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Bumper Cars

Sharpen your strategic thinking skills by driving a bumper car up the road.

Ages: 4+
Players: 2
Time: Under 10 Minutes
Type: abstract games
Location: tabletop
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Ages: 4+
Players: 2
Time: Under 10 Minutes
Type: abstract games
Location: tabletop

Instructions

This is a game of simple rules and complex strategy. Start by drawing a road with some dots on it. (Starting with 8 dots is a good idea.)

Curvy line with dots drawn on it

This is a one-way road. Cars can only move forward.

Place 2 red cars and 1 blue car on dots along the road. We used LEGO bricks, but anything will do. (They don't even need to be blue and red as long as 1 of the tokens is different from the others.) The blue car can't start in front.

Curvy line with dots drawn on it and two blue pieces and one red piece

Now, take turns. On a turn you can choose to do one of the following:

  • Choose any car, and move it forward any number of open dots. Passing or jumping over other cars is not allowed.

A hand moving a red piece up the line
Moving a car forward

  • Or pick up the car that is in front. "In front" means the car that is closest to the front of the road, whether or not that car is on the very last dot.

A hand picking up the front red piece
Picking up the front car

Whoever picks up the blue car wins the game.

The blue piece is the farthest forward
The next player wins by picking up the blue car!

Play multiple games. Try different numbers of red cars, and different arrangements of all of the cars (but remember, the blue car can't start in front), and try roads of different lengths. Take turns going first, and try to decide whether the first or second player has an advantage. To figure this out, try starting small, then moving to games with more cars and more dots. See Learning Notes below for some ideas about the strategy behind this game.

Bumper Cars is a variation of the classic math game Nim (and so are other BA Playground activities, including Anywhere Nim and our upcoming activity, Odd Knights).

Don't forget: it's Beast Academy Playground, not Beast Academy Study Hall. Change the rules, be silly, make mistakes, and try again. The Variations and Learning Notes are here for you if you want to dive deeper, but not all of them apply to learners of every age. The most important thing is to have fun.

Variations d

More Dots or More Cars:

After playing a simple version of the game, try drawing roads with more dots, or try adding more red cars. Also try out different starting arrangements.

An arrangement of red and blue pieces on a curvy line
An arrangement of red and blue pieces on a curvy line
An arrangement of red and blue pieces on a curvy line
An arrangement of red and blue pieces on a curvy line
An arrangement of red and blue pieces on a curvy line

Classroom Tips d

Play Bumper Cars (or any of our abstract strategy games) after a test or quiz. Once students know these games, they can play them together in pairs or groups if they finish a task early.

Discussion Questions

  • Does it matter how many dots and/or cars there are?
  • Does it matter if there are an even or odd number of dots and/or cars?
  • How do you know who will win before the game is over?
  • Can you find a starting arrangement that guarantees Player 1 a win? What about Player 2?
  • Can you think of your own variations for this game?

Alignment with Beast Academy Curriculum

  • Level 2, Chapter 12: Problem Solving
  • Level 3, Chapter 8: Division (students learn Nim in this chapter)
  • Level 4, Chapter 6: Logic

See Variations and Learning Notes for more ideas on how to adapt this activity and incorporate it into your classroom.

Learning Notes d

Strategic Thinking:

Play several rounds of Bumper Cars. As you play, ask your child to tell you when they know who will win. Even young players will notice that whoever picks up the last red car in front of the blue car will lose. Let's call such a car the "Bummer Car" since, if you have to pick it up, then, bummer, you'll probably lose. In fact, the only way you wouldn't lose after picking up the Bummer Car is if your opponent were to make a mistake. Of course, in real life, mistakes happen. To understand how a game works, though, we want to plan how to win without relying on our opponent's mistakes, so let's assume, as we think about Bumper Cars, that both players are playing without making any mistakes.

An arrangement of red and blue pieces on a curvy line
The "Bummer Car," circled in red, will eventually end up in the very front space. Someone will pick it up (and maybe say "bummer"), which will allow their opponent to pick up the blue car.

Since picking up the Bummer Car leads to losing, no one will do so until they have to. They'll drive other cars forward if they can. So the Bummer Car only gets picked up when all other cars are moved up as far as they can go, in a "traffic jam." Whoever creates this final traffic jam arrangement will force their opponent to pick up the Bummer Car.

An arrangement of red and blue pieces on a curvy line
Traffic jam!

Depending on the starting game board, there is a lot of complexity involved in how to create the traffic jam scenario for your opponent. We encourage you to try out each of the starting game boards shown above in Variations, and any others you think of. For each, can you figure out which player will be able to create the final traffic jam?

Let's think about just one scenario: The Bummer Car in the very front space, followed by some empty spaces, then the blue car with another red right behind.

An arrangement of red and blue pieces on a curvy line

If the next player, Player 1, moves the blue car all the way up, then Player 2 will move the rear red car all the way up, creating the final traffic jam and therefore winning. So Player 1 doesn't want to do that.

But if Player 1 moves the blue car only part of the way up, Player 2 can still move the rear red car right up behind the blue car, creating the same situation, but with fewer spaces for the blue car to move. Eventually, Player 1 will be forced to move the blue car directly behind the Bummer Car, and Player 2 will be able to follow, creating the final traffic jam. So, Player 2 will tailgate the blue car as closely as possible, eventually creating the final traffic jam and winning. This means that whoever is able to create an arrangement like the one above will be able to win!

Noticing these arrangements and understanding how they will lead to the final traffic jam can help guide choices you make earlier in the game.

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Materials
  • paper
  • pencil
  • tokens (2 colors)
Learning Goals
  • strategic thinking
Common Core Standards
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Ready to level up?

Keep problem solving with Beast Academy’s full math curriculum for students ages 7–13. Check out our captivating comic book series and immersive online platform.

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Keep your entire class engaged with a full book and online math curriculum, for students ages 7–13. 98% of teachers say they’re satisfied with Beast Academy.

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