This week’s activity highlights a mathematical game demonstrated by Dr. Holly Krieger from Murray Edwards College, University of Cambridge. You can view her presentation of the game on the Numberphile Youtube channel linked here. We will also be using the Dollar Game applet developed by tech startup Hoodies to help explain the game and possible strategies.
Imagine there is a group of people where everyone in the group has either a certain amount of dollars in their pocket or they are in debt (i.e. having a negative amount of dollars in their pocket). Each person is represented by a node (or circle) with either a positive number representing the amount of money they have in their pocket or a negative number representing the amount they are in debt.
Additionally, some people in the group know each other, while some do not. People that know each other are connected by edges (displayed as dotted lines in the applet). People that know each other can give money to each other. But if a person chooses to give money, they must give a dollar to each person they know, even if it that puts them in debt!!
In Example 1, Person A knows Person B and Person C. A can only give to B and C.
In Example 2, Person A gave away dollars two times (giving Persons B and C 2 dollars each). If we wanted, we could continue to have Person A give money to B and C as many times as we want, but Person A would be put in debt.
The goal of the Dollar Game is to get every person to have 0 dollars or more, or in other words, to get everyone in the group to have no debt!
In the Dollar Game applet, there is a tutorial to help students understand the rules of the game and how to use the applet. The first few problems give the player an unlimited number of moves to get everyone out of debt. This gives students a chance to get used to playing the game without much pressure.
Trial-and-error may certainly be used to do the first two problems with infinite moves. But once students get to Example 3, problems must be solved with a limited number of moves, and trial-and-error may not always work. This means students must start thinking more strategically to solve problems.
There are many tips and strategies, including I’m sure many more that aren’t below, that will help students plan ahead in deciding which node is best to give money and where money is allocated. Attempting problems with limited amount of moves may lead to these tips/strategies:
- If there is enough money in a node to get those around them out of debt or closer to being out of debt, it is most likely a great move. In Example 3, the bottom right node can get the node with -2 dollars closer to out of debt.
- The more edges that a node has, the more money it will give to others. The less edges that a node has, the better to give money to a node that is in debt. For example, if a Node A has 3 dollars and it is only connected to Node B that is in debt 2 dollars, A can get B out of debt by themself!
- Using the node with the most edges is not always a bad thing, since lots of nodes can give back to that node. What isn’t so good is to have given to nodes that also have lots of nodes themselves, unless they have a sufficient amount of money. In Example 3, most nodes have three or more edges, but fortunately the bottom right nodes have a fair amount of money.
- Reallocating money to the node with the least amount of edges, which is the node with the 2 edges towards the top in Example 3, may potentially lead to getting one or two nodes out of debt simultaneously!
- Try not to spread the money too thin to each node. Otherwise, it could end up like Example 4 where there is one move left and there is no way to get the middle node out of debt without having another node go into debt.
Thinking ahead and working backwards are great ways to think about the problem presented in Example 3.
First of all, clicking the middle node first isn’t so bad since there are many nodes around it that have a fair amount of money. Second, it looks like the two farthest right nodes can best help the node with -2 dollars to get out of debt. Third, the best way to get the node with -3 dollars out of debt is to use the nodes that are connected above (top left node) or to the right (middle node). Using a mix of both nodes is great since the top middle node has some money.
There are many problems that students can try on the applet. The first “World” in the applet involves nodes giving money. Once students get comfortable with those problems, the second “World” has nodes taking money from each person that they are connected to! The strategies completely change when it is the opposite. You are welcome to explore and find strategies for problems where nodes take money.
Another extension is having students create their own Dollar Game graph and sharing it with their classmates so that they can try to solve it. Make sure they try to solve their own puzzles first to make sure that they are solvable. For example, if all nodes are in debt, there is no solution!
Watch students attempting to solve Dollar Game problems during Math Monday Live.