6
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Let $f(x)$ be a function such that

  • $f(1111)=4$
  • $f(1234)=3$
  • $f(4567)=2$
  • $f(1357)=4$
  • $f(6518)=4$
  • $f(3817)=6$
  • $f(8008)=6$
  • $f(1000)=4$

Then, $f(2014)=\,\,?$

(Hint: Each digit is valuable.)

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6
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Perhaps

$~~~f(2014)=2$

Argument:

The digits $0,1,2,3,4,5,6,7,8$ respectively contribute $1,1,0,2,0,0,1,1,2$ to the function value; that is $f(0)=1$, $f(1)=1$, $f(2)=0$, $\ldots$, $f(8)=2$.
With this, $f(2014)=f(2)+f(0)+f(1)+f(4)=0+1+1+0=2$.

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