In yesterday’s post, I shared some mistakes that I made while working on this problem:
There were some good questions and thoughts in the comments. A few people wanted to know why I added 15 and 333 to get 348 in my second solution to the problem:
I’m also really curious about is “1+2+3+4+5=15″. – Nathan Kraft
Not 100% sure where the 15 comes from. – Mr. T
Allow me to explain where that 15 comes from.
When I started the problem, my first thought was 333. I wanted to know how many 3’s would be in the prime factorization of 1000!, and I figured that I’d just count all the multiples of 3. I entered that without much further thought, and when I saw that it was wrong I realized that the other powers of 3 matter as well. In other words, I realized that I had to consider 3^2, 3^3, 3^4, 3^5, and 3^6. As so many of you pointed out, handling this realization is the key to solving the problem.
Then — and I think that this is the important psychological piece — I tried to figure out a way to use that insight in the problem. I didn’t try to think through the problem again with that insight. I didn’t think of the implications of that insight, or try to count everything. Rather, I was trying to find a use for the new insight that I had discovered.
I realized that if you have 3^2 as a factor of 1000!, that’s also going to be a multiple of 3, so it’s only going to be adding 1 factor of 3. Likewise, 3^6 (the largest power of 3 to divide 1000!) is only going to be adding 5 factors of 3. So, realizing that I needed to account for everything up to 3^6, I simply added the extra powers of 3 that 9, 27, 81, 243 and 729 would be contributing.
That’s where 1 + 2 + 3 + 4 + 5 comes from.
When this turned out to be wrong, for a moment I felt surprise. After all, hadn’t I used the insight? Once I got over myself, it didn’t take very long to get the right answer and realize where I had gone wrong.
Anyway, the point of this all is that in the midst of problem solving, sometimes I seek a way to use an insight rather than use that insight to rethink the problem. Using that insight gives me a feeling of satisfaction and a sense of completeness with the problem.
In yesterday’s comments, a few of you offered strategies for helping me through this:
I think I would ask you about your first strategy (solving a simpler problem) and ask you to explain what you noticed. – Laurel Pollard
I suppose I would push the student to specifically tell me where 3^333 and 3^15 come from? Which integers does 3^15 represent? Are we sure we didn’t miss something? – Nathan Kraft
A quick way to point out this mistake is to simply start with your list above of the first ten integers and see that the 9 creates 2, not 3, instances of a 3 as a factor. – mrdardy
But, honestly guys, anything that would’ve slowed down my thinking and encouraged me to think through the problem (rather than find a nail for my hammer) would’ve helped me get to the right answer.
I think it’s also worth considering what exactly we ought to be concerned about with a mistake like mine. My wrong answer contained the idea that would turn out to be crucial for solving the problem. I understood exponents. I understood factors. Really, the concern that we ought to have is that I’d make a similar sort of mistake when I solve the next problem — rush to use an insight rather than think through its place carefully. But what sort of feedback can a teacher give to help ensure that I am more careful in the future?
Maybe the best thing that a teacher could do would be to give me an opportunity to reflect on interesting questions. Maybe, instead of asking students for revisions or corrections, we should ask them for reflections on parts of their work that interest us, or pique our curiosity.
8 replies on “Problem Solving Strategy: Have hammer, seeking nails.”
The most interesting thing to me about your post is what you have chosen as your mistake:
“I tried to figure out a way to use that insight in the problem. I didn’t try to think through the problem again with that insight.”
It seems to me that even thinking through the problem again with that insight, you are still in too much of a hurry. And certainly if you end the problem where you did, you are missing the key synthesis about what is happening.
The insight to which you refer is that you need to keep track of, that is count, various powers of three. But this should instantly lead you to the idea of place value! — we should be thinking about whether the base 3 numeration system is a more powerful framework in which to couch the problem.
It would take a long time, perhaps too long for whatever course might touch on this problem, but perhaps a good inquiry-based lesson with lots of smaller examples would lead one to the following realization:
The exact power you are looking for is related to the sum of the digits when you express 1000 in base 3! In fact, there is a formula for this power in terms of the sum of these digits If we call the sum of the digits when x is written in base 3 sd(x), then the power of 3 dividing x! is in fact given by
1/2 ( x – sd(x) )
Even if it takes to long to stumble upon the sum of the digits as key, even the outright suggestion that a formula involving sd(x) exists might be enough to design an inquiry session in which the formula is produced by looking at lots of small examples.
Even an idea of proof can be gleaned: such a formula needs to have the property that when we switch from x! to (x+1)!, the exact power of 3 involved must go up by the exact power of 3 dividing x+1. But on the other hand, if the exact power of 3 dividing x+1 is e, then one can believe by looking at examples that sd(x+1) = sd(x) + 1 – 2e (consider first the case that e=0!) and then see that
1/2( (x+1) – sd (x+1))
must then be exactly e more than
1/2( x – sd(x) )
Thus the formula always goes up by the appropriate amount when you switch from one number to the following number. Since it is correct for x=1, it will always be true.
If you don’t at least take some of this additional journey, it seems to me all of you done is an exercise in very careful counting requiring a small amount of familiarity with exponents.
And since I can’t edit the post, let me add the following erratum: I committed the same sin myself, suggesting that the journey is at an end once you have the formula and an idea about why it is true. Perhaps a suggestion that we have reached a conclusion to a journey, rather than a waypoint, too often affords the students the opportunity to ask, “What’s the point?” Instead, leave them asking, does the same formula work if we replace 3 with something else (No.) Does a similar formula work? (Sometimes?) Does a similar formula exist with base 10? (No.) Why not? etc. etc.
Lol, I love this topic! I thought of another fun question! Why should 1/2 (x – sd(x)) always give an integer!!!
Hey Barry,
I’m loving the math that you’ve brought into discussion here, but I think that it’s a bit of a stretch to say that my mistake on this problem is that I didn’t spend a ton of time reflecting on some (beautiful) math that’s related to it.
Personal fault? Maybe. But not a mistake in the way that I solved this problem.
I wasn’t saying your eventual solution was wrong. Does “mathmistakes.org” exist to analyze the mistakes in problem solutions or the trains of thought that lead to those mistakes? It seems to me the goals is to study the way students approach problems and thereby find causes for mistakes — and that one type of desired feedback is suggestions to get students to analyze their approaches and perhaps shift them to have more effective conceptual frameworks.
We spend so much time trying to get students to understand the ideas behind place value, and yet here, where it is a natural outgrowth of the realization that we need to keep track of numbers of different power of 3, you make no mention of place value. It seems to me a huge missed opportunity not to even consider the connection.
“But, honestly guys, anything that would’ve slowed down my thinking and encouraged me to think through the problem (rather than find a nail for my hammer) would’ve helped me get to the right answer.”
-Sounds like making you work in a group or at least partnering with someone to make you explain as you go would be the best teacher strategy?
Laurel, if the goal is to for me to get the right answer on this problem, then slowing me down by having me explain as I go would certainly work.
But should the goal be that I get this problem right?
I agree that the reflections bit is really important here. What should we do after we’ve solved a problem? Check the answer, maybe, but that’s only one part of it. I like to tell students who have solved something hard to think of a sentence that they could send back in time to them-at-the-beginning that would be helpful in getting them over the part of the problem that was an obstacle. Here your sentence might be more of the form “Be careful and patient” or more like “Think of all the multiples of powers of 3” if you’re on a more problem-specific kind of approach. Still, limiting it to one sentence generally prevents it from being too overly specific. Sometimes I ask for naming a strategy, to prevent things that deal too much with this one problem, so care/patience could be it, or “make an organized list” or something like that.