# how to solve it

everyone should know the work of George PĆ³lya on how to solve problems

• Marvin Minsky (in Steps Toward Artificial Intelligence)

wiki

example: how to solve 4x4 dark hex?
• solve a simpler problem ?

simple dark hex
• restriction of dark hex

• first stone must touch own edge

• each subsequent stone must touch own previous stone

• must reach other side within 4 moves

• can we solve this ?

notation: label SDH strategies by 1st move
```*p1   * * * *        p1              p2
~ ~ ~ ~    D+ C+ B+ A+    d-      a+
~ ~ ~ ~                   c-      b+
~ ~ ~ ~                   b-      c+
~ ~ ~ ~    A- B- C- D-    a-      d+
* * * *
```
p1 minimax win probability matrix ?
```     a- b- c- d-  a+ b+ c+ d+
A-  ?  1  ?  ?   ?  1  1  1      D+ C+ B+ A+
B-                                c-       b+
C-                                 b-       c+
D-                                  A- B- C- D-
A+
B+
C+
D+

what should we assume if player sees opponent stone?
minimax ?  equiprobable ?
eg: A-a- .5 if p2 plays minimax = equiprob. {d-,a+}
eg: A-d- .25 if p2 moves 2,3 uniform {left/right}
```
expanding matrix entries
```A-- move 2 towards opponent -
A-+   "  "    "       "     +

a- c-  d-  a+-  a++
A--  ?  0   0    1   .5
A-+  ?  1  .5    1    0

so if p2 plays a- and sees a stone at A-,
then p2 minimax is equiprob {d-, a++}, 1p win rate .25
```