RMO combinatorics DPP 2 - grid based questions

 


Solution:
Total ways: 64C2
Now subtract invalid:
2 main diagonals: 2*8C2
8 rows and 8 columns: 2 * 8* 8C2

Smaller diagonals: 4 of each type
4 * [2C2 + 3C2 + ... 7C2]
Simplify using hocky stick identity.

= 4 * [8C3]

So finally: 64C2 - 4*8C3 - 18*8C2 = 1288



Case 1:
Axis-aligned squares.
1x1 squares  = 15*15
2x2 = 14*14
....
15x15 squares = 1*1
total = 1^2 + 2^2 .... 15^15 = 1240(sum of first n natural number squares)

Case 2:
tilted squares.
1 tilted square in each 2x2 axis aligned square:

2 tilted squares in each 3x3 axis aligned square:


3 in 4x4







14 in 15x15



Total: 1*14^2 + 2*13^2 + 3*12^2 .... 14*1^2
= Sigma(k=1 to 14) (k * (15-k)^2)

Finally: 4200 + 1240 = 5440 = Answer

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