6 of 9 added 136 characters in body

Java 8, 77 93 74 71 69 bytes

int n=1111;int m=1;while(x>m++){n=Integer.parseInt((n*n+"").substring(0,4));}

x->{int n=1111;int m=1;while(x>m++){n=Integer.parseInt((n*n+"").substring(0,4))‌​;}return n;}

x->{int n=1111;for(;--x>0;){n=Integer.parseInt((n*n+"").substring(0,4));}}

x->{long n=1111;for(;--x>0;){n=Long.valueOf((n*n+"").substring(0,4));}}

x->{long n=1111;for(;--x>0;)n=Long.valueOf((n*n+"").substring(0,4));}

Each repetition makes n the first 4 characters of n*n.

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Post history:

  • 77 bytes: initial code (incomplete)

  • +16 bytes, by Olivier Grégoire: completed code by making it a Lambda function.

  • -19 bytes: replace while with for cycle.

  • -4 bytes: used longs instead of ints

  • -2 bytes, by Roman Gräf: removed unnecessary brackets

Thanks to @OlivierGrégoire and @RomanGräf for pointing out some issues!

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