C++ (gcc), 4960 bytes

#include <iostream> using namespace std; string secret = "secret"; void f1(){if(secret.length()!=6){cout<<"1";return;}int a[6]={secret[0]-' ',secret[1]-' ',secret[2]-' ',secret[3]-' ',secret[4]-' ',secret[5]-' '};for(int i=0;i<5;i++){if(a[i]>=a[i+1]){cout<<"1";return;}}for(int i=0;i<6;i++){if(a[i]<=0||a[i]>111){cout<<"1";return;}}for(int i=1;i<=111;i++){for(int j=i+1;j<=111;j++){for(int k=j+1;k<=111;k++){for(int l=k+1;l<=111;l++){for(int m=l+1;m<=111;m++){for(int n=m+1;n<=111;n++){int b[6]={i,j,k,l,m,n};bool yay=false;for(int o=1;o<(1<<6);o++){for(int p=1;p<(1<<6);p++){int asum=0;int bsum=0;for(int q=0;q<6;q++){if(o&(1<<q)){asum+=a[q];}if(p&(1<<q)){bsum+=b[q];}}if(asum==bsum){yay=true;break;}}if(yay){break;}}if(!yay){cout<<"1";return;}}}}}}}} void f2(){if(secret.length()!=6){cout<<"1";return;}int a[6]={secret[0]-' ',secret[1]-' ',secret[2]-' ',secret[3]-' ',secret[4]-' ',secret[5]-' '};for(int i=0;i<5;i++){if(a[i]>=a[i+1]){cout<<"1";return;}}for(int i=0;i<6;i++){if(a[i]<=0||a[i]>111){cout<<"1";return;}}for(int i=1;i<=111;i++){for(int j=i+1;j<=111;j++){for(int k=j+1;k<=111;k++){for(int l=k+1;l<=111;l++){for(int m=l+1;m<=111;m++){for(int n=m+1;n<=111;n++){int b[6]={i,j,k,l,m,n};bool yay=false;for(int o=1;o<(1<<6);o++){for(int p=1;p<(1<<6);p++){int asum=0;int bsum=0;for(int q=0;q<6;q++){if(o&(1<<q)){asum+=a[q];}if(p&(1<<q)){bsum+=b[q];}}if(asum==bsum){yay=true;break;}}if(yay){break;}}if(!yay){cout<<"1";return;}}}}}}}} void f3(){if(secret.length()!=6){cout<<"1";return;}int a[6]={secret[0]-' ',secret[1]-' ',secret[2]-' ',secret[3]-' ',secret[4]-' ',secret[5]-' '};for(int i=0;i<5;i++){if(a[i]>=a[i+1]){cout<<"1";return;}}for(int i=0;i<6;i++){if(a[i]<=0||a[i]>111){cout<<"1";return;}}for(int i=1;i<=111;i++){for(int j=i+1;j<=111;j++){for(int k=j+1;k<=111;k++){for(int l=k+1;l<=111;l++){for(int m=l+1;m<=111;m++){for(int n=m+1;n<=111;n++){int b[6]={i,j,k,l,m,n};bool yay=false;for(int o=1;o<(1<<6);o++){for(int p=1;p<(1<<6);p++){int asum=0;int bsum=0;for(int q=0;q<6;q++){if(o&(1<<q)){asum+=a[q];}if(p&(1<<q)){bsum+=b[q];}}if(asum==bsum){yay=true;break;}}if(yay){break;}}if(!yay){cout<<"1";return;}}}}}}}} void f4(){if(secret.length()!=6){cout<<"1";return;}int a[6]={secret[0]-' ',secret[1]-' ',secret[2]-' ',secret[3]-' ',secret[4]-' ',secret[5]-' '};for(int i=0;i<5;i++){if(a[i]>=a[i+1]){cout<<"1";return;}}for(int i=0;i<6;i++){if(a[i]<=0||a[i]>111){cout<<"1";return;}}for(int i=1;i<=111;i++){for(int j=i+1;j<=111;j++){for(int k=j+1;k<=111;k++){for(int l=k+1;l<=111;l++){for(int m=l+1;m<=111;m++){for(int n=m+1;n<=111;n++){int b[6]={i,j,k,l,m,n};bool yay=false;for(int o=1;o<(1<<6);o++){for(int p=1;p<(1<<6);p++){int asum=0;int bsum=0;for(int q=0;q<6;q++){if(o&(1<<q)){asum+=a[q];}if(p&(1<<q)){bsum+=b[q];}}if(asum==bsum){yay=true;break;}}if(yay){break;}}if(!yay){cout<<"1";return;}}}}}}}} void f5(){if(secret.length()!