Pattern and Sequence Puzzles

One of the harder types of question to answer effectively is a puzzle, which as I define it means that there is no routine way to solve it, so any hint would likely give away the answer. But sometimes these are only “puzzles” to us, because we don’t know the context that would have told us what kind of answer to expect. I have collected here several different sorts of sequence or pattern puzzles, showing various ways we can approach them. (These are often done before learning algebra, so that although variables may be used in the answers, no algebra is needed to solve them.)

Finding a formula for a sequence

Here is a typical question from 2001:

Terms and Rules

My step daughter is in sixth grade and she has been doing a pattern journal where she has two columns of numbers: the first column is the n, and the second column is the term, and she has to find the rule (e.g. n^2), etc. Is there some place I can find out the basics for this concept?

The one example offered may or may not be typical of the particular kind of “rule” she is to look for, so I started with general suggestions and a request for examples:

I don't know that there is much to say in general about this kind of problem. A lot depends on the level of difficulty; most likely she has been given a set of problems that all have a similar kind of pattern, so that a method can be developed for solving them. The problems can differ in the presentation (whether consecutive terms are given, for example), and in the complexity of the rule (a simple multiplication, a more complicated calculation from n, a recursive rule - based on the previous term - or even a weird trick rule like "the number of letters in the English word for the number n"). For that reason, it would be very helpful if you could send us a couple sample questions so we could help more specifically.

I started by offering a particularly simple example:

I don't see any good examples in our archives at the elementary level - probably just because we've never felt that any one such answer was useful to help others. Let's try a few samples. Here's an easy one:

     n | term
    ---+------
     1 | 3
     2 | 6
     3 | 9
     4 | 12

Here you may just be able to see that the terms are a column in a multiplication table (or, more simply, that all the terms are multiples of 3); or you might look at the differences between successive terms (a very useful method at a higher level, called "finite differences") and see that the terms are "skip-counting" by 3's, a clue that the rule involves multiplication by 3. However you see it, the rule (using "*" for the multiplication sign) is

    3 * n

At that level, there is no method needed for solving; mere recognition is often enough.

Here's a slightly more complicated one:

     n | term
    ---+------
     1 | 6
     2 | 11
     3 | 16
     4 | 21

Here the differences are all 5, suggesting multiplication by 5. You might want to add a new column to the table so you can compare the given term with 5n:

     n | 5n | term
    ---+----+------
     1 | 5  | 6
     2 | 10 | 11
     3 | 15 | 16
     4 | 20 | 21

Now you can see that each term is one more than 5n, so the rule is

    5n + 1

I particularly like this method, which I think of as building up the formula step by step. The “skip-counting” tells us the formula should be related to multiplication by 5, so we can just compare the given sequence to 5n, and observe that we have to add 1. Any linear sequence can be found this way.

Now they might get still more complicated:

     n | term
    ---+------
     1 | 2
     2 | 5
     3 | 10
     4 | 17

Here the differences aren't all the same (3, 5, 7), so something besides multiplication and addition must be going on. I happen to know that when the differences are themselves increasing regularly (by 2 each time in this case) that there is a square involved; try adding a column for the square of n and see if you can figure it out.

This just gives a small taste of what these puzzles might be like. They can get much harder. Actually, in many harder cases I think the problems can be very unfair, because in reality (when you don't know ahead of time what kind of rule to expect) the rule could be absolutely anything, such as "the nth number on a page of random numbers I found in a book"! It's really just guesswork, and sometimes it's really hard to say which of several possible answers is what the author of the problem might have had in mind. But at your daughter's level you don't have to worry about that; it may be mostly just a matter of recognizing familiar patterns such as skip counting or squares. It's only where I sit, seeing problems completely out of context, and having to make a wild guess as to what sort of rule to look for, that these problems are always challenging.

Finding a formula for a process

William responded to my request for examples:

Thank you, Dr. Peterson. That is exactly what I was talking about and you gave me great examples. One question I do remember was connected pentagons 1 = 5, sides -2 = 8 - 3 = 11 etc. where n is the number of pentagons and the term is the number of sides exposed, and then we needed to find the rule for 100 pentagons. Your answer definitely helped.

This is really an entirely different type: not just a list of numbers, but numbers generated by a defined process. This is no longer just a puzzle, since we have not just the first few terms, but a way to make all of them. There is definitely one correct answer.

