Total cubes in a staircase pile on a 20 by 20 board

A $20\times20$ board has a single cube on one corner square. Every square one step further south or east carries one more cube than its neighbour, so the square $i$ steps south and $j$ steps east of that corner holds $1+i+j$ cubes. How many cubes are on the board altogether?

$8000$. Sum $1+i+j$ over $i,j\in\{0,\dots,19\}$: $400$ from the ones plus $2\cdot20\cdot\sum_{i=0}^{19}i=40\cdot190=7600$, giving $8000=20^3$. Pairing each square with its mirror image across the anti-diagonal (sums $1+i+j$ and $39-i-j$) shows the average is $20$ per square.