
| Depth                                                                                                       | Std R = 2       | Std R = 3      | Vrfd R = 3     |
|-------------------------------------------------------------------------------------------------------------|-----------------|----------------|----------------|
| 9                                                                                                           | 5,374,275,763   | 2,483,951,601  | 4,848,596,820  |
| (+10.84%)                                                                                                   | (-48.76%)       | -              |                |
| 10                                                                                                          | 16,952,333,579  | 7,920,812,800  | 14,439,185,304 |
| (+17.40%)                                                                                                   | (-45.14%)       | -              |                |
| 11                                                                                                          | 105,488,197,524 | 24,644,668,194 | 51,080,338,048 |
| (+106.51%)                                                                                                  | (-51.75%)       | -              |                |
| Table 3: Total node count of standard R = 2, R = 3, and verified R = 3 at depths 9, 10, and 11, for 869 ECM |                 |                |                |

![0_image_0.png](0_image_0.png)

these trees. We therefore conducted a search to a depth of 11 plies, using 869 positions from the *Encyclopedia of* Chess Middlegames (ECM)4. Table 3 provides the total node counts at depths 9, 10, and 11, using standard R = 2, R = 3, and verified R = 3. See also Figure 4.

As Figure 4 clearly indicates, for depth 11 the size of the tree constructed by verified null-move pruning with R = 3 is closer to standard null-move pruning with R = 3. This implies that the saving from verified null-move pruning will be greater as we search more deeply. This can be explained by the fact that the saving from the use of R = 3 in the shallow null-move search far exceeds the verification cost of verified null-move pruning.

Having studied the effect of verified null-move pruning on the search tree size, we now take a closer look at the resulting tactical strength in comparison with standard null-move pruning with different depth reductions. For this purpose we used 999 positions from the *Winning Chess Sacrifices* (WCS) test suite, and 434 positions of "mate in 4" and 353 positions of "mate in 5" from the test suites of the *Chess Analysis Project* (CAP); see the Appendix.

The WCS positions were searched to depths of 8, 9, and 10 plies, using standard R = 2, R = 3, and verified R = 3. Table 4 provides the total node counts, and Table 5 gives the number of correctly solved positions for the WCS test suite. For each position of "mate in 4" we conducted a search to a depth of 8 plies, and for each "mate in 5" position a search to a depth of 10 plies. The search was conducted using standard R = 1, R = 2, R = 3, and verified R = 3. Table 6 provides the number of positions that each method solved (i.e., found the checkmating sequence).

4Because of the large number of errors in ECM's suggested best moves, we did not check here for number of solved positions.