=6){cout<<"1";return;}int a[6]={secret[0]-' ',secret[1]-' ',secret[2]-' ',secret[3]-' ',secret[4]-' ',secret[5]-' '};for(int i=0;i<5;i++){if(a[i]>=a[i+1]){cout<<"1";return;}}for(int i=0;i<6;i++){if(a[i]<=0||a[i]>111){cout<<"1";return;}}for(int i=1;i<=111;i++){for(int j=i+1;j<=111;j++){for(int k=j+1;k<=111;k++){for(int l=k+1;l<=111;l++){for(int m=l+1;m<=111;m++){for(int n=m+1;n<=111;n++){int b[6]={i,j,k,l,m,n};bool yay=false;for(int o=1;o<(1<<6);o++){for(int p=1;p<(1<<6);p++){int asum=0;int bsum=0;for(int q=0;q<6;q++){if(o&(1<<q)){asum+=a[q];}if(p&(1<<q)){bsum+=b[q];}}if(asum==bsum){yay=true;break;}}if(yay){break;}}if(!yay){cout<<"1";return;}}}}}}}} void f6(){if(secret.length()!=6){cout<<"1";return;}int a[6]={secret[0]-' ',secret[1]-' ',secret[2]-' ',secret[3]-' ',secret[4]-' ',secret[5]-' '};for(int i=0;i<5;i++){if(a[i]>=a[i+1]){cout<<"1";return;}}for(int i=0;i<6;i++){if(a[i]<=0||a[i]>111){cout<<"1";return;}}for(int i=1;i<=111;i++){for(int j=i+1;j<=111;j++){for(int k=j+1;k<=111;k++){for(int l=k+1;l<=111;l++){for(int m=l+1;m<=111;m++){for(int n=m+1;n<=111;n++){int b[6]={i,j,k,l,m,n};bool yay=false;for(int o=1;o<(1<<6);o++){for(int p=1;p<(1<<6);p++){int asum=0;int bsum=0;for(int q=0;q<6;q++){if(o&(1<<q)){asum+=a[q];}if(p&(1<<q)){bsum+=b[q];}}if(asum==bsum){yay=true;break;}}if(yay){break;}}if(!yay){cout<<"1";return;}}}}}}}} void f7(){if(secret.length()!=6){cout<<"1";return;}int a[6]={secret[0]-' ',secret[1]-' ',secret[2]-' ',secret[3]-' ',secret[4]-' ',secret[5]-' '};for(int i=0;i<5;i++){if(a[i]>=a[i+1]){cout<<"1";return;}}for(int i=0;i<6;i++){if(a[i]<=0||a[i]>111){cout<<"1";return;}}for(int i=1;i<=111;i++){for(int j=i+1;j<=111;j++){for(int k=j+1;k<=111;k++){for(int l=k+1;l<=111;l++){for(int m=l+1;m<=111;m++){for(int n=m+1;n<=111;n++){int b[6]={i,j,k,l,m,n};bool yay=false;for(int o=1;o<(1<<6);o++){for(int p=1;p<(1<<6);p++){int asum=0;int bsum=0;for(int q=0;q<6;q++){if(o&(1<<q)){asum+=a[q];}if(p&(1<<q)){bsum+=b[q];}}if(asum==bsum){yay=true;break;}}if(yay){break;}}if(!yay){cout<<"1";return;}}}}}}}} int main() {f1();f2();f3();f4();f5();f6();f7();cout<<6;f1();f2();f3();f4();f5();f6();f7();} 

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Intended solution only modifies the characters in secret, and also takes very long to run.

The hidden n is \$6\$. If unchanged, the program outputs \$111111161111111\$.