Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original post. Figure them out for yourself — and then check the answers just to prove that you got them right.
Euclid’s Geometric Algebra
Picture from MacTutor Archives.
After the Pythagorean crisis with the square root of two, Greek mathematicians tried to avoid working with numbers. Instead, the Greeks used geometry to demonstrate mathematical concepts. A line can be drawn any length, so straight lines became a sort of non-algebraic variable.
You can see an example of this in The Pythagorean Proof, where Alexandria Jones represented the sides of her triangle by the letters a and b. These sides may be any length. The sizes of the squares will change with the triangle sides, but the relationship 
The Pythagorean Proof
[In the last episode, Alexandria Jones received a letter from archaeologist Sofia Theano, asking for help with a Pythagorean puzzle.]
The Mosaic Tile Mystery
Dear Alexandria Jones,
We continue to excavate the ancient building complex, which I believe may have been Pythagoras’s school. Yesterday, one of our digging crews uncovered a mosaic tile floor in the courtyard. The pattern of the tiles alternates between two square designs. (See enclosed sketches.)
During your family’s recent visit, you expressed an interest in the mathematical ideas of Pythagoras. Could you or your father offer us any insight into what these tile designs may represent?
I look forward to your response.
Sincerely,
Sofia Theano, Ph.D.
Crotone, Italy
An Ancient Mathematical Crisis
[When Alexandria Jones and her family visited an excavation in southern Italy, they learned several tidbits about the ancient school of mathematics and philosophy founded by Pythagoras. Here is Alex’s favorite story.]
It hit the Pythagorean Brotherhood like an earthquake, a crisis of faith which shook the foundations of their universe. Some say Pythagoras himself made the dread discovery, others blame Hippasus of Metapontum.
Something certainly did happen with Hippasus. The Brotherhood sent him into exile for insubordination, or for breaking the rule of secrecy — or was it for proving the unthinkable? According to legend, Hippasus drowned at sea, but was it a mere shipwreck or the wrath of the gods? Some say the irate Pythagoreans threw him overboard…
Alexandria’s Dog is Now a Teacher
Photo by alex-s.
A reader of Indian descent has been kind enough to write to me about a difference in our cultures. In the US, or at least in the parts with which I am familiar, it is common to name one’s pet after a famous person. In India, however, to name a dog after a human is a very deep insult.
I am sorry! It was not intended that way.
Therefore, I have re-named Alexandria Jones‘s dog after a Westerner. I would like to keep the tradition of naming the characters in my Alex stories after people in math history, so I am hoping this one will not offend anyone.
Game: Avoid Three, or Tic-Tac-No!
Math concepts: slope, logical strategy
Number of players: 2 or more
Equipment: 4×4 or larger grid, pebbles or other tokens to mark squares
Set Up
Alexandria Jones and her brother Leon played Avoid Three with pebbles on a grid scratched in the sand, but you can also use pencils or markers on graph paper. You need a rectangular playing area at least 4×4 squares large. The bigger your grid, the longer your game.
Answers to Leon’s Figurate Number Puzzles
Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original post. Figure them out for yourself — and then check the answers just to prove that you got them right.
Puzzle: Figuring Out Figurate Numbers
Photo by frumbert.
Alexandria Jones‘s parents decided that the family needed to relax after the excitement of tracking Simon Skulk, so they spent the next day at a beach on the Mediterranean coast. Leon collected pebbles and tried to build up figurate numbers — numbers that make a figure, or shape — the way Dr. Theano had shown them.
The Puzzling Pythagorean Pebbles
Alexandria Jones and her family flew to Italy for spring break. Her father, the famous archaeologist, had to visit an excavation.
It was late when their plane landed in Crotone, a small coastal city near the instep of Italy’s boot. Dr. Jones had used the Internet to find a hotel that allowed pets, so Alex was able to snuggle down with her favorite pillow — her trusty dog, Ramus.
The Original School of Mathematics
The next day, Dr. Jones introduced his family to Sonya Theano, a former student of his and the director of this dig. “Come, let me show you around,” Dr. Theano said. “We’ve uncovered several buildings of a small compound, set apart from the city of Crotona, as it was called then. From the pottery and trade goods, we estimate these buildings were in use around 550-500 BCE.”
Denise Gaskins' Let's Play Math 