function Solution $=$ SudokuNxN$($A$,$ T$)$

$\%$ Get the size of the Sudoku grid

gridSize $=$ size$($A$,1)$;

boxSize $=$ sqrt$($gridSize$)$; $\%$ Assumes the grid is a perfect square

$\%$ Initialize player existence and move arrays

players $=$ zeros$($gridSize$,$ gridSize, $2)$;

for i $=1$:gridSize

for j $=1$:gridSize

if A$($i$,$ j$)==0$

players$($i$,$ j$,1)=1$; $\%$ Mark cell as active if it's empty

end

end

end

S $=0$;

N $=1$;

maxIterations $=250000$;

while S $==0$ && N $<=$ maxIterations

$\%$ Each player picks a move according to the learning rule

for i $=1$:gridSize

for j $=1$:gridSize

if players$($i$,$ j$,1)==1$

Test $=$ A;

payoffs $=$ zeros$(1,$ gridSize$)$;

for move $=1$:gridSize

Test$($i$,$ j$)=$ move;

payoffs$($move$)=-0\mathrm{.}5*$ MatchesNxN$($Test$,$ i$,$ j$,$ gridSize, boxSize$)$;

end

players$($i$,$ j$,2)=$ PickMove$($payoffs$,$ T$)$;

end

end

end

$\%$ Update the matrix

for i $=1$:gridSize

for j $=1$:gridSize

if players$($i$,$ j$,1)==1$

A$($i$,$ j$)=$ players$($i$,$ j$,2)$;

end

end

end

T $=$ T $/1\mathrm{.}00005$; $\%$ Reduce the temperature

S $=$ IsSolvedNxN$($A$,$ gridSize, boxSize$)$; $\%$ Check if the puzzle is solved

N $=$ N $+1$;

end

Solution $=$ A;

if N $==$ maxIterations $+1$

fprintf$(\text{'}$Fail

$\text{'})$;

elseif S $==1$

fprintf$(\text{'}$Resolved in $\%$d iterations at temp $\%$f

$\text{'},$ N$,$ T$)$;

end

end

function M $=$ MatchesNxN$($A$,$ i$,$ j$,$ gridSize, boxSize$)$

M $=0$;

$\%$ Check the row

for n $=1$:gridSize

if n ~$=$ j && A$($i$,$ j$)==$ A$($i$,$ n$)$

M $=$ M $+1$;

end

end

$\%$ Check the column

for n $=1$:gridSize

if n ~$=$ i && A$($i$,$ j$)==$ A$($n$,$ j$)$

M $=$ M $+1$;

end

end

$\%$ Check the sub$-$box

boxRow $=$ floor$(($i$-1)/$boxSize$)*$boxSize $+1$;

boxCol $=$ floor$(($j$-1)/$boxSize$)*$boxSize $+1$;

for n $=$ boxRow:boxRow$+$boxSize$-1$

for m $=$ boxCol:boxCol$+$boxSize$-1$

if ~$($n $==$ i && m $==$ j$)$ && A$($i$,$ j$)==$ A$($n$,$ m$)$

M $=$ M $+1$;

end

end

end

end

function S $=$ IsSolvedNxN$($A$,$ gridSize, boxSize$)$

S $=1$; $\%$ Assume solved unless a conflict is found

for i $=1$:gridSize

for j $=1$:gridSize

if MatchesNxN$($A$,$ i$,$ j$,$ gridSize, boxSize$)>0$

S $=0$; $\%$ If any conflicts, mark as unsolved

return;

end

end

end

end

function move $=$ PickMove$($payoffs$,$ T$)$

annealed $=$ exp$((1/$T$)*$ payoffs$)$; $\%$ Calculate annealed probabilities

sumAnnealed $=$ sum$($annealed$)$;

probabilities $=$ annealed $/$ sumAnnealed; $\%$ Normalize to create probability distribution

r $=$ rand$()$;

cumProb $=0$;

move $=1$; $\%$ Default move if none other selected

for k $=1$:length$($payoffs$)$

cumProb $=$ cumProb $+$ probabilities$($k$)$;

if r $<=$ cumProb

move $=$ k;

break;

end

end

end

$\%$ Test with an example Sudoku puzzle and temperature

A $=[$

$5300700000000000$;

$6001950000000000$;

$0980000600000000$;

$8000600030000000$;

$4008030010000000$;

$7000200060000000$;

$0600002800000000$;

$0004190050000000$;

$0000800790000000$;

$0000000000000000$;

$\%$ Add more rows to complete the grid

$]$;

T $=10$;

Solution $=$ SudokuNxN$($A$,$ T$)$;

disp$($Solution$)$;

Extend it to larger settings

Implement fictitious play in Matlab for rock$-$paper$-$scissors game, show that it converges to the equilibrium.

Then, extend it to rock$-$paper$-$scissors, superman, kryptonite game and demonstrate again that it converges to the equilibrium