Nim Square

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 $0 \le a \le b \le c \le 29$ then the number of losing positions for the next player is $1160$.

Find the number of losing positions for the next player if $0 \le a \le b \le c \le 100\,000$.

This sounds like a very complex combinatorial math problem! From a mathematical perspective, it is a problem of Mexican standoff games. To get the solution, we can build a sequence for $0 \le a, b, c \le N$ where we increase $N$ one by one. As the maximum of $a, b, c$ increases by 1, the new losing positions are those where the new $c = a \oplus b$, where $\oplus$ is the bitwise XOR operation, for some existing losing position $(a, b)$.

We can build a computer program to simulate the game and find the number of losing positions. Such a program will have at least these components:
1. A function to perform the XOR operation on two numbers.
2. A loop to iterate over each possible value of ‘a’ and ‘b’, calculate the ‘c’ for each pair, and update the list of losing positions.
3. An outer loop to increment the maximum value ‘N’ and repeat step 2 for the new range of values.

Given the enormity of the number (i.e., 100,000), it would be unrealistic to attempt to manually calculate the sequence. The computational approach is recommended for its efficiency.

In summary, to find the number of losing positions for the next player if $0 \le a \le b \le c \le 100,000$ would require a computational program to iterate over all potential numbers within a set range and perform a series of logical operations to simulate the game of Nim Square. Please note that this kind of problem could be out of reach without the use of a computer to perform the calculations.

Remember, the detailed response in terms of specific algorithm or code is highly recommended to be performed by a computer scientist or software engineer, as it involves recursion, bitwise operations, and a detailed understanding of computational efficiency.

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