2, 4, 6 Puzzle

The 2-4-6 Puzzle teaches an invaluable lesson about inductive reasoning, confirmation bias, and how “wrong” answers lead to deeper knowledge. It can be adapted with different rules for endless play options and difficulties.

Also see Josh Golden’s writeup of Eleusis Express for a card game version of the game,
perfect to play with standard cards, SET cards, or Tiny Polka Dot cards.

The 2-4-6 Puzzle comes from an experiment by Peter Watson done in 1960 on confirmation bias. It makes a remarkably good activity for a classroom, and one that can help set a very positive tone.

In particular, the 2-4-6 Puzzle reveals confirmation bias, and shows how making mistaken conjectures is the critical way to find out what’s really going on.

The teacher challenges the class to guess a rule that she knows. To find it, they can offer triples of numbers and she’ll tell them whether they fit the rule or not. To start, she puts forward the triple 2, 4, 6, and announces that it fits the rule. The class guesses for as long as they want. When everyone is convinced that they’re sure what the rule is, they can guess. However, they’re only allowed one guess per day, so if they’re wrong, they have to wait until tomorrow to have another shot at finding out what the rule is.

What’s the rule? The numbers have to be in ascending order. So 8, 10, 12 follows the rule, but so does -1, 121, 130.5. On the other hand, 2, 2, 3 and 3, 2, 1 fail. Sometimes students will guess the rule in 10 minutes, sometimes it will take much longer, and sometimes people will be convinced that there’s a much more complicated.

When students are ready to make their guess, let them. If it’s right, tell them. If it’s wrong, stop the game for the day, and say they can come back to it later. This will teach them to take their guess seriously.

What this game shows is that we all have a tendency to avoid “wrong” guesses, and favor safe guesses, which will follow the rule. Having a discussion about which guesses were most useful can be fascinating. Often, it was the guess that seemed ludicrous at the time that actually proved the most helpful. Being willing to be wrong, and taking feedback as data rather than as judgment, in other words, lets us learn faster. This lesson is so valuable in a math class that it can be worth doing this game early to set a positive tone.

1. Record every guess. Have a board set up for guesses that work, and another set up for guesses that don’t.
2. Don’t give hints.