What’s the pattern? – 8, 16, 77, 154, 603, 909, 1818, …

What are the next three terms?

If you’d like to take a crack at figuring it out, I recommend grabbing a pencil and some paper and pausing your reading here.

Patterns like this get my blood pumping. These little puzzles require dexterous use of mathematical reasoning, problem solving skills and creative thinking. And they do this all while remaining both accessible to younger children and challenging to adults. The spectrum of mathematical operations and methods is wide open, we aren’t restricted in what we could use to begin trying to tease apart this pattern. You can have the students work individually, in groups, or have students begin alone, but have groups grow as certain milestones are hit. The discussion between students as they engage with a puzzle like this tends to be rich and fascinating. Furthermore, from the teaching point of view, a pattern like this has a lot of flexibility in the types of questions that could be asked about it and that means we can dig deep into a single pattern with students, or proceed through more patterns rapidly depending on the skill we want our students to achieve. We could ask questions such as:

• What are the next X terms?
• What are the differences between numbers? Is there a secondary or underlying pattern present?
• What is the 12th (or any number of your choice) number in the pattern?
• What should we do first to analyze this pattern?
• Is the pattern consistent from one step to another?
• Can we organize this pattern in a better way?
• Do we spot any initial similarities or differences?
• What are the characteristics of the numbers in the pattern? Do we see any common items we could

Each of the questions above can lead the students to reason through the problem in different ways or focus on certain skills you may target.

Number sequence patterns like this also tend to have a funny quality to them. If we get sufficiently mathematical (think layers of operations), we can likely shoehorn a complicated calculation that may be able to fit. There are likely a large number of different calculations that could be performed to do this. It’s good to recognize that they do have some drawbacks, as modelling a pattern like this concretely tends to be difficult (since the numbers get large).

Let’s go back to the original pattern of 8, 16, 77, 154, 603, 909, 1818, … In my opinion, the pattern above is interesting for a few reasons (you may have more than this):

• It originally gives you a significant number of terms to work with.
• It features some ambiguity about what the core of the pattern could be. After a quick glance, it appears like some terms might be doubled (8 going to 16, 77 going to 154, and 909 going to 1818) but there are also strange changes between certain numbers where we don’t see an initial pattern.
• It seems to be increasing.
• Does the pattern begin at 8? Or does the pattern begin at 0? Zero isn’t identified in the pattern, but we’d likely be safe to assume that’s the reference point.
• We have several options for beginning our analysis of the pattern above.

What follows is an account of students working on this problem. They begin to break this pattern down by examining how the numbers change from one step to another and organize the information in a table. (why might this have been a good place to start?)

Term	Number	Difference From Previous Term (Term(n+1) – Term(n))	Difference Between Differences
1	8	N/A, Students may assume Term 0 is 0.	N/A
2	16	8	N/A
3	77	61	53
4	154	77	16
5	603	449	372
6	909	306	-143
7	1818	909	603
8	??	??	??
9	??	??	??

Student’s decide that examining the column on the far right appears to be a bit fruitless. There doesn’t seem to be a relatively accessible pattern there at first glance (or even upon closer inspection), so we may abandon engaging with that set. We might be back later, but for now let’s focus on the the first column of differences (8, 61, 77, 449, 306, 909, …).

At this stage, you see some students begin examining the sequences of differences. They notice the numbers 8, 77 and 909 appear in both sequences. They may notice that the numbers in the differences are mostly increasing but not always increasing (the jump to 449 from 77 and then back down to 306). They may begin to notice (after examining both the original pattern and the differences) that the numbers of 16 vs. 61, 603 vs. 306 and 909 vs. 909. They go back and see that this is the case for the 8 vs. 8 and the 77 vs. 77. The students are sure they’ve spotted the pattern. You are always adding the reverse of the original number to get the next number. One student even begins typing the pattern into search engines and discovers the RATS sequence. They really want to say that they have figured it out and the only thing they need to do, is ignore that pesky difference of 449 which isn’t the reverse of 154. So that may be exactly what happens.

You are accused of making a mistake in the pattern. I am reminded of images of Principal Skinner “it’s the children who are wrong”, but in this instance it’s the students saying “the teacher must be wrong”. While teacher’s make mistakes like everyone else, you assure the students that there is no typo or error in the sequence you’ve provided them. You mention that if you had made that mistake, then the next terms are all wrong as well – 603 would turn into 605 as you would be adding 451 to 154 instead of 449. This would then extend through the rest of the pattern and they wouldn’t be able to use the pattern they determine in that case. Some of them groan and mutter, threatening to give up on the problem. Others furrow their brows, stepping back from the problem and trying to make new meaning in the numbers. You mention to the students that they should continue their investigation.

