knights on the same colour as the central square; in the second case we
place them all on black, or all on white, squares.


THE TWO PIECES PROBLEM.

On a board of n² squares, two queens, two rooks, two bishops, or two
knights can always be placed, irrespective of attack or not, in ½(n^{4}
- n²) ways. The following formulæ will show in how many of these ways
the two pieces may be placed with attack and without:--

With Attack. Without Attack.

2 Queens 5n³ - 6n² + n 3n^{4} - 10n³ + 9n² - 2n
------------------- ------------------------------
3 6

2 Rooks n³ - n² n^{4} - 2n³ + n²
----------------------
2

2 Bishops 4n³ - 6n² + 2n 3n^{4} - 4n³ + 3n² - 2n
-------------------- -----------------------------
6 6

2 Knights 4n² - 12n + 8 n^{4} - 9n² + 24n
--------------------
2

(See No. 318, " Lion Hunting.")