Given a list of integers $[a_1, a_2, \dots a_n]$, I want to find the number of $n$-tuples $(x_1,\dots,x_n)$ of integers such that the following three conditions are satisfied:$x_1 \oplus x_2 \oplus \dots \oplus x_n = 0$.$x_k \in [0,a_k]$ for each $k$.$x_k=a_k$ for at least one $k$.For instance, if the given list is [1,2,3], the answer is 4, and the specific solutions are[0,2,2],[1,0,1],[1,1,0], and[1,2,3].A brute-force approach would just compute the Cartesian product of all $x_k$s or simply count through the whole number space starting with [0,0,0]. But how can I tackle this problem with a more sophisticated algorithm? What paradigm is a good choice? Dynamic programming? If I only consider condition 2, then I could think of denoting the sub-problem by$$E[n,i]$$Where $n$ defines the number of integers and $i$ the current slot. Also given the caps $[m_1, m_2, \dots,m_n]$ then the number of combinations could be computed with:$$\begin{align*}  E[0,0] & = 0 \\  E[0,i] & = 1 + m_i \\  E[n,i] & = (1 + m_i) \cdot E[n, i-1] \\  E[n,i] & = \text{invalid if } i + 1 > n\end{align*}$$But how can I integrate the XOR constraint here?On the other hand it seems this is a classical Constraint Satisfaction Problem which could be backtracked.
How can I find all numbers for which the XOR-sum is 0?
co.combinatorics;gt.game theory;dynamic programming;ds.algorithms