While some of the students need some extra encouragement to continue, they dig back into struggling with the pattern. They are talking with their classmates, comparing notes and different things that they have tried. New strategies emerge and die out. Leads are pursued and ruled out. Finally, a breakthrough! One of the groups thinks they have it. The classroom hums with anticipation. Their productive struggle has paid off! Well, maybe.

• Term One = 8
• Term Two = (Term One x 2) = 16
• Term Three = (Term Two x 4) + 13 = 77
• Term Four = (Term Three x 2) = 154
• Term Five = (Term Four x 4) – 13 = 603

The student’s postulate that there are two concurrent patterns running through this. Odd terms are simply multiplied by 2 to get the next term. Even terms are multiplied by 4 and then 13 is added or subtracted. Unfortunately, the students didn’t test Term Six = 1206 to recognize that it fails the pattern, or maybe they did test it but decided to “solve a simpler problem” and then hopefully use it to garner some additional clues about the pattern from the teacher. At least, they found something that fits the majority of the pattern and undeterred, the students bring the (incomplete) pattern to your attention.

You ask them what they think about their pattern when it comes to the last two terms in the sequence. The students indicate that they follow it. Then they backtrack. Oh wait. Ugh. The students head back to the drawing board, but they feel like they’ve made some progress on the pattern. With some prompting, they begin to try and combine the reverse and add sequence discovered earlier with the new pattern. Discussions continue that move the students further down this path.

However, other students start taking some different approaches. What if the pattern isn’t in the individual numbers, but in the digits themselves? They begin adding the digits together to attempt to figure out if there is a pattern present there.

• 8 – digits sum to 8
• 16 – digits sum to 7
• 77 – digits sum to 14
• 154 – digits sum to 10
• 603 – digits sum to 9
• 909 – digits sum to 18
• 1818 – digits sum to 18
• New pattern of 8, 7, 14, 10, 9, 18, 18

There doesn’t immediately appear to be a pattern there either. Others begin examining the characteristics of the numbers, looking at whether it’s the number of digits, value in a specific position or if the number is even or odd.

• 8 – even, one digit, 8 groups of one
• 16 – even, two digits, 1 group of ten, 6 groups of one
• 77 – odd, two digits, 7 groups of ten, 7 groups of one
• 154 – even, three digits, 1 group of one hundred, 5 groups of ten, 4 groups of one
• 603 – odd, three digits, 6 groups of one hundred, no groups of ten, 3 groups of one
• 909 – odd, three digits, 9 groups of one hundred, no groups of ten, 9 groups of one
• 1818 – even, four digits, 1 group of one thousand, 8 groups of one hundred, 1 group of ten, 8 groups of one

Now an interesting pattern has emerged. We see one 1-digit number, two 2-digit numbers, three 3-digit numbers, followed by a single 4-digit number. The student’s hypothesize that the next numbers should be 4-digit numbers, and following the pattern, all three of terms they were trying to locate should have 4-digits. They quickly tell the other students in the class that they’ve discovered something. You quiz them a bit on how them came to that conclusion and they explain their process to both you and the rest of the students in class.

Now the class has more information to work with. The students also note that the numbers keep increasing. Eventually a small hand is raised and they say, “I think that’s the pattern. The next three terms will be four digit numbers, and they must be increasing, but we don’t have enough information about the gaps between the numbers to figure out exactly what the numbers will be. So the next term must be larger than 1818, then the next term must be larger must be number than that one. The pattern would then have five 5-digit numbers after the four 4-digit numbers.” You smile and nod. The student’s have gotten somewhere on the problem, after much deliberation and struggle. But is the pattern completed? Did the students get the correct answer? Or is there more to discover? The students discuss for a moment and decide that this pattern has satisfied the answer to the question.

Did they get the right answer? Are we satisfied with the pattern that the student’s found? Or does the right answer even matter in the face of the brilliant display of mathematical reasoning and problem solving? How will we assess the complex skill building that just happened in this room?

Studying puzzles and patterns provides a fantastic opportunity for students to develop their mathematical reasoning muscles. But it remains a difficult item to assess for mathematics teachers.