It looks like there may be a way for mathematicians to figure out what to do when they encounter a problem that has several correct solutions, but what of the rest of us? What should we do? And how that does that happen anyway? Problems with multiple, possibly positive, outcomes?
Another possible reason for multiple answers is that, when conditions in a problem are unspecified or unknown, mathematicians make use of something called the “principle of indifference” or “principle of insufficient reason.” If there are several different, mutually exclusive scenarios that could take place in a situation, mathematicians assign equal probability to each of them, at least as a first guess. In problems that involve picking numbers from known ranges, this amounts to choosing uniform distributions. Sometimes this is just not possible, as we saw in the previous puzzle. And sometimes there are several equally valid ways of applying the principle of indifference that may yield different probabilities.
Mind you, we are talking probability problems here. I’m doubt more day to day issues stand to be solved like this.
Writing about an unpleasant experience and then sealing the completed account in an envelope has been found to go along way in easing the upset and memory of the event.
Sealing a disturbing news story in an envelope reduced the negative emotional impact of the story and reduced participants’ memory of it. By contrast, sealing an unrelated piece of paper did not have these effects, thus showing that it’s the act of containing the emotional material that’s important, not the mere act of putting anything in an envelope.
The situation whereby a solution to problem – especially one we’ve been thinking over for a while – comes to us when we’re giving it the least amount of thought, is probably pretty familiar. The challenge therefore is to place yourself in situations where you are not giving such problems direct thought:
You can’t directly control where your thoughts drift. If you’re controlling them, they’re not drifting. But you can control them indirectly, by controlling what situations you let yourself get into. That has been the lesson for me: be careful what you let become critical to you. Try to get yourself into situations where the most urgent problems are ones you want think about.
Someone who writes brain teasers and puzzles would also surely know to solve an array of other problems as well, or at least has ways of trying to.
Glad it’s not just me that finds the WWW a tad frustrating at times, though I hadn’t compiled a list 65 gripes long… yet.
The reset button. Do we really need this? I especially love it when I accidentally press “reset” instead of “submit”. It’s especially satisfying when it’s a long form.