Answers

3.1 CHESS

For the given position (Fig. 3.6) to be legally realised the white king must have made the trip round the board shown by the squares marked D. Therefore black cannot castle and the mate is simple. There are other two-move mate problems which rely on showing that a pawn can be captured en passant for solution, and these also the program cannot solve except by default.

3.3 KALAH

50.    50-2    48 mod 49 
49.    48-2    46 mod 48 
48.    46-2    44 mod 47 
  . . . . .
27.    4-2      2 mod 26 
26.    2-2      0 mod 25 ~ 25 mod 25 
25.   25-4     21 mod 24 
24.   21-4     17 mod 23 
23.   17-4     13 mod 22 
22.   13-4      9 mod 21 
21.    9-4      5 mod 20 
20.    5-4      1 mod 19 
19.    1-4     -3 mod 18 ~ 15 mod 18 
18.   15-6      9 mod 17 
17.    9-6      3 mod 16 
16.    3-6     -3 mod 15 ~ 12 mod 15 
15.   12-8      4 mod 14 
14.    4-8     -4 mod 13 ~  9 mod 13 
13.    9-10    -1 mod 12 ~ 11 mod 12 
12.   11-12    -1 mod 11 ~ 10 mod 11 
11.   10-14    -4 mod 10 ~  6 mod 10 
10.    6-16   -10 mod 9  ~ -1 mod 9  ~ 8 mod 9 
9.     8-20   -12 mod 8  ~ -4 mod 8  ~ 4 mod 8 
8.     4-24   -20 mod 7              ~ 1 mod 7 
7.     1-30   -29 mod 6              ~ 1 mod 6 
6.     1-40   -39 mod 5              ~ 1 mod 5 
5.     1-56   -55 mod 4              ~ 1 mod 4 
4.     1-84   -83 mod 3              ~ 1 mod 3 
3.    1-140  -139 mod 2              ~ 1 mod 2 
2.    1-280  -279 mod 1              ~ 1 mod 1 
1.    1-840  -839 mod 0              ~ 0 

Note last (839 mod 0) because there are 839 stones.