## Mission 000 : : January 25, 2014 : : Counting Greetings

LEVEL 2 SNOW EMERGENCY. Our headquarters (HQ) afforded a terrific view of the snow storm. We wondered:

We greeted each other with handshakes. There were 7 LEAGUErs + Javier and they all shook hands. How many handshakes was that? Some said $8\times 7\times \cdots \times 2\times 1$. Others said $8+7+\cdots +1$. Who was right?

Eventually the group latched on to a formula. LEAGUERs believe that $\frac{n(n+1)}{2}$ represents the number of handshakes that $n$ people engage in if everyone shakes everyone else's hand.

Every good LEAGUEr knows the importance of keeping the mind but also the body fit. Javier led the group in an exercise in "Ally and Nemesis." Silently, we chose one person in the room to be our ally and one person to be our nemesis. When Javier gave the command, we had to arrange ourselves so that our chosen ally (who didn't know they were our ally) was between us and our nemesis (who didn't know they were our nemesis). The casual observer would have thought that we were running around chaotically, moving outward toward the corners of the room.

When Javier shouted "SWITCH! Your ally is now your nemesis and vice-versa!" the chaos shrank to the center of the room. Why? This is one example of what mathematicians call a "dynamical system." Interesting.

This led to some questions.

We came to a truly cool solution to the above questions and made some powerful realizations. Then, our time came to a close. Now that we were a closer LEAGUE of friends, we now had the option of either shaking hands with a person or hugging a person. We could make that choice independently for each person we said goodbye to. How many handshake-or-hug possibilities were now possible for all eight people collectively if they all said goodbye to each other in one of those two ways? THIS questions was left for LEAGUErs to ponder for next meeting. Javier pointed out that we might start with just two people, then three people, and then build to eight people (or beyond!). He's using a powerful strategy we might call "Start with a simpler case. Then try to increase and generalize."

Great job, MATH LEAGUErs. See you on the 8th of February. We'll have new LEAGUE friends to meet and welcome to our family of problem solvers.

Signing out. - BK -

- Are there more snowflakes in the sky over the City of Athens (at that instant) or birds on Planet Earth?

We greeted each other with handshakes. There were 7 LEAGUErs + Javier and they all shook hands. How many handshakes was that? Some said $8\times 7\times \cdots \times 2\times 1$. Others said $8+7+\cdots +1$. Who was right?

Eventually the group latched on to a formula. LEAGUERs believe that $\frac{n(n+1)}{2}$ represents the number of handshakes that $n$ people engage in if everyone shakes everyone else's hand.

- Is that right? How do you know?
- If there are
*at least*20,000 residents of the City of Athens then what is a*low estimate*of the number of handshakes if**everyone**in the City of Athens shook everyone else's hand?

Every good LEAGUEr knows the importance of keeping the mind but also the body fit. Javier led the group in an exercise in "Ally and Nemesis." Silently, we chose one person in the room to be our ally and one person to be our nemesis. When Javier gave the command, we had to arrange ourselves so that our chosen ally (who didn't know they were our ally) was between us and our nemesis (who didn't know they were our nemesis). The casual observer would have thought that we were running around chaotically, moving outward toward the corners of the room.

When Javier shouted "SWITCH! Your ally is now your nemesis and vice-versa!" the chaos shrank to the center of the room. Why? This is one example of what mathematicians call a "dynamical system." Interesting.

This led to some questions.

- If we were to pick two people (out of the eight total) to be an ally and nemesis, how many ways could we do that?
- What if, instead, we wanted to pick six people (from the eight total) to be our "team." How many ways?
- Is there a relation between the questions above? Why or why not?

We came to a truly cool solution to the above questions and made some powerful realizations. Then, our time came to a close. Now that we were a closer LEAGUE of friends, we now had the option of either shaking hands with a person or hugging a person. We could make that choice independently for each person we said goodbye to. How many handshake-or-hug possibilities were now possible for all eight people collectively if they all said goodbye to each other in one of those two ways? THIS questions was left for LEAGUErs to ponder for next meeting. Javier pointed out that we might start with just two people, then three people, and then build to eight people (or beyond!). He's using a powerful strategy we might call "Start with a simpler case. Then try to increase and generalize."

Great job, MATH LEAGUErs. See you on the 8th of February. We'll have new LEAGUE friends to meet and welcome to our family of problem solvers.

Signing out. - BK -