Monday, October 5, 2009

Another Sample Contest Problem - Counting...

There is still time to register for the upcoming MathNotations Third Online Math Team Contest, which should be administered on one of the days from Mon October 12th through Fri October 16th in a 45-minute time period.

Registration could not be easier this time around. Just email me at dmarain "at" "gamil dot com" and include your full name, title, name and full address of your school (indicate if Middle or Secondary School).

Be sure to include THIRD MATHNOTATIONS ONLINE CONTEST in the subject/title of the email. I will accept registrations up to Fri October 9th (exceptions can always be made!).

* Your school can field up to two teams with from two to six members on each. (A team of one requires special approval).
* Schools can be from anywhere on our planet and we encourage homeschooling teams as well.
* The contest includes topics from 2nd year algebra (including sequences, series), geometry, number theory and middle school math. I did not include any advanced math topics this time around, so don't worry about trig or logs.
* Questions may be multi-part and at least one is open-ended requiring careful justification (see example below).
* Few teams are expected to be able to finish all questions in the time allotted. Teams generally need to divide up the labor in order to have the best chance of completing the test.
* Calculators are permitted (no restrictions) but no computer mathematical software like Mathematica can be used.
* Computers can be used (no internet access) to type solutions in Microsoft Word. Answers and solutions can also be written by hand and scanned (preferred). A pdf file is also fine.

Ok, here's another sample contest problem, this time a "counting" question that is equally appropriate for middle schoolers and high schoolers:

How many 4-digit positive integers have distinct digits and the property that the product of their thousands' and hundreds' digits equals the product of their tens' and units' digits?

The math background here may be middle school but the reading comprehension level and specific knowledge of math terminology is quite high. This more than counting strategies is often an impediment. If this were an SAT-type question, an example would be given of such a number to give access to students who cannot decipher the problem, thereby testing the math more than the verbal side. On most contests, however, anything is fair game!

Beyond understanding what the question is asking, I believe there are some worthwhile counting strategies and combinatorial thinking involved here. Enjoy it!

Click More to see the result I came up with (although you may find an error and want to correct it!)

My Unofficial Answer: 40
(Please feel free to challenge that in your comments!!_


mathmom said...

I came up with 40 also. I ended up just enumerating a list of the pairs of digits whose product equals the product of another pair of digits. I didn't feel like I had a great way of thinking about that (though some rough ideas about primes and squares and cubes emerged) so it felt kinda brute-force-ish to me. Of course the list was reasonably short in the end, so that was ok. But like you, I wasn't terribly confident of my answer in the end. (At least I suspect you weren't since you invited folks to disagree.)

Dave Marain said...

It's always comforting when we agree on an answer! I felt pretty sure of the result but I believe in hedging my bets, particularly since i was doing this mentally around 6:30 AM!

It would be interesting to "kid-watch" on this one, although the younger set may need clarification or one example. Do you think high schoolers' approaches would be more sophisticated than middle schoolers or about the same?

I started by considering all pairs in which "1" is the least digit. "1" could not be paired with any primes so that left "14" (no good), "16" (that paired with "23"), "18" (paired with "24") and "19" (no good).

Each pairing like "16" with "23" leads to 2x2x2 = 8 outcomes.

Then I considered pairs in which "2" is the least possible digit, i.e., no "1's." This produced "23" (already covered above), "24" (already covered), "26" (paired with "34") and "29" (pair with "36").

This leads to 2 more sets of eight results.

I'll stop the process here to allow students to check these and complete them.

Since we think in similar ways, everything you described in your comment applies to me as well!

mathmom said...

I find that students with more experience with this type of problem tend to better handle the "organized" part of "make an organized list". Since I don't think there are really any high school level skills that enter into this problem, and the hard part (IMO) is making the organized list of matched pairs, I think experience will win out over school level here.