A math puzzle to start the day
November 20th, 2008 by SplineGuy
I’m on a math puzzle and modeling kick. I’ve gotten back into writing code (in MATLAB, at least) and am having fun with a few blogs that post problems regularly. For example
- MAA Minute Math (http://maaminutemath.blogspot.com/)
- Project Euler (http://projecteuler.net/)
- Monday Math Madness (http://wildaboutmath.com/)
Below is a problem that I thought was fun and interesting to solve. I found it interesting because I could write a very short program that would execute quickly to do a brute force search for the solution. However, a quick algebraic approach yields the answer just as quickly.
Let 
and 
be two-digit numbers such that is formed by reversing the digits of . The integers and satisfy the equation 
for some positive integer 
. Find 
.
My solution can be found below. (Spoiler alert!! - Try it yourself before you peek)
Solution:
Let 
where 
. Thus, 
. We now have that
and
Thus,
If we set this equal to 
we see that 3 and 11 must be factors of . In other words, 
for some 
. Thus,
This give us that
or
Now, because and 11 does not divide any such 
, we know that 
. Additionally, we know that 
. There are only two possible values, then, for k. Either 
or 
. If , then the solution is 
, which is a contradiction. Thus, , and this implies that 
.
Thus, 
.
So, 
weehoo!
I’ve got to say your solution suffers from a bit of “Professoritis”. That’s what I call it when the explanation also has pieces that make little sense.
Although I get why and what you did, I remember sitting in classes hearing the professor (and let’s be honest, it wasn’t just you. hehe) explain things and feeling like I could have never thought of that because at some point in the derevation, there is an “assumption” or and “assertation” that comes out of left field.
Like this for example:
If we set this equal to m^2 we see that 3 and 11 must be factors of m. In other words, m=(3)(11)k for some k that is in the set of positive integers.
This would be where your students raise their hand and go “Wha?”
Explaining that the 3 and 11 came from factoring 99 and why you eliminated 9 and 33 would go a long way. Otherwise, a nice little problem to solve.
Also you got an error in your last paragraph, k=1 not 2.
While I appreciate your comment, I respectfully disagree. There is more than just one place where I have taken leaps. The one that stood out to you here is based on the fact that you are more than capable of seeing why k must be 1 or 2, or why I can set x = 10a +b, or why 65+56+33 = 154. The deeper you go in mathematics, the clearer the “missing” steps become. At some level, providing the extra statement becomes spoon feeding the readers. The challenge I have as a blog writer is choosing my audience. Is it my students, is it my fellow math bloggers, is it members of my extended family who were lost before I finished the statement of the problem?
In any mathematics writing exercise, there is a balance somewhere between too much detail and not enough. It is amusing to me that this would be attributed to my status as a professor, or even attributed to me at all, when I complained loudly and often when reading my graduate textbooks for the same fault. Now, going back and reading many of the beginning graduate texts, I reflect on how there may have been too much detail, thus robbing me of the opportunity to “discover” it for myself.
“Professoritis” is such an ugly word. Why can’t I suffer from “over-capacity for motivating metacognitive development”? Doesn’t have the same ring, I guess…
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