I want to select models with regsubsets. I have a dataframe called olympiadaten (data uploaded: http://www.sendspace.com/file/8e27d0). I first attach this dataframe and then start to analyze, my code:

    attach(olympiadaten)
    
    library(leaps)
    a<-regsubsets(Gesamt ~ CommunistSocialist + CountrySize + GNI + Lifeexp + Schoolyears + ExpMilitary + Mortality +
    PopPoverty + PopTotal + ExpEdu + ExpHealth, data=olympiadaten, nbest=2)
    summary(a)
    plot(a,scale="adjr2")
    
    
    summary(lm(Gesamt~ExpHealth))



screenshot of the plot: http://tinypic.com/r/2pq8agy/6 

The problem is now, that I want to fit the best model again "manually" and have a look at it, but the value of the adjusted R squared is not the same as in the regsubsets output? This is also the case for the other models, e.g. when I do the simplest model in the graphic:

    summary(lm(Gesamt~ExpHealth))

The graphic says, it should have an adjusted R squared of about 0.14, but when I look at the output, I get a value of 0.06435

output of : summary(lm(Gesamt~ExpHealth))

    Call:
    lm(formula = Gesamt ~ ExpHealth)
    
    Residuals:
        Min      1Q  Median      3Q     Max 
    -18.686  -9.856  -4.496   1.434  81.980 
    
    Coefficients:
                Estimate Std. Error t value Pr(>|t|)  
    (Intercept)  -3.0681     6.1683  -0.497   0.6203  
    ExpHealth     1.9903     0.7805   2.550   0.0127 *
    ---
    Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 
    
    Residual standard error: 18.71 on 79 degrees of freedom
      (4 observations deleted due to missingness)
    Multiple R-squared: 0.07605,    Adjusted R-squared: 0.06435 
    F-statistic: 6.502 on 1 and 79 DF,  p-value: 0.01271 

I don't know which mistake I did, any help would be nice, thanks. (I could not attach both files correctly, maybe a mode can attach them correctly, thanks).

And last but not least, some more questions: What is the difference between selecting models by AIC and by the adj. R squared? Both measure the fit and recognize the number of variables, so is the best model choosen by AIC also the model with the highest adj. r squared? 

When I have 12 variables, this means, there are 2^12 possibilites of models, right? So does the regsubsets command calculate each model and shows the two best (nbest=2) of each size? So do I really get the 'best' model? And when I do stepwise AIC backwards selection (starting with the model which contains all variables), does this also ends with the same model as regsubsets says it is the best?