My daughter brought home the "problem of the week" last night and it was explained to me as this:
Given the following digits: $$1\ \ 1\ \ 2\ \ 3\ \ 3\ \ 4\ \ 5\ \ 6\ \ 6\ \ 7$$
Arrange them in this equation to make a correct answer: $$\_\ \_\ \_\ \times \_\ \_ = \_\ \_\ \_\ \_\ \_$$
I did find 4 possible solutions, and will put them at the bottom of this post, don't look if you want to try it yourself first. My question is more about the method to solve this, not the answer.
So I did a little homework, and found out that there are about 3.6 million ways to arrange 10 characters ($10!$). However, since the $1$, $3$, and $6$ are repeated, it knocks it down to a much more manageable ~$470,000$ possibilities.
Here teacher told her that there was only one correct answer.
I tried to just use some common sense to narrow the possibilities down but it still seemed like there were way too many permutations to try.
So I decided to brute force it with a Java program. After spending some time working on a few performance tweaks, I can attempt all ~$470,000$ combinations in about 2 seconds on my mac. I found that the teacher was incorrect and there are 4 possible permutations that are correct.
How would an 11-year old student with no programming experience solve this problem? Is there some "new math" ;) that I don't understand?
$$617 \times 43 = 26531$$ $$667 \times 23 = 15341$$ $$653 \times 37 = 24161$$ $$637 \times 23 = 14651$$