Here is your problem of the week to work that beautiful muscle called the brain. Remember: for those of you who submit a correct answer, you will be eligible for discounts on my collection of ebooks, as well as recognition on my website. So good luck, and here's to problem solving. Remember, you must work your brain in order that it get stronger.
Scientific research done at Stanford University has shown that regular mental exercise, such as that performed in doing brain teasers, riddles, and the like, improves cognitive functioning and brain health. Come work out your brain with these fascinating problems and brain teasers.
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The solution of the last Problem of the Week is as follows: Irrational numbers have decimal representations that neither terminate nor repeat. All we have to do is come up with some kind of decimal which exhibits both behaviors. This is quite easy. For example, one such number would be 0.010010001... where this pattern continues forever. Obviously this neither terminates nor repeats since we insert an extra 0 each time into the pattern. Once you get this idea, you can easily construct an infinite array of irrational numbers. Another example would be 0.12123123412345... in which case you count to the next number each time in the pattern. Obviously, this number neither repeats nor terminates. By the way, this little brain teaser gives a clue as to how you can show that there are more numbers between 0 and 1, than all the counting numbers. Think about it.
Now for this week: You are offered two choices by a friendly genie who wants to give you money. The genie says that for the next thirty days you can have either $1 on the first day, then $2 on the second day, then $3 on the third day, and so on until the end of the thirty days; or you can have 1 penny on the first day, then two pennies on the second day, then four pennies on the third day, and so on until the end of the thirty days. Which option would you choose and why?
