Stone Game II

A game is played with two piles of stones and two players.
On each player’s turn, the player may remove a number of stones from the larger pile.
The number of stones removed must be a positive multiple of the number of stones in the smaller pile.
E.g. Let the ordered pair $(6,14)$ describe a configuration with $6$ stones in the smaller pile and $14$ stones in the larger pile, then the first player can remove $6$ or $12$ stones from the larger pile.
The player taking all the stones from a pile wins the game.
A winning configuration is one where the first player can force a win. For example, $(1,5)$, $(2,6)$, and $(3,12)$ are winning configurations because the first player can immediately remove all stones in the second pile.
A losing configuration is one where the second player can force a win, no matter what the first player does. For example, $(2,3)$ and $(3,4)$ are losing configurations: any legal move leaves a winning configuration for the second player.
Define $S(N)$ as the sum of $(x_i + y_i)$ for all losing configurations $(x_i, y_i), 0 \lt x_i \lt y_i \le N$.
We can verify that $S(10) = 211$ and $S(10^4) = 230312207313$.
Find $S(10^{16}) \bmod 7^{10}$.

This question is a discrete mathematics problem, involving game theory, combinatorics and modular arithmetic. Specifically, it’s about winning and losing configurations in a game where two players take turns to move stones from two piles, subject to certain rules.

This is an example of a problem that occurs in the ‘Project Euler’, a series of challenging mathematical problems that require a good knowledge of algorithmic ways of processing numbers. To solve the problem completely requires understanding a relationship between the indices of the losing configurations and combining this knowledge with a fast way to calculate these values.

The solution involves recognition of behaviour in the differences between the indices of the losing positions, and specifically their tendency to converge to the golden ratio, which provides a formula for their generation. Representing the indices in binary form provides a way of calculating them faster.

This relationship can be noticed either from graphing the differences or from recursive nature of the differences in indices. The problem then reduces to calculation of a formula for the index-from-value and value-from-index.

With the formula for the losing configurations defined as the Fibonacci sequence with index as the floor of the golden ratio times the previous index, the calculation of the values themselves becomes a question of efficiently calculating the Fibonacci numbers mod 7^10.

This is accomplished by recognizing that powers of two are involved in the binary representation of the numbers and thus a form of the Lucas numbers, companion to the Fibonacci numbers, can be used to calculate the Fibonacci numbers quickly.

Thus, the general structure of the solution is:

1. Understand losing configurations behaviour using the golden ratio.
2. Write an efficient formula to calculate Fibonacci numbers using Lucas numbers and binary representation.
3. Calculate Fibonacci numbers up to a certain limit.
4. Sum all those numbers and take modulo 7^10.

Given the complexity of this problem, it likely requires a strong background in discrete mathematics, number theory and computer algebra to compute S(N) for large values of N and further to compute it modulo 7^10. To find the exact procedure, you would need to develop a detailed algorithm or write a computer program, which typically involves the Python or C++ languages, or a similar high-level language with strong mathematical libraries.

It is important to note that these are challenges designed for those with considerable mathematical and computational expertise, intended to provide a rigorous intellectual challenge.

More Answers: