A math game called 0-20

I take my 4 year old to a math circle at a nearby university. If you don't know what a math circle is, check this out: https://www.cut-the-knot.org/books/Reviews/BerkeleyMC1.shtml.

Math circles are weekly math programs that attract middle and high school students to mathematics by exposing them to intriguing and intellectually stimulating topics, rarely encountered in classrooms.

Maybe a good summary is that it's a program design to present children with fun, failure, and puzzles. All in the aim of distinguishing mathematics from merely arithmetic or doing things computers do, but by hand.

Anyways, our circles begin with a game. It looks like this:

• Make a straight line of 21 paper plates on the floor.
• Next, label the plates beginning with 0 and ending with 20.
• Select two players.
• Determine who goes first.

Now, before I explain a few more instructions, I'll make a board.

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

The two players alternate choosing how many steps to take together. In other words, the two players move together. The objective of each player is to be the one responsible for landing each other on plate 20. Finally, the last instruction is that you can only choose to move 1, 2, 3, or 4 steps at a time.

Like all games, this will be easier to grok after watching it. In fact, I'm shocked that my 4 year old ever understood this game.

To demonstrate, I'll show you exactly what my son did when playing the game for the first time. Keep in mind that this game rewards courtesy (more on that later). Below is a faithful reproduction of his first game. The names have been changed to protect identity.

Players: Dr. S, PhD & Daniel

Dr. S : Would you like to go first or second?
Daniel: First.
Dr. S : Ok, how many steps should we take? 1,2,3, or 4?
Daniel: 4.
Dr. S : Ok.

(Now it becomes obvious that Daniel has selected 4 because it provides him with the biggest number of plates to jump. He jumps, somewhat impressively, from 0 to 4. Dr. S says that he doesn't think he's that good at jumping, so he takes four steps.)

Dr. S : Now it's my turn to choose. I'll choose to go 1 step.

(Now they're at plate #5.)

Dr. S : Alright, how many steps should we take?
Daniel: 4.

(He jumps again and they make it to #9.)

Dr S : I'll take 1 step again.

(They arrive on plate #10.)

Dr S : How many steps?
Daniel: 4.

(At this point a couple of the brighter children realize that Daniel has picked a losing strategy. He doesn't notice them, blinded by the radiance of his long jumping ego. They arrive at plate #14.)

Dr S : How many steps do you think I'll take?
Daniel: 1.
Dr S :Yep.

(They arrive at plate #15.)

Daniel: 4 steps.

(They arrive at p late #19.)

Dr S : I'll take my one step. I win. Thanks for playing, Daniel.

(They exchange a high five and Daniel walks back to me, proud of his jumping.)

Here's a sequence of their steps: 0, 4, 5, 9, 10, 14, 15, 19, 20. As mentioned previously, this game rewards courtesy, not impressive hops. Dr. S won the game by going second and by knowing where to go. So long as you know what you're doing, you cannot lose the game if you go second.

I wanted my son to grok this game. I took a piece of paper and drew 20 squares on it, labelled 1-20. Then we played the game with two little people toys. We played a round and he was mildly interested in it. To get some skin in the game, I placed a mini marshmallow on square 20. After that, he was motivated. Unfortunately, I had to win three games in a row. His eyes were a bit watery, so I encouraged him that he could beat me once he understands the game.

To facilitate this eureka moment, I drew another board. This time it was with numbers from 0-5. Instead of a square, I drew a star around spot #5.

0
1
2
3
4
5

The same rules applied.

Me: Who should choose steps first?
Daniel: Me. I choose 4.

(We step to spot #4. He sees what happened and gets a bit sad again.)

Me: Let's go 1 step. Good game, Daniel. Let's play again. Do you want to go first or second?

(Daniel uses his finger to count squares: 1, 2, 3, 4. He pauses.)

Daniel: I want to go second.

(I smile.)

Me: Why do you want to go second? Isn't this game about jumping really far? Aren't you nervous that I'm going to jump 4 and you will only jump 1?
Daniel: I want to win.

Me: Hmm... I guess I'll go 1 step.

(This must've thrown a wrench in Daniel's plan. He must've assumed I was going to jump 4 squares. He's a bit rattled. He moves to #1 and then counts: 2, 3, 4, 5. He thinks.)

Daniel: I will jump 4 steps.

I congratulate him and then draw a board from 0-10, 0-15, and then I bring back the 0-20 with a marshmallow. The long story short is that it actually took him multiple days of repeatedly mastering smaller boards to see that the 0-20 was just a 0-15 which was just a 0-10 which was just a 0-5.

Nowadays the game is something he likes to play whenever we have grown-up visitors. Most of the time he announces that he's going to win when he lands on spot #5.

(We go to step

No matter which plate they take you too, you can win.

Plate #16, take 4 steps
Plate #17, take 3 steps
Plate #18, take 2 steps
Plate #19, take 1 step

But how can you guarantee landing on plate #15? By landing on plate #10. And how can you guarantee that? Land on plate #5. And how can you guarantee that? Go second. As I said, the game rewards courtesy, not mad hops.

Imagine a game board that goes from 0-5. Doesn't that make obvious the opening question: would you like to first or second? If you understand that, then you'll see that the same principle applies to a board with 0-10 plates, 0-15, and 0-20. By going second, your opponent will never get to the plate divisible by 5, but they will always make it possible for you to get to it.