More Dice

Discrete Mathematics Level pending

We have \( n^2 \) dice labelled from \( d_{1} \) to \( d_{n^2} \), where every dice has the numbers \( 1,2,3,...,n^2 \) on its faces. We roll all the dice simultaneously, and denote \( f(d_{i}) \) as the number which comes up on \( d_{i} \). It is given that \( d_{i} \ne d_{j} \) for all \( i \ne j \). Define: \[ g(n) = \displaystyle\sum_{k=1}^{\lfloor \dfrac{n^2}{2} \rfloor} f(d_{k}) \] Given that \( n \) is the smallest positive integer such that it is possible that \[ \displaystyle\sum_{k=1}^{\lfloor \dfrac{n^2}{2} \rfloor} f(d_{k}) : \displaystyle\sum_{k=\lfloor \dfrac{n^2}{2} \rfloor+1}^{n^2}f(d_{k}) = 2 : 5 \] What is \( g(n) \)?

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