228 On the Relative Values of the Pieces in Chess. 



This result must be multiplied by 2(2r — 1), tlie number of the 

 positions of A on the given row, and the product summed for 



values of r between 1 and 4 ->. The numerator of the chance- 

 fraction, therefore, 



= 2 r =, 2 (±r-2){-rn 2 + -± g '•) 



= _4n 2 . ?,r +V~ (10/i 2 -9* + 2) + 2/i 2 J-Sr- j • j (10/i 2 - 



f(i+ 1 )(*+ 1 ) sis* 1 )/*. 



n 2 



-^(W-9n + 2) 



=:?L(4n 3 -6n+2). 



The chance, therefore, 



^(4rc 2 -6/i + 2) 



■'o 4 



rr n l 



Ifn=x8. 



= 4 (w-)(2? i-l) 

 3* fc(rc 2 -2) ' 



the chance = 1 fj^. 



Simple check from two rooks. 



We will call the rooks A and B. A can be placed on n 2 

 squares, Bon n 2 — 1 squares for each position of A, and the 

 king on n 2 — 2 squares for each position of A and B. If A 

 and B defend each other, they check altogether 3n— 4 squares, 

 and there are 2n — 2 squares on which B will defend A on a 

 given square. If B does not defend A, they check 4w — 6 

 squares, and there are (n— l) 2 squares on which B will not de- 

 fend A on a given square. The chance of the king being in 

 check 



' _ n 2 (8n~4)(2n-2) + n 2 (4n-6)(n-l) 2 _ 2(2n 2 -2n-l ) 

 n\n 2 -l)(n 2 .-2) ~(n + l)(n 2 -2) ' 



If 71=8, 



the chance = -§§-. 