He’s describing (a little cryptically) a table:

n | term
--+-----
1 |  5
2 |  8
3 | 11

Presumably the problem was to count the “exposed sides” of a sequence of pentagons like these:

Even without a formula, by continuing to add more pentagons in a row we could get as many terms as we want. I responded:

My answer is most helpful if you are just given a list of terms, and have to guess the rule. Your example is actually easier in some respects, and harder in others. It's common to just make a table of terms based on the geometry of the problem, and then try to guess a rule from that. But how do you know that the rule is really the right one, when it's not just an arbitrary list of terms, but one generated by a real situation? You really need to find some reason for connecting the rule to the objects you are counting. For that reason, I recommend trying to find a rule in the counting process itself.

How do you count the exposed sides? You start with 5; then when you add a second pentagon, you add 5 more sides, but one side from each pentagon becomes "hidden"; so the new number is 5 + (5 - 2) = 8. When you add another (presumably always on the "opposite" side of the pentagon, to avoid overlapping with those already placed), you again add 5 and take away 2; so the rule seems to be that you start with 5 and add 3 for each pentagon after the first. This leads directly to the rule, with no need to guess.

So you can use the guessing approach if you want, and I suspect many texts and teachers expect that, but it's much more satisfying to skip the tables and really know you have the right answer!

So the rule, from the process I described, is \(a_n = 5 + 3(n – 1)\), which can be simplified to \(a_n = 3n + 2\).

Using my table method, we compare to 3n because terms increase by 3, and get

 n | 3n | term = 3n + 2
 --+----+-----
 1 |  3 | 5
 2 |  6 | 8
 3 |  9 | 11

This is correct; but it does not in itself guarantee that it will work for all n.

Tricky sequence puzzles

Here is another question (2002) requesting general guidelines for sequence puzzles, but turning out to need an entirely different type of answer:

Finding the Next Number in a Series

I haven't been able to find much information about an approach or method in determining the "next" number is a given series of numbers, e.g., 

  9, 5, 45, 8, 6, 48, 6, 7...   What is the next number?

I can usually figure it out but if there is a formal way that makes it easier I would love to know about it!

In the previous question, we were looking at fairly routine types of sequences. This one looked different. I started with some general comments about non-routine puzzles:

Mathematically speaking, problems like this are impossible. Literally!

That's because there is no restriction on what might come next in a sequence; ANY list of numbers, chosen for no reason at all, forms a sequence. So the next number can be anything.

A question like this is really not a math question, but a psychology question with a bit of math involved. You are not looking for THE sequence that starts this way, but for the one the asker is MOST LIKELY to have chosen - the most likely one that has a particularly simple RULE. And there is no mathematical definition for that.

I meant that literally: the next number could be anything at all, and we could find a rule to fit the sequence:

If you just wanted _a_ sequence that starts this way, but can be defined by _some_ mathematical rule, there is a technique that lets you find an answer without guessing. This is called "the method of finite differences," and you can find it by searching our site (using the search form at the bottom of most pages) for the phrase. It assumes (as is always possible) that the sequence you want is defined by a polynomial, and finds it. Sometimes this is what the problem is really asking for.

As we explain in detail elsewhere, any list of n numbers can be generated by a polynomial with degree \(n-1\) or less. But the coefficients will often be ugly, and it is very unlikely that it is the intended answer for a puzzle of this type. That, however, is not a mathematical conclusion, but a psychological one!

But often, especially when many terms are given, there is a much simpler rule that is not of polynomial form. Then you are being asked to use your creativity to find a nice rule. Sometimes starting with finite differences gives you a good clue, even if you don't end up with a polynomial; just seeing a pattern in the differences can reveal something about the sequence. Other times it is helpful to factor the numbers, or to look at successive ratios. Here you are doing a more or less orderly search, in order to find something that may not turn out to be orderly.

Some puzzles like this are really just tricks. The "rule" may be that the numbers are in alphabetical order, or that each number somehow "describes" the one before, or even that they are successive digits of pi. In such cases, you have to ignore all thoughts of rules and orderly solutions, and just let your mind wander. This is sometimes called "lateral thinking," and it's entirely incompatible with "formal methods"!

I’ll be looking at some of these ideas in a later post. In this case, after suggesting things that might work, I got around to trying the specific sequence we’d been given, and got a surprise:

I first assumed the specific sequence you gave was just a random list of numbers, rather than a real problem, so I shouldn't bother looking for a pattern. But glancing at it, I see that it is not random:

    9, 5, 45, 8, 6, 48, 6, 7, ... 

I see some multiplications here:

    9 * 5 = 45, 8 * 6 = 48, 6 * 7 = __

I can't recall what chain of reasoning my mind went through to see that, but it may have helped that my kids asked me to go through a set of multiplication flash cards an hour or two ago. And focusing on the few larger numbers, thinking about how large numbers might pop up (multiplication makes bigger changes than addition), probably led me in the right direction. I don't recall seeing anything quite like this presented as a sequence problem before, but seeing factors, one of my usual techniques, was the key.

