Math with Many Right Answers

Torsten, math teacher
The discussion matters more than the final answer.
The discussion matters more than the final answer.

One of the most persistent math myths in popular culture is the idea that mathematics is primarily about getting right answers.

The truth is, the answer doesn’t matter that much in math. What really matters is how you explain that answer. An answer is “right” if the explanation makes sense.

And if you don’t give an explanation, then you really aren’t doing mathematics at all.

Try This Number Puzzle

Here is a short sequence of numbers. Can you figure out the rule and fill in the next three blanks?

2, 3, 5, 7, ___, ___, ___, …

Remember, what’s important is not which numbers you pick, but rather how you explain your answer.

Possibility #1

Perhaps the sequence is the prime numbers?

2, 3, 5, 7, 11, 13, 17, …

The prime numbers make a wonderful sequence, though it isn’t the one I was thinking of.

Possibility #2

Perhaps the sequence is the odd numbers?

2, 3, 5, 7, 9, 11, 13, …

But how do you explain that even number 2 at the beginning of the row?

This is the answer I thought of. As far as I know, no one else has described this exact sequence, so I get to name it. Letting kids name their math is a great motivator for creativity‌—‌everyone likes to name something after themselves.

Let me introduce you to the Gaskins Counting-By Sequence. The first number tells you what interval you are counting by, and the second number tells you where to start counting. From there on, it’s automatic.

The natural numbers 1, 2, 3, 4… form a basic Gaskins Counting-By Sequence. Other examples would include any list of multiples that starts with the n×1 term.

Possibility #3

I liked my counting-by answer. Still, I wondered if I could come up with at least one more possibility, so I kept thinking. And then I found:

2, 3, 5, 7, 10, 13, 16, …

Again, I couldn’t find any reference online to this exact sequence. The OEIS had eleven almost-matches, but each list veered off after a few more numbers.

Hey, that means I got two bits of original mathematics from a single puzzle. Cool!

I named this Denise’s Differences Sequence. You count the differences between the numbers, and each difference is repeated according to its value: +1 once, +2 twice, +3 three times, etc. You can start with any two numbers, and then the difference between them determines the rest of your list.

Incidentally, the sequence of differences in my list (1, 2, 2, 3, 3, 3…) is called the Self-Counting Sequence.

Possibility #4?

What do you think? Can you come up with yet another rule to explain the numbers in this puzzle?

Or invent a new sequence. Give it a name. I’d love to hear how you explain it!

Please share your answers in the comment section below.

Math-with-Many-Right-Answers


Feature photo (top) by Blondinrikard Fröberg and math partners photo (above) by woodleywonderworks via Flickr (CC BY 2.0).

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24 thoughts on “Math with Many Right Answers

  1. 2, 3, 5, 7, 8, 11, 11, …

    This sort of sequence is very common on standardized and IQ tests: It’s interleaved, so the even numbers are irrelevant to the odd numbers. Hence there are really two sequences: 2, 5, 8, 11… (add 3) and 3, 7, 11… (add 4).

    I’ll dub it the ACT Double Helix.😀

    1. ACT Double Helix—Wonderful name! I remember seeing puzzle sequences like that. They can be very frustrating, until you look at them cross-eyed.

      I figured there would be a curve-fitting solution, but this puzzle started as something to do in my head while I was otherwise indisposed. My mental math skillz weren’t up to that…

  2. 2, 3, 5, 7, 8, 7, 3, …

    These are the y values of y = -x^3/6 + 3x^2/2 – 7x/3 + 3. This is the Cubic Curve Fitting Approach.

  3. I did use an online matrix solver for the curve fitting. I tried to do it by hand, but I messed up.🙂

    Incidentally, it appears that y is always an integer when x is an integer for this particular function. I didn’t think that was usually the case.

  4. I left a comment similar in spirit to the discussion around your initial sequence (though I began with a 1) at an earlier blog entry:

    https://mathmindsblog.wordpress.com/2015/05/26/growth-patterns-the-beginning/#comments

    For this post, I thought you could try the Fibonacci sequence with some subtraction:

    Start with 2 and 3, and then find the next entry by taking the sum of the previous two, but subtracting 0 for the first pair, subtracting 1 for the second pair, subtracting 2 for the third pair, etc:

    2+3 -0 = 5
    3+5 -1 = 7
    5+7 -2 = 10
    etc.

    Indeed, the sequence is already listed in the OEIS with an essentially equivalent formulation (“nth Fibonacci number + n” at http://oeis.org/A002062).

