22 May 2013

More Domino Puzzles: Middle Secondary Mathematics Competition Questions


Like yesterday’s Domino Puzzle, you are given a full domino set of 28 tiles. Firstly, remove all the tiles that have a blank end, leaving you with 21 tiles. Now, by looking at each tile as a vertical arrangement of numbers, we are going to use them as fractions. For example, the tile [5,2] can be thought of as the fraction 5/2 or 2/5.

Question A

This time we are going to add together two domino fractions, giving the sum of 6/5. The tile [6/5] is already used as our answer so, using the 20 remaining tiles, how many different sums can you find? The order of the fractions is not important.

Algebraically,

[A/B] + [C/D] = [6/5] = 6/5

where the square brackets [A/B] merely serve as a reminder that these are domino tiles being used as domino fractions.



Question B

How many more combinations are there if we also allow the sum of three domino fractions to equal 6/5?


Question C

Given the same construction as the previous questions, but this time looking at the general case where the sum of distinct domino fractions is equal to one other domino fraction, what is the maximum possible number of tiles that can be used from one set to make such a calculation?


I know, having three questions may seem over-the-top! However, there is a serious point to this [drum roll]. One of the best ways of learning mathematics is to take an existing problem and find ways of creating your own problems. How do textbook writers come up with questions? How do mathematics competition setters come up with questions? I would wager that they do pretty much what I do; they play around with existing questions and try to find interesting alternatives. They may also go back to existing theorems and turn them into original questions.

Sometimes, especially in textbooks, it is enough to merely change a few parameters and a new question can be created in seconds! However, most competition questions, and especially mathematical puzzles, try to be a bit more ingenious, a tad more challenging than the average problem. There aren’t really any traps – these are not lateral thinking puzzles – but there may be subtle conditions that only the smart students pick up on. This is where language, English in our case, is as important as the mathematics, and learning what the technical terminology actually means is often enough to turn what may at first appear a vague question into crystal clarity.

Creating your own problem is not just about rewriting an existing one, but about investigating the source, removing some of the restrictions and letting the key idea swim around in an infinite sea of possibilities. Then you might discover just why the problem setter chose those particular conditions and, if you’re lucky, you may find other islands with different conditions but equally interesting solutions.

If you think that solving problems takes a long time, I can assure you that setting them takes far far longer. On the plus side, though, you will be learning far more by deconstructing a problem and then reconstructing a similar one, than just by solving one question. With this in mind, I wish to announce that one of the Prize Maths Quizzes coming up will actually be about you submitting your own quiz question! I’m not sure if this will be popular or not, but I think it’ll be worth a try.



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