Essentially, if you take low cards (let's call the Ace a 1) and put them in the order 1-5-2-4-3, and then flip the first card down (Ace), and bury the next card (put it at the bottom), flip the next card (2), put the next card at the bottom, and continue until one card is left that you put down, you end up with the cards face up in the order 1-2-3-4-5. The question Alex asked is, can you do this with one suit (Ace through King) so that when you flip and bury, you get them in correct order 1-2-3-4-5-6-7-8-9-10-J-Q-K?
When given this problem at Twitter Math Camp, I immediately thought I knew how to do it...my group mates and I wrote it out on the board (thus, the vertical non-permanent surface), and then tried to prove it...with an epic fail. Unfortunately, there were so many great things to talk about that I did not have time to figure it out during the limited time Alex gave us.
Although I don't have the book with me at home to reference it now, I bought the book Crossing the River with Dogs to help a bit with the problem solving course I am starting this year, as I have no text book and am just using 25 years of experience and a book I wrote 17 years ago to guide the course. I had forgotten about the card problem from TMC and was flipping through the book before school started--and am pretty sure I saw it and got an idea of a solution. I closed it immediately so as not to see the entire answer, but just to get the idea.
So, I thought this would be a perfect problem to give to my new Honors Problem Solving Seminar class on the first day. I actually like to give students problems without fully knowing the answer, so that I can't lead them in a direction and can learn with them. Like me, these students immediately used their intuition, only to find out they were WRONG!! They are a very bright bunch, but with only 10 minutes left in class to start, just like I had at TMC, they did not figure it out. I let the students take home 13 cards of mine each so they could play.
I got a message from a student asking me if, after an hour and a half during his free periods he did not figure it out, was it OK? I said it was.
A teacher came up to me and told me that he almost got said student in trouble for playing cards and not working in the library...till the student told the teacher he was working on a math problem for the problem solving class.
SOMEWHAT OF A SPOILER ALERT...
Before I went over it in the next class, I asked my colleague Liz next door (who also went to TMC with me) to help me with the solution. What I saw in the book was a circle with the cards, where the ace was at the top, and then there was a space, and then there was a 2, and then there was a space, and then a 4, etc...so the space represented the card that would be skipped. We thought we could do it right away. It still took us about 10 minutes, as we were almost right several times...I tell you, when we did get it, we screamed like giddy little girls and did the nerdiest high-fives ever!! We even took a picture of the answer, but I don't want to show it if you want to try it.
So, back in class the next day, we had A LOT to go over in our hour and a half block class. We started class with http://corbettmaths.com/5-a-day/, we watched the Grit video (a must watch) and then took a quiz on How Gritty Are You and talked about grit when solving problems. Then we created our interactive notebooks for the course...maybe I will blog about that at a later date. Before we knew it, there was 10 minutes left. So I opened up the card question again.
A student went up to the board to explain his solution. Basically, he said that you take the amount of cards and divide them in half. So there were 13 cards, divide in half, and round up, and there are 7 that will be flipped (1-7) and the remaining would be buried. He then divided the remaining 6 by 2 and said that half of those would be buried, so he separated 8-9-10 from J-Q-K and then figured it out from there. Some students got this method, some did not.
I showed them the circle method that I used to solve it...they hated that!?
I knew a another student had gotten it, so I asked her if she did it the same way. She said she figured it out in two minutes. I was impressed. I said how did you do it? She said she worked backwards (genius!) like this:
1. Take k
2. Take q and put it on the top
3. Take k from the back to the front
4. Take j and put it on the top
5. Take q from the back to the front
Then just repeat the same steps:))))))
When you do this, you get, face up, the cards in the correct order with the Ace on top, ready to put down with the next card to bury. What!!!! Cray. So easy.
I loved that:
1. They did not get the answer at the end of class and had to think about it on their own for two days.
2. They had no way to Google the answer.
3. They organically came up with unique ways to solve, and that it clicked differently for different students.
4. They hated my way.
5. A student the answer in a very simplistic way (an "elegant" solution) that was different from the way the first student got it.
I love this class, and I am so happy to be teaching it this year...I expect to have a lot of unexpectedness!