High School Strategies for Solving Mathematical 'Riddles' in Scholarship Tests

[Isaac0427]

I took a test for a scholarship that had mathematical "riddles" just like Micromass' challenges. It was multiple choice and I guessed for some, but others I was able to do or at least use process of elimination. I didn't think I did that well, but I advanced to the second round. Could I have some strategies for doing these problems?

Problems were in the form:
There exists (some amount) of numbers that satisfy (some condition). What is the sum of the numbers?

How many ways can (this set) be arranged so (this condition) is satisfied?

And other things like that.

Thanks!

 

[mfb]

Could I have some strategies for doing these problems?

Practice is the best help. The interesting problems (something beyond the typical homework) are those that cannot be solved with the "one obvious step at a time" approach. You have to try something, if it does not seem to work stop and try something else, if that also does not work go back to the first approach or try a third approach, and so on. Sometimes multiple approaches can be combined to find a solution.

 

[fresh_42]

Hard to tell without an example at hand. One thing that often works, is to look out for symmetries of all kind. Either to reduce the amount of possibilities by identifying basically equal solutions, or to use them to analyse the objective function to have an additional condition in the toolbox.
Also Diophantine equations and integer problems may be solvable by passing them to modules $p\mathbb{Z}$.

 

[jedishrfu]

Another strategy is to explore using simple numbers like 0, 1 and 2 or 3 to see what an equation produces. Sometimes the insight gained is enough to eliminate some multi choices.

Also sometimes graphing things may help. I once took a test on the roots of complex numbers and didn't recall the formula but did remember the rotational symmetry of the roots. I graphed the problem with one root that I knew off hand and by symmetry was able to draw the others and to write down all of the roots.

For math genius level tests like the Putnam and MAA tests, be prepared to have worked a lot of different math problems and to have the ability to think outside of the box.