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A function of one variable $y = f(x)$. However, I suspect that the function does not need to necessarily be strictly mathematical.

If $x = 1$, then $y = 0$.

If $x = 2$, then $y = 0$.

If $x = 10$, then $y = 9$.

If $x = 3$, then $y = 3$.

If $x = 20$, then $y = 18$.

If $x = 1990$, then $y = 1989$.

What function is it? (A puzzle from an old informatics textbook which I was unable to solve for years.)

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  • $\begingroup$ The "old informatics textbook" seems to be from around 1990, since the authors of such problems often use the current year as a "sufficiently large" integer. $\endgroup$
    – trolley813
    Dec 15, 2020 at 17:32

2 Answers 2

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It looks like

$y = 3 \lfloor \frac{x}{3} \rfloor$

where

$\lfloor . \rfloor$ is the floor function indicating the greatest integer less than or equal to the argument.

Similarly

$y$ is the greatest integer $\leq x$ which is divisible by $3$.

Given that it comes from an informatics textbook Bubbler make an important point that

some programming is relevant here (and it must be C if it's from 1990): int f(int x) { return x / 3 * 3; }

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  • $\begingroup$ Great! You beat me for less than a minute. $\endgroup$
    – trolley813
    Dec 15, 2020 at 17:31
  • $\begingroup$ Given that it's from an informatics textbook, I guess some programming is relevant here (and it must be C if it's from 1990): int f(int x) { return x / 3 * 3; } $\endgroup$
    – Bubbler
    Dec 15, 2020 at 23:21
  • $\begingroup$ @Bubbler that's a good point, I may add that into the answer (with credit) if that's okay as it provides good context. $\endgroup$
    – hexomino
    Dec 15, 2020 at 23:31
  • $\begingroup$ @hexomino Sure, no problem. $\endgroup$
    – Bubbler
    Dec 15, 2020 at 23:33
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An alternative (and very similar) answer (which still fits):

$f(x) = x - L_{10}(x) \bmod 3$, where $L_{10}(x)$ is the leftmost (nonzero as usual) digit of $x$ written in decimal (i.e. $L_{10}(534)=5, L_{10}(6207)=6$ etc.). (The hexomino's answer is simply $f(x)=x-x\bmod3$).

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