Question

the smallest number of keys that will force a b-tree of order 3 to have a height 3 is

Answer 1

Single median is based on among leaf's elements and the new element and after that the split the node into two nodes.

Median value is shifted to parent node. Split node with the two nodes after m/2-1 where m is order of tree position.

The median value is shifted for parent node and inserts value to the appropriate position.