What is the parity of $$$s$$$ after an operation?

What is the condition for four edges to form an isosceles trapezoid?

How do you find two bases after fixing the length of the legs?

How do you choose two legs?

How can you solve the problem in $$$O(n^2)$$$?

No two liars can stand next to each other is important.

Thinking outside the box.

How is a number $$$x$$$ merged from?

Approach the problem in reverse.

Consider the minimum number that can be obtained by operating $$$0, 1, 2, \dots, m$$$ times on a number $$$x$$$.

Do these numbers have any other properties?

Consider the cost of each operation and each number independently.

What is the property for the final string?

Consider the immovable blocks.

For the easy version, fix the immovable blocks greedily.

For the hard version, use dynamic programming to determine the immovable blocks.

Use a data structure to speed up dynamic programming.

What is the size of the maximum possible matching? What can be the "most dense" graph with maximum matching $$$k$$$?

Mixed-color triplet($$$u, v$$$ with edge $$$0$$$ and $$$v, w$$$ with edge $$$1$$$) is a flexible structure to find a matching for both types.

Add vertices one by one. Try to get a "mixed-color triple" or a larger set of the same color.