When did you learn that there are still unsolved problems in mathematics?
Do you have a favorite unsolved problem?
For me, it’s a close tie among the Goldbach Conjecture, the Twin Prime Conjecture, and the 3N + 1 Conjecture. It’s thrilling to play around with these problems that, despite being so simple to state, have stumped mathematicians for decades or centuries. Yet many students (and adults!) never learn that mathematics is a living, growing subject.
The 3N + 1 Conjecture is especially appealing because all you need to play around with it is a little bit of arithmetic. Thanks to this inviting and accessible conjecture, even six or seven year olds can encounter their first unsolved problem—while having fun playing this printable math board game for all ages.
The 3N + 1 Conjecture
Start with any positive integer. If it’s even, divide by 2. If it’s odd, multiply by 3 and add 1. Continue the process. So if we start with 6, we get the sequence 6, 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, …. From here, 4, 2, 1 will cycle indefinitely, so we might as well just stop if we hit 1.
Let’s try starting with 7. Then we get the sequence 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.
Try some more examples. What do you notice?
The 3N + 1 Conjecture proposes that no matter what positive integer you start with, you will always get to 1 eventually. The conjecture has stumped mathematicians for close to a century, although the mathematician Terence Tao recently made exciting progress.
What do you get when you cross Sorry! with the 3N + 1 Conjecture? 3N + FUN.
3N + FUN is accessible to early elementary students, and fun for older students and grown-ups too. (Click to download if you haven’t already.)
For the younger players, it’s a great opportunity to develop concepts of odd and even numbers, halving, and multiplication. While players should have at least a decent understanding of two-digit numbers, they can develop fundamental ideas of dividing and multiplying as they play.
To play, you’ll need two standard dice, 6 game pawns or tokens in two colors (three of each), and the assembled game board.
The game starts with all 6 pawns placed randomly on the squares from 2 to 12. To place the pawns, take turns rolling the two dice. Place a pawn on the sum of the two dice. If there is already a pawn in that space, roll again.
The two players take turns moving one pawn on each turn. The pawn moves with the following rules:
- If the pawn is on an even number, move the pawn to half that number.
(e.g. 8 ⟶ 4)
- If the pawn is on an odd number, move the pawn to triple the number plus one.
(e.g. 5 ⟶ 3×5 + 1 = 16)
On your turn, choose which one of your pawns to move. Move that pawn one step, following the rules above. Your pawn may not land on a number already occupied by one of your own pawns.
And here’s the fun part! If your pawn lands on a number occupied by your opponent’s pawn, bump that pawn to an available number from 2 to 12 of your choice. Choose wisely!
If your pawn lands on 1, place that pawn on one of your home base spots. Your goal is to get all three of your pawns to 1 and safely home!
Asking Good Questions
3N + FUN is an invitation to start exploring this amazing unsolved conjecture. Questions should come up naturally as you are playing. While there’s no need to force any questions, here are some examples of questions you could listen for or model yourself:
I wonder if you can land on two even numbers in a row? How about two odd numbers in a row?
I wonder what number I should put this bumped piece onto?
I’m noticing that all our pieces are getting to 1! I wonder what would happen if we started on a different number?
I wonder what would happen if we change the rules? How about 5n + 1 for odd numbers / divide by 2 for even numbers or … ?
I wonder if we could draw some sort of picture to keep track of where all the numbers go?
There’s plenty to explore here! What will you and the little ones discover?
Tips for Playing with the Youngest Players
Finally, here are some tips to try out if you are playing with very young children. Kids can enjoy 3N + FUN long before they master division and multiplication in school. A set of snap cubes or unifix cubes will help the youngest kids work successfully with the trickier calculations as they develop concepts of multiplication and division.
For example, one of the more challenging even numbers that comes up in the game is 34. A young player might not know how to split it in half, or even whether it does split in half or not. But they can figure this out with snap cubes and perhaps a little guidance.
Look, you’re on 34. Where should you move your pawn?
I wonder if we can split 34 in half? What do you think?
Here are 34 cookies. Can we share them fairly? Try giving some to me and keeping some for you.
Let’s check if that’s fair. How many cookies do we each get?
So what is half of 34?
Great! Go ahead and move your pawn to 17.
Similarly, you can support any of the multiplication calculations with snap cubes as well. The most challenging multiplication that comes up in the game is 3 x 17. (For the youngest players, you might even say things like “three 17s” instead of using the language of multiplication.) Here’s how that might go with snap cubes.
Look, you’re on 17. Where should you move your pawn?
Let’s make 17. Does that split in half or not?The child can try “sharing the cookies” as above. Since we are working with integers, one cookie cannot split into halves here. And a snap cube does not split in half either.
Since 17 is odd, we need to make three 17s and then add 1.
How much are three 17s?
How many 10s do you see? What could we do with the three 7s?They might regroup in tens, or count one-by-one, or add 7 + 7 = 14 and then add 7 more, or count three (5 + 2)s, or use even another strategy. Just follow their lead.
So how many is this? How many 10s do you see? How many 1s?
Now remember to add 1.
Great! Go ahead and move your pawn to 52.
With some practice and support, young players will soon feel comfortable enough with the game play to start noticing, wondering, and investigating new math of their own.
(Standards: 3N + FUN addresses second and third grade Common Core Standards CCSS.MATH.CONTENT.2.OA.C.3 / Work with equal groups of objects to gain foundations for multiplication and CCSS.MATH.CONTENT.3.OA.C.7 / Multiply and divide within 100 .)