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Nim Square

Problem 310

Published on Saturday, 13th November 2010, 07:00 pm; Solved by 579

Alice and Bob play the game Nim Square.
Nim Square is just like ordinary three-heap normal play Nim, but the players may only remove a square number of stones from a heap.
The number of stones in the three heaps is represented by the ordered triple (a,b,c).
If 0≤a≤b≤c≤29 then the number of losing positions for the next player is 1160.

Find the number of losing positions for the next player if 0≤a≤b≤c≤100 000.