IBM Ponder This #064 · August 2003

This month's chess puzzle is from Gyozo Nagy.

It is well known that exactly 8 queens can be placed on a chessboard
without attacking each other; similarly 8 rooks, or 14 bishops,
or 32 knights, or 16 kings.
What if we are allowed to mix pieces?

So assign values of 1/8 to a queen (Q); 1/8 to a rook (R);
1/14 to a bishop (B); 1/32 to a knight (N); and 1/16 to a king (K).

Problem (Part 1): Construct the most expensive army on a chessboard,
using any number of these pieces, so that no piece attacks another.
(Here "any number" includes 0.)

Problem (Part 2): Same thing, but disallowing kings (so our army
consists only of queens, rooks, bishops and knights).

Can these mixed armies achieve values greater than 1?
In each case give the largest possible value (and a setup achieving it).

For example, the following setup

. . . . Q . . .
R . . . . . . .
. . . R . . . .
. . . . . . N .
. . . . . . . R
. K . . . . . .
. . . . . K . .
. . R . . . . .

would have value
1/8 + 4/8 + 0/14 + 1/32 + 2/16 = 25/32 = 0.78125.

To be listed among the correct solvers,
you need to submit both a solution for part 1 with value exceeeding 1.3,
and a solution for part 2 with value exceeding 1.06.
Extra credit if you prove optimality.

Solution

Best opened after a real attempt

To be added.