Knighted!
A graph with 64 nodes represents the squares of a chessboard. The nodes are joined appropriately by edges, each representing a move of the knight. In this graph, every node will be of degree two, three, four, six, or eight. Let the values
and
be the number of nodes with each respective degree such that
is the number of nodes with degree two,
is the number of nodes with degree three, etc. What is the value of
The answer is 40.
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1 solution
Each corner represents a node of degree 2 . There are 4 of them.
The middle 4 on each edge and the four that "connect" those 1 6 diagonally represent nodes of degree 4 . There are 2 0 such squares.
The sixteen that are at least 2 away from any edge represent nodes of degree 1 6 .
4 + 2 0 + 1 6 = 4 0