SetOptions[InputNotebook[],NotebookEventActions->{
"LeftArrowKeyDown":>(stat=Coalesce[stat];AddNew[]),
"RightArrowKeyDown":>(stat=Reverse/@Coalesce[Reverse/@stat];AddNew[]),
"UpArrowKeyDown":>(stat=Coalesce[stat\[Transpose]]\[Transpose];AddNew[]),
"DownArrowKeyDown":>(stat=(Reverse/@(Coalesce[Reverse/@(stat\[Transpose])]))\[Transpose];AddNew[])
}
];

n=4;
bgcolor=GrayLevel[0.84];
colorfunc=Blend[{{0,Gray},{1/2,Red},{1,Blend[{Yellow,Orange}]}},#]&;

ClearAll[AddNew,PrintStat,Coalesce,SubCoalesce,AddRandomNumber]
AddNew[]:=(stat=AddRandomNumber[stat])
PrintStat[stat_]:=Module[{gr1,gr2,gr3,dr=0.2,cols,nstat=stat,positions},
gr1={bgcolor,Rectangle[-dr{1,1},n+dr{1,1},RoundingRadius->dr]};
cols=Map[If[#==0,0,Log2[#]]&,nstat,{2}];
cols=Map[If[#==0,Lighter@bgcolor,colorfunc[#/Max[cols]]]&,cols,{2}];
positions=Table[{i,n-j+1},{j,n},{i,n}];
gr2=MapThread[{#2,Rectangle[#3-{1,1}(1-dr/3),#3-{1,1}dr/3,RoundingRadius->dr/2]}&,{stat,cols,positions},2];
gr3=MapThread[If[#1>0,Style[Text[#1,#2-0.5{1,1}],20,White],{}]&,{stat,positions},2];
Graphics[{gr1,gr2,gr3},PlotRange->{{-0.5,n+0.5},{-0.5,n+0.5}},ImageSize->500]
]
Coalesce[stat_]:=SubCoalesce/@stat
SubCoalesce[statlist_]:=Module[{st=statlist,n=Length[statlist],pairs},
st=Split[DeleteCases[st,0]];
st=Partition[#,2,2,1,{}]&/@st;
st=Map[If[Length[#]==2,Total[#],#]&,st,{2}];
Join[Flatten[st],ConstantArray[0,n-Length[Flatten[st]]]]
]
AddRandomNumber[stat_,n_:2]:=With[{pos=Position[stat,0,{2}]},If[Length[pos]>0,ReplacePart[stat,RandomChoice[pos]->n],stat]]

stat=Nest[AddRandomNumber[#,RandomChoice[{2,4}]]&,ConstantArray[0,{n,n}],4];
Dynamic[PrintStat@stat]