C++ (gcc), 5204 bytes

#include <iostream> using namespace std; string secret = "secret"; void f1(){if(secret.length()!=6){cout<<"1111111";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<<"1111111";return;}}for(int i=0;i<6;i++){if(a[i]<=0||a[i]>111){cout<<"1111111";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<<"1111111";return;}}}}}}}} void f2(){if(secret.length()!=6){cout<<"1111111";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<<"1111111";return;}}for(int i=0;i<6;i++){if(a[i]<=0||a[i]>111){cout<<"1111111";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<<"1111111";return;}}}}}}}} void f3(){if(secret.length()!=6){cout<<"1111111";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<<"1111111";return;}}for(int i=0;i<6;i++){if(a[i]<=0||a[i]>111){cout<<"1111111";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<<"1111111";return;}}}}}}}} void f4(){if(secret.length()!=6){cout<<"1111111";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<<"1111111";return;}}for(int i=0;i<6;i++){if(a[i]<=0||a[i]>111){cout<<"1111111";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<<"1111111";return;}}}}}}}} void f5(){if(secret.length()!=6){cout<<"1111111";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<<"1111111";return;}}for(int i=0;i<6;i++){if(a[i]<=0||a[i]>111){cout<<"1111111";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<<"1111111";return;}}}}}}}} void f6(){if(secret.length()!=6){cout<<"1111111";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<<"1111111";return;}}for(int i=0;i<6;i++){if(a[i]<=0||a[i]>111){cout<<"1111111";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<<"1111111";return;}}}}}}}} void f7(){if(secret.length()!=6){cout<<"1111111";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<<"1111111";return;}}for(int i=0;i<6;i++){if(a[i]<=0||a[i]>111){cout<<"1111111";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<<"1111111";return;}}}}}}}} int main() { f1(); f2(); f3(); f4(); f5(); f6(); f7(); cout<<6; f1(); f2(); f3(); f4(); f5(); f6(); f7(); } 

Try it online!

Same as my previous submission, hopefully it is a bit more radiation hardened.

\$n = 6\$.