One of my favorite Beast Academy problems is the Honeycomb Path Puzzle. It’s a simple concept with incredible flexibility, and gives many opportunities for challenge. The goal is to create a continuous pattern through a honeycomb grid by counting or skip-counting. Let’s see a solved example:

This type of puzzle taps into the excitement I feel when designing math problems - exploring a concept deeply and figuring out how to make it more engaging and more challenging.

Let’s consider a typical skip-counting problem:

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Skip-count by 5’s from 5 to 50:
5, ____, ____, ____, ____, ____, ____, ____, ____, 50

We may want to take a question like this, and make it more challenging. And I try to avoid certain common pitfalls I often see in traditional math lessons:

1. Making the numbers awkward or large: While this can make a problem more difficult, it doesn’t necessarily make it more interesting. The following examples might be more complex, but they lack the fun and intrigue that keeps students engaged.
   • Skip-count by 7’s from 7 to 70.
   • Skip-count by 13’s from 52 to 195.‍
2. Switching to a different topic: Adding complexity by moving to a new skill—such as switching from skip-counting by 4’s to counting by 1/4’s - shifts the focus away from mastering the current concept.‍
3. Contrived real-life scenarios: Consider this word problem:
   "A football team start on the 5 yard line, and move 5 yards each play until they are on the 50 yard line. What other yard lines will they start plays on?"
   While real-world problems play a crucial role in a math curriculum, they don’t always add depth. And if this question follows a series of basic skip-counting questions, it’s pretty easy for a student to guess what’s happening without even engaging with the question.

I believe any topic can be made more challenging without relying on these common tactics. Honeycomb Path Puzzles are a perfect example, especially since they often target one of the most basic early math concepts - counting! These puzzles can be adapted in various ways to meet students at different levels while encouraging deeper mathematical thinking. Some adaptations include:

·      Show a completed puzzle and ask students to trace the path.

·  Provide a path and have students fill in the missing numbers.

·  Offer a partially completed grid and let students deduce the rest.

·  Remove both the starting and ending numbers and challenge students to figure out where the sequence begins and ends.

So, let’s try and solve my favorite Honeycomb Path Puzzle.

Complete the Honeycomb Path Puzzle, skip-counting by 4’s!

In this case, I don’t know the smallest or largest number, but I do know some missing values like 36, 44, and 52. If I’m using this puzzle in a classroom, I might pause here to ask, “Which number should we place first?” Since 36 is the smallest of those, that might be a good place to start – often trying the most extreme example is a good place to find an opening. But in this particular puzzle 44 and 52 are better starting points, as they’re more restricted in terms of possible placement – 52 must go in the one hex that touches 48 and 56. Once they’re placed, 36 becomes easy to locate.

Now we have a complete sequence from 32 to 56, but we still don’t know if the sequence continues above 56, below 32, or both. The hexagon that catches my attention next is the one that touches both 32 and 56 - this number could be 28 or 60. I might first guess 60, but that creates a dead-end. So, I go back and try 28, which opens up the path and leads to a successful solution.

Solving a problem like this brings a special sense of satisfaction that I believe is unique to math. It encourages students to engage with specific math skills while also practicing broader problem-solving techniques. On the surface, this task asks students to practice skip-counting, but a deeper look lets us see how students can engage with key mathematical habits - analyzing knowns and unknowns, checking guesses, eliminating options, and strategically narrowing their focus to a productive starting point. I’ve heard problems like these compared to piano etudes - designed to develop a particular skill while letting students enjoy the music and see how their abilities fit into the bigger picture of the domain.

The only downside to solving a puzzle like this is that once it’s done, it’s over. But what makes this honeycomb path puzzle my absolute favorite is that the next challenge is exactly the same, with a small twist:
Complete the same honeycomb path, this time skip-counting by 2’s!

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