A Simple Arithmetic Puzzler ...

Gifts for David

Solving mathematical and recreational puzzles have always been a fun pursuit to me, and scribbling solutions to puzzles  that I had come across was often a great way to keep myself busy.  Interestingly enough, I also realized that making puzzles is also a very fun pursuit.  There are some creative people , like Scott Kim, who have made a career out of simply that: creating puzzles for a living!  Cool!  So, here is a very simple and somewhat interesting puzzle that I came up with and shared with my 9-year-old son a short while ago.  It is pretty easy to solve.  So, I have a follow up question to make it a bit more challenging once you have solved the easier version.  It goes like this:

In the following, add enough parentheses as needed, and a single operator in {+, -, *, /} before each operand or between two operands in the left hand side of the equal sign (=) to make each equation work:

    1     2     =    3

    1     2     3     =     4

    1     2     3     4     =     5

    1     2     3     4     5     =     6

    1     2     3     4     5     6     =     7  

    1     2     3     4     5     6     7     =     8

    1     2     3     4     5     6     7     8     =     9

Gifts for Daniel

Note that the parentheses and/or operators are allowed only on the left hand side of each equation to make it work.  As I said, this is a pretty easy problem to solve.  My 3-rd grader son solved it in one sitting.  If you liked the above, try the following extension to it:   Let's refer to the i-th equation that evaluates to i+2 as P(i) .  Could you make P(i) work for arbitrarily large positive integer i?  For example P(15) is shown below:

    1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   =   17

Once you have figured this one out and are able to prove it (hint: it is possible to do so, for arbitrarily large i), other related questions can be posed as well.  For example, consider the following questions whose solutions may be much more difficult, if at all possible:

      . In how many different ways can each of the above equations 

        be made out to work, if we stick to the problem requirements?

      . How does the number of possible ways to make each equation  

        P(i) work grows asymptotically as a function of i?

Gifts for Ken

Notice how we could go from a simple question to more complicated ones.  Often, deep mathematical theories can start from a simple question and curiosity in solving one interesting problem or puzzle, and then asking a little bit deeper question, each time challenging oneself to answer more and more questions about the problem, its extensions, generalizations, etc.  In the process, one can often use the insights gained from solving simpler versions of the problem to seek deeper understanding about the fundamental nature of the problem at hand.  It all is like a game if you look at it that way.  Try it some time.  It is fun!

It has been observed that solving problems and posing problems are interconnected and can lead to life-long love of, and adventures in mathematics.  Mathematician Paul Halmos in an article entitled "The Heart of Mathematics" published in 1980 in the American Mathematical Monthly observed the following:

Another testament to the interconnectedness of problem solving and problem-posing is the  life and the legacy of Paul Erdős legendary mathematician who is often described as:

In addition to a great many problems that he helped solve and many papers that he co-authored, there are a great many open problems that are part of his legacy to the mathematical world.  There are over  a hundred unsolved problems attributed to Paul Erdős, in only one small (but important) branch of Mathematics called Graph Theory.  A collection of these open problems are listed in this book.

Gifts for Samantha

If you enjoy mathematical puzzles, try a few more problems in here, here or here.  Also, this one is pretty neat! Those interested in challenging problems targeted at high school and mathematics Olympiads might want to check this page out. If you are looking for even more challenging math problems, you could try tackling some of the unsolved mathematical problems listed here and here, or even taking a shot at some of the unsolved Millenium Prize Problems, each carrying a $1,000,000 cash prize.  Good luck!  If you enjoy mathematical discussions, I highly recommend Dick Lipton's Blog on the Theory of Computation and Theoretical Computer Science.  Highly readable and very educational.