The display shows the %age complete, and also shows counts of the number of changes on the previous step, and the current number of cells containing guessed numbers. These were originally included for debugging purposes but they seemed interesting enough to leave there.

At each stage the program goes through a crossing out process where it looks at the current set of numbers in the grid and unflags the possibility of those numbers occurring in other cells in the same row, column or box. If only one possibility exists for a cell that number is entered.

If at the end of the stage no changes have taken place the program takes a guess at the value of one of the unfilled cells with the lowest number of possibilities remaining. If during a stage the program sees that a cell has zero possibilities it then considers the current number of guesses.

If there are no guesses it declares the starting position insoluble and stops. If there are guesses it restores the situation to that immediately preceding the last guess, unflags the guess as a possibility (which usually leaves just one other possibility for that cell) and carries on as before.

I believe that these two processes are properly called deduction and bifurcation, but as this is the non-tech list we probably ought to stick with the terms crossing out and guessing. Easy puzzles require no guesses whereas really hard ones seem to need 4 or 5.