## Friday, March 15, 2013

### Pi Day Puzzle Party - Join in, a bit late...

My friend Sharon invited me to head over to the city (San Francisco) on Thursday evening to meet her at a Pi Day Puzzle Party, sponsored by Ask a Scientist. Contestants could form teams of up to 6 people or work alone. The venue was a fascinating spot. SOMA Streatfood Park is an open lot with a pre-fab metal building in the center, surrounded by food trucks (some gourmet, some not so much), buried in a very industrial part of the city. It made me think of a gypsy camp.

I arrived first and grabbed us a (picnic) table. Sharon arrived soon after and we got food from the Peruvian food truck. When the party started, there were about 40 teams, with about 100 people packed into the cozy space.

We were handed the first question face down, and the MC, Wes Carroll, read it it to us all. Then we could turn it up and work on it for ten minutes, before getting the next question. There would be seven questions in all.

Question 1: There is a rare disease, called Anti-Mathitis (AMI for short), that afflicts 1 person in 10,000. There's a 90% accurate test that tells whether you have AMI. This means that 90% of the people who have it test positive and 90% of the people who don't have it test negative. You have just found out that you tested positive. What are the chances that you have the dreaded AMI?
Well, I knew all about this from teaching statistics. Back in the nineties, the state of Illinois required an HIV blood test for people applying for a marriage license. It turned out not to be as helpful as expected, along with being a big expense.

My teammates were working away together, while I wrote out what I needed. Sharon had emailed us messages about how she wasn't sure her 8-year-old friend Pai would like it, because we would be working hard at solving the puzzles and not slowing down to explain. So I knew ... winning was our goal, and my new buddies would be ok with me jumping in to explain my answer. Pai did come, and didn't mind our intense focus at all. We named our team The Pai Team.

Explaining my method showed me my arithmetic mistakes. (No calculator, oh no!) As I was cleaning those up we were handed the next question:
Question 2: There are 100 ants in a line on a 1 meter log. They are all walking on the same straight line. When one reaches the end of the log, it falls off. If two collide, they immediately reverse direction and continue on their way. Each is going at the speed of 1 centimeter per second. How long until all of them are guaranteed to have fallen off the end?
I knew I had seen one like this before, but had no idea how to proceed. I knew I needed to think about making the problem simpler, though... I suddenly had a flash of insight. This time, I waited until my team had finished discussing other ideas before I threw my idea in the ring.

And before we had time to take a breath, we were on to the next:
Question 3: Pat calls out 4 consecutive integers. Chris divides each by her age (a whole number), and notes the remainder each time. She adds the remainders and gets 40. Now Chris calls out 4 consecutive integers, Pat divides each by his age (a different whole number), and notes the remainder each time. He also adds the remainders and also gets 40. What are their ages?
This ended up being my favorite problem. I'd never seen one like it. We were still explaining it to one another when we got the crazy logic problem ...
Question 4: Stuck on an island there are 100 people with blue eyes, 100 people with brown eyes, and one guru with green eyes. Each of these people is a perfect logician, and will figure out their own eye color as soon as it is possible. They cannot speak, or communicate in any way, except for the one time the guru makes a statement. Nor can they find any way to see their own eye color. They each can easily see the eye color of every other person on the island. Each night at midnight, a good fairy comes and takes away anyone who has figured out their own eye color. One day the guru says, "I see someone with blue eyes." Does anyone leave the island? If so, who and when?
We got it...

Then came the red, white, and blue balls. I thought this problem would be our downfall...
Question 5: We have 6 balls that are visually identical except for color. There are two of each color, red, white, and blue. One of each color is heavier and one is lighter weight. All the lightweight balls weigh the same, and all the heavy balls weigh the same. We also have a balance scale. We need to determine which is which with only two weighings.
We came up with lots of good ideas, but they all ended up requiring three weighings. We had to leave #5 blank on our answer sheet and go on to #6...
Question 6: In the picture at right the square has area 64, and the triangle is equilateral. What is the diameter of the circle?
Hew got us started on this one, and we worked feverishly on the algebra. We got it... (At the end, the MC shared with us all another  way to solve it that requires no algebra. This other way is elegant and beautiful.)

