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I'm teaching problem solving to my students and I'm looking for a problem at the high school level to demonstrate a problem-solving technique that appears as though it might work, but doesn't. I want to be able to show my students that a strategy may look like a good fit for a problem, but in fact a different strategy will work better. Any help will be appreciated! Thanks!

luvteachingmath
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    This is absurdly broad. – Rushabh Mehta Feb 29 '20 at 14:37
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    I think the question is far too broad, and needs more focus. A better question might focus on algebra, or geometry, or trig, or precalc, or calculus, or (etc.). But to ask about problem solving techniques used by students in any/all of those subjects, and at all grade levels is biting off too much in one question; certainly too much for anyone to answer adequately. – amWhy Feb 29 '20 at 14:39
  • Sorry, I am new to this exchange. Perhaps a problem in which finding a pattern appears as though it may work, but working backwards is better suited? – luvteachingmath Feb 29 '20 at 14:44
  • @luvteachingmath Still, too broad. Are you teaching "problem solving"? What does mean? Problem solving is vastly different in different areas. – Rushabh Mehta Feb 29 '20 at 14:48
  • Yes, teaching problem-solving techniques. For students who are interested in competing in math competitions. I'm just hoping to aid them in not getting stuck focusing on the same technique for similar problems that they see, and hoped to give an exemplar problem in which we attempt and approach that we think may work, but it doesn't. – luvteachingmath Feb 29 '20 at 14:51
  • @luvteachingmath Maybe try this problem. – Rushabh Mehta Feb 29 '20 at 14:52

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Two (similar) examples that come to mind are the "tennis tournament problem" and the "two trains and a fly" problem.

The tennis tournament puzzle is as follows: Suppose there are 100 participants in a knock-out style tennis tournament. How many games must be played until there is a winner?

A brute force approach would be to calculate how many games are in each round, and how many players are eliminated in each round, all the way to the quarter-finals, semi-finals and the final. On the other hand, you could easily solve the puzzle by noticing that exactly one player is eliminated per game, so there is a winner after exactly 99 games (i.e. when 99 players have been eliminated).

The classic two trains and a fly problem is similar in that there is an obvious 'brute force' approach but there is also a trick to solving the problem which one might initially miss.

There are, of course, lots of other neat puzzles in which a simple trick/'strategy' gives a solution (e.g. the mutiliated chessboard problem), but often these don't have an 'obvious' but inefficient strategy that one could also use to solve the problem.

As noted in the comments, your question is a bit vague - I'm not sure if these examples are the sort of thing you are looking for; whilst the 'correct' strategy in each case is making an observation that immediately simplifies the problem, crucially, there is also a more obvious 'brute force' strategy that one may be tempted to use in each case. I've also tried to avoid simply giving examples of fallacious proofs since these really aren't 'problem-solving strategies'.

John Don
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