Math problem cracked!

I have to brag about a solution I outlined for a fun math problem that was posted on the Facebook page Free Weekly Math Problems.  That page, by the way, was started by my friend Soce to promote Math Bee week at the Chelsea Mind Games.

The problem was this:

Solve this problem, using each of the digits 1 through 9 exactly once:

_ _ _ + _ _ _ – _ _ _ = 0

**Note: There’s more than one answer. Can you come up with all of them?? Or at least one of them?**

Below is my solution.  Now, I understand the what of this solution, but not the underlying why.  Any math fiends are welcome to offer their wisdom.

I got a few answers, and I noticed that in all of them, the three digits of the largest number added up to 18, while the remaining six digits added up to 27 (still teasing out the exact reason why…of course, it just HAD to be something involving multiples of 9!). So I started looking at all the three-digit numbers whose digits add up to 18.

For this problem, the hundreds digit of the largest number has to be at least 3, since the smallest that the remaining hundreds digits could be is 1 and 2. None of those possibilities works, I’m guessing because of the restrictions on the hundreds digits.

There are 34 other three-digit numbers whose digits add up to 18:

981 972 963 954 945 936 927 918
891 873 864 846 837 819
792 783 765 756 738 729
693 684 675 657 648 639
594 576 567 594
495 486 468 459

For each one of these, there are 4 unique ways to solve the problem (or 8, if you discount additive commutativity). This is because you can switch the hundreds, tens, and units digits of the two smaller numbers. For example:

235 + 746 = 981
236 + 745 = 981
245 + 736 = 981
246 + 735 = 981

746 + 235 = 981
745 + 236 = 981
736 + 245 = 981
735 + 246 = 981

I’m guessing most wouldn’t count the second four as unique solutions.

So, with 34 numbers whose digits add up to 18, my guess is that there are 136 (or 34*4) unique solutions to this problem. Or 272 solutions, if you count repeats.

Extra note: Just to show that I did work to find some solutions to this problem, here are a few more…

659 + 214 = 873
546 + 192 = 738
478 + 215 = 693
394 + 182 = 576
293 + 175 = 468

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1 comment so far

  1. […] Posted March 16, 2009 Filed under: Math | Tags: Math, Proof | As you recall from my blogpost a few days ago, I found a few solutions to the following fun math […]


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