Build a (normal?) distribution from $n$, quartiles and mean? I have some data that is described by $n$, quartiles (+ additional quantile point) and the mean.  Is it possible to rebuild or model this distribution from these statistics?  As the median and the mean are not the same, there is at least some skew, but otherwise, I would assume the data to be normal like.
Edit:  This was marked as a duplicate, but in the other questions I found while searching, none of them included the information regarding the mean as a data point to recreate the distribution.  Because of that additional parameter, I wondered if it made the estimation possible.  In short, the affect of having the mean was not apparent from the other answers related to the question.
 A: The answer is No, not exactly anyhow.
If you have two quartiles of a normal population then you can find $\mu$ and $\sigma.$ For example the lower and upper quantiles of $\mathsf{Norm}(\mu = 100,\, \sigma = 10)$ are $93.255$ and $106.745,$ respectively.
 qnorm(c(.25, .75), 100, 10)
 [1]  93.2551 106.7449

Then $P\left(\frac{X-\mu}{\sigma} < -0.6745\right) = 0.25$
and $P\left(\frac{X-\mu}{\sigma} < 0.6745\right) = 0.75$
provide two equations that can be solved to find $\mu$ and $\sigma.$
qnorm(c(.25,.75))
[1] -0.6744898  0.6744898

However, sample quartiles are not population quartiles. There is not
enough information in any normal sample precisely to determine $\mu$ and
$\sigma.$  
And you are not really sure your sample is from a normal population.
If the population has mean $\mu$ and median $\eta,$ then
the sample mean and median, respectively, are estimates of these two
parameters. If the population is symmetrical, then $\mu = \eta,$ but
you say the sample mean and median do not agree. So you cannot be sure
the population is symmetrical, much less normal.
A: It is possible to estimate the parameters based on this information, but to construct some (approximate) likelihood function based on the given information $n,Q_1, Q_3, \bar{X}_n$ do not see easy. In a way this is a followup on the answer by @BruceET, trying to formalize ideas in that answer.
Using the theory of order statistics we can construct a likelihood function based on $n,Q_1, Q_3$. How to also incorporate the observed mean do seem more difficult. To simplify I will assume $n=4k$ and that $X_{(k)}\le Q_1\le X_{(k+1)}$ and $X_{(3k)}\le Q_3\le X_{(3k+1)}$. An exact analysis (if at all possible) for small $n$ would need to know exactly how the quartiles was computed (different methods can give quite different answers for small $n$). Then we can find the likelihood
$$
L(\mu,\sigma) \propto \Phi(\frac{Q_1-\mu}{\sigma})^k \left[\Phi(\frac{Q_3-\mu}{\sigma})-\Phi(\frac{Q_1-\mu}{\sigma})  \right]^{2k}\left[1-\Phi(\frac{Q_3-\mu}{\sigma})\right]^k \cdot \phi(\frac{Q_1-\mu}{\sigma})\phi(\frac{Q_3-\mu}{\sigma})/\sigma^2
$$ where $\phi, \Phi$ are the standard normal pdf, cdf respectively. This can now be used as any other likelihood function.
But to extend this to a likelihood also using the mean, we need the joint distribution of $Q_1, Q_3, \bar{X}_n$ and that would probably have to be approximated in some way. That seems like a nice little project! 
Other ideas to look into here is ABC (approximate bayesian computation) which seems a good fit to estimation based on (insufficient) summary statistics. Or maybe simulated maximum likelihood. I will come back here to look at that. 
