Solving a Riddle

*It is not knowledge, but the act of learning, *

*not possession but the act of getting there, *

*not being but the act of becoming, *

*which grants the greatest enjoyment.*

*C. F. Gauss*

Throughout history, many math riddles captured the minds of amateurs and master mathematicians alike. A riddle is a question that seems to make no sense but that has a clever answer. For example: *What has four wheels and flies?* Answer: *A garbage truck*. Of course, math riddles have a similar connotation, but they refer to logical questions with clever and logical answers. It is of great satisfaction to solve a riddle because it implies challenge. This Activity is about riddles.

**Instructions**

- When solving a riddle, it is very important to understand properly the described situation. In order to understand, we have to read. But reading just one time does not guarantee that we have understood what the challenge is about. For that reason, please read several times to make sure the situation is completely clear.
- Once you have understood it, you must use the power of the combination of your knowledge, a bit of talent, and, of course, persistence. I am sure that you are now armed with the three required elements.
- Remember the steps you have to follow when solving a riddle: be armed with pencil and paper, draw a diagram, act it out, make a model, guess and check, work backward, account for all possibilities, find a pattern, make a table or graph, solve a simpler problem, use a formula, write an equation, mix applications. At the beginning, do your job using a draft, but once the riddle is solved, in an organized way using at all times detail explanations, please write the procedure used towards the solution of the riddle.
- Following the instructions 1 to 3, success is guaranteed (100%). This schema is general when solving math problems including math riddles. Congratulations! You have solved the riddle.

**THE RIDDLE OF “ROW, ROW, ROW YOUR TREES”**

How can seven trees be planted so that there are 6 rows of trees in a straight line with each row having 3 trees? For example, if you plant them the following way you get only 4 rows of 3 (two horizontal, one vertical, one diagonal).

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**Extra:** With 10 trees, make 5 rows of 4 trees each. **Hint:** remove one tree from the previous case. I think the next is 19 trees, 9 rows with 5 in each row.

**A RIDDLE OF GAME THEORY**

In 1944 John von Newman cowrote the book *The Theory of Games and Economic Behavior*, in which he coined the term *game theory*. Game Theory was von Newman’s attempt to use mathematics to describe the structure of games and how people play them. Mathematical Game Theory has become a basic tool to test military strategies by treating battles as complex games of chess. An illustration of the application of game theory in battles is the riddle of the *truel*.

A truel is similar to a duel, except there are three participants rather than two. One Morning Mr. Black, Mr. Gray, and Mr. White decide to solve a conflict by trueling with pistols until only one of them survives. Mr. Black is the worst shot, hitting his target on average only one time in three. Mr. Gray is a better shot, hitting his target two times out of three. Mr. White is the best shot, hitting his target every time. To make the truel fairer Mr. Black is allowed to shoot first, followed by Mr. Gray (if he is still alive), followed by Mr. White (if he is still alive), and around again until only one of them is alive. The question is this: Where should Mr. Black aim his first shot?

**THE RIDDLE OF DIOPHANTUS’S AGE**

Euclid’s Book, *The Elements*, was his greatest contribution to Geometry. *The Elements* would form the geometry syllabus in schools and universities for two thousand years.

The mathematician who compiled for number theory the equivalent text book, *Arithmetica*, was Diophantus of Alexandria, the last champion of the Greek mathematical tradition. Virtually nothing is known about this formidable mathematician. Appropriately for a problem-solver, the one detail of Diophantus’s life that has survived is in the form of a riddle said to have been carved on his tomb:

God granted him to be a boy for the sixth part of his life, and adding a twelfth part to this,

He clothed his cheeks with down; He lit him the light of wedlock after a seven part, and

five years after his marriage He granted him a son. Alas! late-born wretched child;

after attaining the measure of half his father’s full life, chill Fate took him.

After consoling his grief by this science of numbers for four years he ended his life.

The challenge is to calculate Diophantus’s life span.