SPOJ Problem: Adjacent Bit Counts

combinatorics, number-theory

A Little Background

I was trying to solve this problem on SPOJ. It’s a dynamic-programming problem. I tried to find some recursive relation. I spent hours. But no breakthrough.

I even tried looking for hints in comments and I came to know it’s a 3D dynamic-programming problem. Some of the users were even able to reduce it to a 2D technique.

Then I realized I need to rethink the entire approach. So somehow, I started treating it like a combinatorics problem and I landed on a solution — a formula that guarantees linear time execution. I love moments like these when you solve an interesting problem with an uncommon approach. And that’s why I am writing this post.

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Beggar's Method

combinatorics, number-theory

Problem

Find the number of integer solutions i.e. ordered pairs such that

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