As I had suggested, I just let my mind wander. And once I saw it, I realized I couldn’t just give a hint, unless it was as vague as, “Think about multiplication”. But I also realized something else:

In this case, you were apparently just asked to find the NEXT number, so we're done as soon as you fill in my blank. It may well be that there is no pattern beyond that; the choice of 9, 5, 8, 6, 6, 7 may be random. That's a good reminder that we have to read the problem carefully and not try to solve more than we were asked. We weren't told that there was any pattern beyond the next number!

A general procedure

Here’s another question that elicited a collection of ideas from Doctor Wilko, in 2004:

Thinking about Number Pattern Problems

How do I figure out the next 2 numbers in the pattern 1, 8, 27, 64, ____, ____?

Using addition, the pattern doesn't add up.  If I use multiplication, I'm stumped.  I get 1x1; 4x2; 9x3; 16x4;, but I don't understand what the next number would be to use for ___x5 and then ____ x6.

Teresa was familiar with some particular kind of pattern; this didn’t fit. What do you do when you run out of ideas? Doctor Wilko started with something like what Teresa had probably done:

Sequence or pattern problems can be anywhere from very challenging to fairly easy.  This is a matter of perspective and depends on how many problems like this you've been exposed to.  The more you do and see, the less challenging these types of problems become.

I'll tell you how I approach a problem like this. 

The first thing is that I know there is a pattern, so that is my goal,to find the pattern.

I might see if there is a pattern using addition.  

          1, 8, 27, 64, __, __, 
           \/ \/  \/  \/  \/
           +7 +19 +37  ?   ?

Hmmm, I don't see any pattern using addition.

He is not necessarily looking for the same number being added, but for some sort of pattern in the differences. This is a classic first step.

Next, I might try to use multiplication.  

The question is:  Is there a number that I can multiply each term by to get the next term?  To get from 1 to 8, there is only one number that I can multiply 1 by to get 8 - it's 8.  But I also notice that I can't multiply 8 by any other number to get exactly 27 (no remainder).

            1, 8, 27, 64, __, __, 
             \/ \/  \/  \/  \/
             *8  *?

So far, I don't see a pattern with using addition from term to term and I don't see a number that I can multiply each term by to get the 
next.

I would quickly do a similar check for subtraction and division, but I don't find anything that works there, either.

Very often the ratio of one term to the next is not nice, so we can give up this idea quickly.

Now, I'm going to look at the position of each number, as sometimes this will reveal a pattern.  1 is in the 1st position, 8 is in the 2nd position, ...(see below)

            1,   8,   27,   64,   __,   __, 
            |    |     |     |     |     |
            1    2     3     4     5     6  

I might ask,
 
  Does 1 have anything to do with 1?
  Does 2 have anything to do with 8?
  Does 3 have anything to do with 27?
  Does 4 have anything to do with 64?

This might go somewhere!

  1 is a divisor of 1.
  2 is a divisor of 8.
  3 is a divisor of 27.
  4 is a divisor of 64.

So maybe multiplication will work, but not in the way I tried above!  I'm going to factor each term in the sequence to look at the divisors.

   1:    1     =      1
   8: (4 * 2)  = (2 * 2 * 2)
  27: (9 * 3)  = (3 * 3 * 3)
  64: (16 * 4) = (4 * 4 * 4)  

The 4s could be factored further, but I'm starting to see a pattern emerge, so I'll keep it as (4 * 4 * 4).

If this pattern continues, can you find the 5th and 6th term of the sequence?  (I rewrote the 1's to match the pattern.)

   1  =  (1 * 1 * 1)  
   8  =  (2 * 2 * 2)
  27  =  (3 * 3 * 3)
  64  =  (4 * 4 * 4)
   ?  =  (? * ? * ?)
   ?  =  (? * ? * ?)

Some people will immediately recognize the numbers in this sequence as cubes; others will need to go through the factoring and notice that factors come in threes. Your level of experience determines how hard this will be.

Notice, though, that Teresa had done almost exactly what Doctor Wilko did. If she had noticed that the multipliers 1, 4, 9, and 16 she found are squares — in fact, the square of n — she might have had the answer.

Doctor Wilko concluded:

Does this help?  The basic idea in number pattern problems is to keep trying possible patterns.  As I said at the top, the more of these you do the easier it gets to see the patterns.  As with most things, experience and practice pays off!

Once you have seen enough patterns, you will start recognizing those you have seen before, and have a longer list of things to try in the future.