One last question... (Although I think they're out of order...)
Question 7:  You have a pile of coins. Ten of them are heads up, and the rest are tails. You are blindfolded and cannot see them, but you can flip as many of them as you'd like. Can you put them into two piles, so that there are the same number of heads in each pile?
I had heard of this problem before, and probably heard the answer. But I have a terrible memory, and remembered nothing. As we worked on it, nothing made sense. Finally, I saw what needed to happen. And they were giving us twenty more minutes to work on our answers! Back to those dratted balls. I scribbled out my final asnwer just as we were asked to pass our answer sheets to a neighboring team to check. They took my scribbles along with our answer sheet. Whew!

The MC gave the first answer. It was different than ours. I had explained ours so carefully to the rest of our team, I was sure we were right. And the team whose answers we were checking had the same answer. So I raised up my hand, and the MC said, "Ahh, we have some dissent." He called me up to the microphone. Of course, there was a chance I was wrong, and I was shaking with nervousness.  But I explained my thinking to him, and he agreed, and there were cheers. (When have I ever been cheered before for the answer to a math problem?)

We answered all 7 questions correctly, as did the team sitting right next to us. It turned out they and we were the only two teams to get them all correct. So our two teams were asked to send a representative up front for a tie-breaker question. We had to stand on chairs to be seen by the crowd. The MC warned us that a wrong answer would give the other person plenty of time to think.

The Final Tie-Breaker Question: What is the smallest prime factor of 317+523?
I saw the answer, said "I know it", and won the contest for our team! I was stoked. I have never done a math competition before, and I'm not particularly fast, so I hadn't dreamed that we'd win. My teammates came up, and photos were taken. It was quite a scene.

I asked the other team if they had recognized any of the questions. No, they hadn't. I told them they had really won then. Even so, we were full of the thrill of victory as we made our way homeward, and delighted to have shared it with 8-year-old Pai. None of our questions had had any relation to the number pi, so we were very satisfied to hear about her part in the  Pi Day parade at the Exploratorium. Pai and her mom held up the 95th and 96th digits of pi, both 1's.

Celebrate Pi Day a bit late, and answer these 8 puzzle questions. (Please, no answers in the comments.) I'll send a fun math book to the first person to get all 8. (No fair using Google. You are on your honor.) Teams are fine. Send your answers to mathanthologyeditor on (use the letter after f, along with mail). Your last puzzle is to make sense of my email, that I'm hoping to keep away from the scambots.

1. Awesome, nice job! Question 3 is new for me, too, or at least new-ish though I've done some similar number theory problems. Question 5 is a nice twist on a familiar-sounding problem. And I'll have to think some more and find the pretty way to do question 6.

Thanks for sharing the problems and your excitement from the event! Sounds like a really fun day. Wish I had been there!

That would be a "ferry", not a "fairy". ;-)

3. And I thought it was a good fairy. My image of the puzzle has been shattered. I hope it's a nice ferry they get to board, after that ordeal. ;^)

4. Thanks for these Sue. These look fun. In the future, will you be posting some of your thought processes behind how you or others solved them?

1. Jason, I never did write any of these up. Which one would you like me to write up?

5. Sure. But questions 2, 4, and 7 are mostly an 'aha!' insight. In a way, number 5 was too. We kept experimenting with different combinations, and finally stumbled on a good one. I forgot to mention that, even there, I had to claim my right answer, because the MC had a different answer. This time both were right.

6. We have a winner. Naomi Klarreich figures she's likely "the only person who thought that drawing endless pictures of infinitesimal ants and red, white and blue balls was a productive use of her day." Naomi can come by my house to pick a book, or I'll send her one.

There's still the satisfaction of puzzle solving...

7. Speaking of coming in a little bit late... very cool contest. #4, I published a version of it a few years ago. And #1 I've done. But the rest were new to me. I'm usually not good at weighing problems, so I sat down with 5 first, and it was not so bad (or I have improved). I'm not sure where I'm going with 7.

Jonathan

8. #4 was my least favorite. I'd seen something like it before, so I had some idea of how it went.

#7 definitely needs an Aha! moment. Maybe after you've slept on it. It sure seems like there's not enough information to get anywhere, doesn't it?