Why doesn't the Floyd-Warshall algorithm work if I put k in the innermost loop The Floyd-Warshall algorithm is defined as follows:

   for k from 1 to |V|
      for i from 1 to |V|
         for j from 1 to |V|
            if dist[i][k] + dist[k][j] &lt; dist[i][j] then
               dist[i][j] ← dist[i][k] + dist[k][j]


Why doesn't it work if I simply use

for i from 1 to |V|
  for j from 1 to |V|
     for k from 1 to |V|
        if dist[i][k] + dist[k][j] &lt; dist[i][j] then
           dist[i][j] ← dist[i][k] + dist[k][j]


In this case, the intermediate node k is iterated in the innermost loop. I expect it will make the same comparisons, but maybe different order. Why is the result different and incorrect?
Take $i - 1$ and $j = 2$. The algorithm is trying to find the shortest path between $1$ and $2$, by considering intermediate vertices. But at this point, most of the array dist is infinity, so we don't even find that $2$ is reachable from $1$ unless the distance is $1$ or $2$.