#3629
If there were no transpositions, then width w = 4 and depth d = 34 would lead to
1 + w + w² + w³ + ... + w^d = (w^(d+1) - 1) / (w - 1) positions
If all moves were interchangeable [ . . . ]
As @Elroch already mentioned, if there are 10⁴⁴ legal different positions (but he and @MARattigan correctly note that the the same position can produce different states, depending on the rules used), and one intends to search only a subset of 10¹⁷ of them, these are different positions too, and one already has to rely on transposition tables to not expand the same node twice. Now, if using TT the branching factor is 4, only a depth of 28 plies, i.e. 14 full moves, can be reached after a search of 10¹⁷ nodes; if, again using TT, the branching factor is 2, you can more or less double the depth, no more.
A typical engine can reach great depths thanks to late move reductions and other techniques, which make the engine actually very selective: it is not unusual that at the end of the search they check only one candidate move. That is obviously inadequate for a game-solving program.
Checkers and antichess have been solved after a relatively small search because captures are compulsory, so often players have just one legal move to play.
#3635
w = 4 is a reasonable width. As shown earlier this would make 1 error in 10^20 positions.
d = 34 is an everage depth after a predecessor game
The calculation shows the number of positions with 0 transpositions and with full permutation.