We enjoy reading, pondering, and solving math problems. Such problems can be important and directly related to your work or field of study (engineering, science, business, etc) or they can be "recreational" (ie, just for fun). They can be short with quick and definite answers or they can be long and vaguely defined such that the "answer" may simply be guidance on how best to refine or approach the underlying topic.
Here are a few wide-ranging examples:
(1) Differentiate the function f(x) = x^x (x raised to the x power).
(2) Prove, if possible, whether the number of "twin primes" is finite or infinite. (Twin primes are a pair of prime numbers whose difference is 2. The numbers 11 and 13 are twin primes as are 29 and 31.)
(3) For any triangle, draw inside its medial triangle (by joining the midpoints of the three sides of the outer triangle). Given the line joining the incenter I of the outer triangle to the centroid C of the outer triangle, extend this line segment beyond C to another point X such that IC = 2CX. Prove that X is the incenter of the medial (inner) triangle.
(4) Solve the "Egg Problem": given two eggs and a building with many floors, find the floor such that an egg dropped from this floor will NOT break when it hits the (possibly well padded) ground whereas the same egg WILL break when dropped from the next higher floor. The challenge is to find this floor with just 2 eggs and a minimal number of egg drops. Clearly, when an egg breaks, you can’t use it again. An unbroken egg can be re-used.
(5) You are given six IDENTICAL LOOKING coins each with a different weight equal to 1, 2, 3, 4, 5, or 6 oz. You have six labels that read "1", "2", "3", "4", "5", and "6". You give the labels to your child to place on the corresponding coins. You come back later to see if he/she has labeled each coin correctly (eg, the label "3" should be on the coin that weighs 3 oz). Here is the problem: all you have is a balance scale which you may use only twice. How do you arrange the two weighings so you can verify the correctness of the labeling of ALL the coins?
(6) Consider the "Coin Placement" game in which you play against an opponent who is seated across a table from you. You both have a large number of identical coins. The goal is to take turns placing a coin flat on the table in an open space without disturbing any coins already on the table. The first player who cannot find open space on the table on which to put the next coin loses. To win, then, you want to place the very last coin that can fit on the table (so that your opponent has no move). Question: What is your strategy to win this game? You are given the choice of going first or going second. Are there any conditions you must place on the size or shape of the table? (Don't take the trivial solution of saying "the table is only large enough for one coin and I go first".)
(7) Find the most practically intelligent answer to Bernoulli's Paradox stated in the following manner. You must pay a price upfront to a dealer to play a game. The dealer flips a coin and pays you $1 if the coin lands tails and the game is over. If the coin lands heads, the dealer flips the coin again and will pay you $2 if this second coin toss lands tails. Again, the game ends at this point when the coin lands tails, but if the coin lands heads, the game continues. With each subsequent coin toss, the dealer pays double the amount he/she would have paid in the prior round when the coin lands tails. The game always ends with the first coin toss that lands tails. When you pay to play this game, your hope is that there are MANY consecutive tosses that land heads until the first tails is reached. If there are two heads before the first tails, you receive $4. If there are three heads before the first tails, you receive $8 and so on. Question: What is the FAIR PRICE for you to pay upfront to play this game?
This last problem will seem to have no answer. But invoke the idea of finding the "practically intelligent answer". This problem is mostly "just for fun", but the correct solution has important application to the financial world. I have seen many very smart people get the WRONG answer. (If you need utility functions, you have the wrong answer. This statement may be arrogance on my part since it was Daniel Bernoulli himself who first proposed the utility function solution in 1738.) I have only encountered one person with the "practically intelligent answer".