When the program completes its search, it makes the best move it has found, and, if the Show principal continuation option has been set, it displays one of the following in a dialog box:
- This problem has no solution
- The program has conclusively proved that there is no solution to the problem (or the program has a bug).
- Found no mate after n moves
- You set the search depth to n, but the program was unable to either prove or disprove the problem. Increase the depth by 2 (or 4,6,8 etc.) and try again.
- Found a mate in n:
- The program has found the fastest (or a fastest) mate, and it takes n moves (by the Japanese count). These moves are then displayed, on sucessive lines. Side variations are not displayed.
The main problem is to correctly set the search depth. The following should be borne in mind:
- You must set the search depth to an even number (this number should be double the number of moves it takes to mate by the Western count).
- The time taken by the program increases exponentially with the
search depth, even if the solution is shorter than the search
depth. This is because most positions have a very large number of
irrelevant checks, by which Black can chase the White King round the
board, getting nowhere very slowly.
- if there is a known/suspected/guessed solution to a problem, then that is the best search depth to set (remember, double the Western count, or add one to the Japanese count).
As a guide, I studied the search times required for two of the historic problems. D84 has a known solution of mate in 9 (5 by the Western count), and so requires a search depth of 10. My computer solves it in less than 30 seconds.
C49 has a known (but not proven by a computer - see below) solution in 43 (22 by the Western count), and so requires a search depth of 44. This poses a problem for current computer technology.
I measured the search time (using Show Statistics) for both these problems, at search depths of 4 to 20. the results were (these were obtained using the transposition table - currently, the program does not use the transposition table in mating mode, resulting in approximately 40% longer times, but the principal continuation is then correct):
| Search Depth | D84 | C49 |
|---|---|---|
| 4 | 0.493 | 0.539 |
| 6 | 1.61 | 1.323 |
| 8 | 5.913 | 3.136 |
| 10 | 22.89 | 11.994 |
| 12 | 52.186 | 44.991 |
| 14 | 112.904 | 200.848 |
| 16 | 316.90 | 727.475 |
| 18 | 1629.585 | 2345.392 |
| 20 | 12986.59 | 8157.602 |
I then plotted these data as a graph (using XploRe 4.1). The curves looked like exponentials to me, so I then modelled them as such. The resulting regression lines, along with the data points, can be seen in the graphs below (Y is the search time in thousands of seconds. X is the result of raising e to the power of the search depth, and dividing by 100,000,000):
As can be seen, for D84, the fit looks very good indeed, confirming my guess. For C49, it is not so good, but is clearly approximately right.
Using the values computed for the regression equation, I then calculated the time for C49 assuming a search depth of 44. The result was Six and three quarters million years! Does anyone have a fast computer? :-)
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