    2, 3, 5, 7, 10, 14, 20, 29, 43, 65, 100, …

    Here’s one not in the OEIS, and left nameless for the time being:

    2, 3, 5, 7, 8, 8, 61, …

    After starting with 2 and 3, we are multiplying two consecutive terms, subtracting off an increased cube, and then taking the absolute value.

    2*3 – 1^3 = 5
    3*5 – 2^3 = 7
    5*7 – 3^3 = 8
    7*8 – 4^3 = -8, and its abs. val. is 8
    8*8 – 5^3 = -61, and its abs. val. is 61
    etc.

    Note that we have now invoked the absolute value portion twice in a row in order to keep our sequence positive. (Does the absolute value ever rear its head again moving forward in the sequence?)

    1. I love your comment at Math Minds blog. What a great way to get kids thinking creatively about math!

      I’m curious why you chose to limit your nameless series to positive numbers, rather than letting it dip its toe into the negatives. The series grows so fast that it seems like the cube term would have a hard time catching up, but I’ve been fooled often enough that I don’t trust my intuition.

  5. RE: nameless sequence limited to positives

    The main reason is that I enjoy the idea of the same number appearing twice in a row. I also find that absolute values are under-utilized. I have had them arise twice with students somewhat recently:

    First, in discussing the meaning of “f(x) = x for all x” as differentiated from “f(f(x)) = f(x) for all x.” The former is necessarily the identity function, whereas the latter constraint is satisfied by the identity function as well as a few other examples, one of which is the absolute value function. (Another example would be a constant function, e.g., f(x) = 7 for all x; yet another example would be the ceiling or floor function…)

    Second, in coming up with function machines that involve the composition of two functions, a game of “guess the rule” proves to be not so tough if you have a line: in particular, any two points determine a line (so check the rule for x = 0 and x = 1 and you’re just about done). But try something like g(f(x)) where f(x) = 3x-2, and g(x) = -|x|. When one begins by evaluating at x = 1, then x = 2, and so forth, students may be (quite reasonably) led to believe that the rule is ‘2-3x’; however, this rule fails when evaluating at x = 0, and a bit more thought is suddenly required.

    Back to the main story: The sequence suggested earlier grows rapidly, yes, and becomes unwieldy shortly after the previous cut-off of 61. More precisely, it continues with 272 and 16249 before simple calculators begin their betrayal!

    1. Good point about the function game. Step functions are also useful for making students think.

      I followed your nameless sequence up into the millions, which didn’t take many steps, and then decided not to move to a spreadsheet. Too many items on the To Do list to get distracted with that rabbit trail…

  6. Here’s another silly way to think about it.🙂

    First you choose two consecutive numbers, in this case 2 and 3. Then you take the next consecutive number, 4, and divide 23 by 4. That is 5 and remainder 3. Ignore the remainder, and pick up 5 as the next number in your sequence.

    2, 3, 5

    Now look at the number formed by the last two items in the sequence, 35, and divide that by 5. We get 7. That’s our next number. We have

    2, 3, 5, 7

    Now take the number formed by the last two items, 57, and divide that by 6, ignoring the remainder. We get 57 / 6 = 9 R3, so 9 is the next number.

    2, 3, 5, 7, 9

    Now, take 79 / 7 = 11 R2, so the next number is 11.

    2, 3, 5, 7, 9, 11.

    Now, take 911 / 8 = 113 R7 , so the next number is 113.

    We get
    2, 3, 4, 7, 9, 11, 113, …

    This was just a silly idea that came to my mind, yet it worked… LOL :^)

  7. 2, 3, 5, 7, 10, 13, 50

    Start with 2 sticks and make the lowest Roman Numeral you can (II).
    Add 1 stick and make the next Roman Numeral you can (III).
    Take 1 stick away and make the next Roman Numeral you can (V).
    Add 2: VII
    Take 2: X
    Add 3: XIII
    Take 3: L
    Add 4: LVII
    Take 4: C (as <)
    Add 5: CVIII
    The sequence ends here, since there are no numbers higher than C that can be formed with just two sticks.

    Call it… uh… Roman Stick Count.🙂

  8. New maths, eh? Let’s see how far we can push this…

    Let D_n be the n’th element in Denise’s Differences Sequence. Consider:

    lim x->inf (D_n – 2n) / n^1.5

    I conjecture that the limit exists, and is a transcendental number. Assuming this conjecture is true, I hereby name it Gaskins’ constant.

  9. Hi, Michael. Those are included in the OEIS, and I’d seen your first one before. But they seem a bit like cheating to me — too gimmicky to count as “real” sequences, since recognizing them depends on non-mathematical coincidences like where one lives or the language one speaks.

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