The Bitcoin Game is an interesting puzzle that requires strategic thinking and careful planning. Here's a step-by-step guide to help you achieve your goal of making all nodes non-negative:

Understanding the Game

Goal: Your objective is to adjust the values of nodes in the graph such that none of them are negative.

Controls:

Left-click: When you left-click on a node, you take $1 from each of its directly connected nodes. This means the value of the clicked node increases by the number of its connections, while the values of its connected nodes each decrease by 1.

Right-click: When you right-click on a node, you give $1 to each of its directly connected nodes. This decreases the value of the clicked node by the number of its connections, while the values of its connected nodes each increase by 1.

Sum Invariant: A crucial aspect of the game is that the total sum of all node values remains constant throughout the game. This means that any increase in one node's value must be balanced by a corresponding decrease in others.

Strategy

Identify Negative Nodes: Start by identifying all nodes with negative values. These are the ones you need to focus on to make non-negative.

Examine Connections: Look at the connections of each negative node. Nodes with more connections offer more potential for adjusting their values efficiently.

Prioritize Nodes with Fewer Connections: If possible, start with nodes that have fewer connections, as they are easier to adjust. However, keep in mind the overall strategy and how your moves will affect other nodes.

Use Both Click Types Strategically:

Left-click: Use left-clicks on nodes with many connections that are connected to negative nodes to help bring those negative nodes up. This is especially effective if the node you're clicking on can withstand the decrease in value (i.e., it's already positive or can remain non-negative after the click).

Right-click: Right-clicks are useful when you need to decrease the value of a node that's excessively high and can afford to "share" some of its value with its connected nodes, potentially bringing some of them up to non-negative.

Plan Ahead: Because the sum of all node values is constant, every move you make has consequences. Think about how your actions will affect not just the immediate nodes but also the overall state of the graph.

Iterate and Adjust: It's unlikely that you'll get it right on the first try. Be prepared to undo moves (if the game allows) and try different strategies until you find one that works.

Example Scenario

Let's say you have a graph with the following node values:

Node A: -2

Node B: 3 (connected to A)

Node C: 1 (connected to A and B)

One possible strategy is:

Left-click on Node B (which is connected to the negative Node A). This will increase Node B's value to 2 and bring Node A up to -1.

Right-click on Node C (which is now connected to a less negative Node A). This will decrease Node C's value to 0 but bring Node A up to 0, achieving your goal.

Remember, this is just one possible solution, and the optimal strategy will depend on the specific layout and values of your graph.

With these tips and strategies in mind, you should be well on your way to mastering The Bitcoin Game!