and seeing if there is a contradiction doesn't help, because if there is one, one doesn't know which one is the culprit, or if both are culprits.

A solution is to take all occurances of pairs, ie where a number appears only twice in a unit. If each of the pair is taken one at a time to be true, and a second number is then taken to be true and all chains possible are generated, then if a contradiction occurs both times then we are in business. Since one or other of the pair must be true, but combining with the other number causes a contradiction each time, then the other number has to be false and can be removed.

Contradictions based on Naked Sets

In a partially solved sudoku, naked sets are sets of n cells containing n candidates, within a single unit. The final placement of the n candidates must be one of the possible combinations of the digits. For example, with the naked triplet of 12, 123, 123 the 3 cells could end up with one of the following 4 combinations: 1-2-3, 1-3-2, 2-1-3, 2-3-1. Each of these can be tested to see if the derived forcing net creates contradictions, and those that do removed. For more detail, refer to the appropriate help page.