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Suppose two arbitrary integer numbers $a$ and $b$. I'm looking for some function $f(a,b)$ with the following properties:
Any example of such a function?
Improvement: Any solution avoiding "floor/ceiling-like" functions?
$$f(a,b)=\frac{a+b-\sin^2\left(\pi(a+b)/2\right)}{2}$$
Yes, it's just the floor function in disguise :-)
\begin{equation} f(a,b) = \begin{cases} a & \text{ if } a=b \\ a+1 & \text{ if } a<b \\ b+1 & \text{ if } b<a \end{cases} \end{equation}
This does ignore the verbal connotation of “integer average”. Because nobody likes connotations.
How about $f(a,b)=\lfloor \frac 12(a+b)\rfloor$?
Any solution avoiding "floor/ceiling-like" functions?
Maybe you were hoping for a polynomial? Sadly, that's impossible. If $f$ is a polynomial, then $g(a)=f(a,0)$ is also a polynomial. We need $|g(a)|\leq |a|$ for all $a$, so the degree of $g$ can't be greater than $1$. But if $g$ is linear and $g(0)=0$ and $g(2)=1$, then $g(1)=\frac12\not\in\mathbb Z$.
I would use "scientific rounding" (round to even) on the arithmetic mean. So if $(a+b)/2$ is an integer, use that, otherwise use the even integer out of $(a+b+1)/2$ and $(a+b-1)/2$. So that would likely be something like $$\left\lfloor a+b+1\over4\right\rfloor + \left\lceil a+b-1\over4\right\rceil$$ The advantage of this kind of rounding is that it is overall unbiased while preferredly rounding to points of greater stability, possibly reducing the amount of followup errors with repeated operations since averaging two values calculated using the "round to even" tie-breaking will have an exact result not needing another tie breaker.
How about
$$ \frac{a + b + ((a + b)\ \text{mod}\ 2)}{2} $$
This works because $a + b + ((a + b)\ \text{mod}\ 2)$ is guaranteed to be a multiple of 2. Note that you can also use $(a - b)\ \text{mod}\ 2$, if you prefer, with no change in behavior.
Both of the modulo expressions are more straight-forward descriptions (IMO) of what Peter was doing in his answer where he used $−sin^2(π(a+b)/2)$, except that he rounds down where I round up, and of course my solution remains completely in the integer domain. (I'm not familiar with any implementation of sine that is integer specific.)
A more complex answer avoiding floor and round that includes the concept of scientific rounding in user198082's answer is:
$$ \frac{a + b + ((a + b)\ \text{mod}\ 4)((a + b)\ \text{mod}\ 2) - 2((a + b)\ \text{mod}\ 2)}{2} $$
Examining the four cases:
What you're looking for is the following:
(1) You want a binary operation on $\mathbb{Z}$. That is, you want a function $f: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$. It's common to use multiplicative notation for any binary operation (even if the operation is not literally multiplication). Thus we could write $x * y = f(x,y)$.
(2) This operation should be idempotent: for all $a \in \mathbb{Z}$, we require $a * a = a$.
(3) This operation should be commutative: for all $a, b \in \mathbb{Z}$, we require $a *b = b*a$.
(4) This operation should satisfy $\textrm{min}\{ a, b \} < a * b < \textrm{max} \{a, b \}\ldots$except in the cases where it can't. For example, there is no integer $z$ that satisfies $6 < z < 7$, so you're forced to accept a value of either $6$ or $7$ for $(6 * 7)$. Similarly, you accept $a * a = a$ for all $a$, as in (2).
Condition (4) is a bit complicated. Things might get a little simpler if you're willing to RELAX condition (4) in favor of just always allowing non-strict inequalities. So, your question didn't ask about this, but you might be interested in what happens if we replace (4) with (4'):
(4') For all $a, b \in \mathbb{Z}$, we require $\textrm{min}\{ a, b \} \leq (a * b) \leq \textrm{max} \{a, b \}$.
Or equivalently:
(4'') If $a \leq b$, then $a \leq (a*b) \leq b$.
This allows for some new examples. To begin with, the operations $x * y = \textrm{min}(x, y)$ and $x * y = \textrm{max}(x, y)$ are obviously two functions that would satisfy properties (1-3) and (4''). Notice that $\textrm{min}$ and $\textrm{max}$ are the meet and join, respectively, of the partially ordered set $(\mathbb{Z}, \leq)$.
In general, we could replace $(\mathbb{Z}, \leq)$ with any lattice; the meet and join functions will always satisfy conditions (1-3). It should be clear that in any lattice, if $a \leq b$, then $\textrm{meet}(a, b) = a$ and $\textrm{join}(a, b) = b$. Therefore the meet and join operations in any lattice will always satisfy condition (4'') also.
http://en.wikipedia.org/wiki/Lattice_(order)
FYI, the meet and join will also always be associative. Notice that for example the arithmetic mean operation, $x*y = \frac{x+y}{2}$, does not have this property.
I understand that the $\textrm{min}$ and $\textrm{max}$ functions are not answers to your question; they do not satisfy your original condition (4). Still, I thought it was worth mentioning that $\textrm{min}$ and $\textrm{max}$ are sort of "degenerate averages", and that you can generalize this to any lattice (which includes any totally-ordered set).
How about:
$$ f(a,b) = min(a,b) + |sgn(b-a)| $$
Where $sgn$ is the sign function, returning $-1$, $0$ or $1$ in agreement with the sign of its argument?
