Like a Good Puzzle? Try the TED-Ed Dark Coin Riddle

Have you seen the new Dark Coin Riddle? Give it a try, but also check out the TED-Ed lesson to go even further with it.

Want to try some more? Give these a try with your classes!

Can You Solve the Locker Riddle? A Ted-Ed Lesson and How to Write One

Two of my favorite problems have become Ted-Ed riddle videos. I wrote about the Ted-Ed Lesson "Einstein's Riddle" here. I worked with the creator, Dan Van der Vieren, the week before his lesson came out when he taught my class how to solve the Rubik's Cube. I was so impressed with his riddle that I asked him how he was able to create it. He told me about the nomination form, and I filled it out and wrote about the problem that I do every year in my classes, The Locker Problem.

The idea, of course, is not to write out all 1000 lockers, but to create a smaller, similar problem and look for a pattern. I use this to teach my students about tenacity, collaboration, and the value of using the solution to a small problem to solve a bigger one.

Within a few weeks, I heard back and was scheduled for a 30-minute interview with someone from Ted-Ed, and we talked about how to change the problem to have a story line, and having something in the lockers that remain open. I brought it up to my Problem-Solving class, and we ended up making a riddle that would involve having to solve a problem to earn an inheritance. Once that was done, I sat one night and wrote the riddle, very excited that the number of words in the sentence that I wanted in the lockers was ten, and that there were exactly ten lockers open. Often when I give this problem, I ask, "Which lockers are touched exactly twice?" So I threw that piece in their as well. As an author of a Ted-Ed riddle, you also need to come up with 5 multiple-choice questions, 3 open-ended questions, additional resources to explore, and a guided discussion question. 

The people at Ted-Ed are awesome! They are extremely professional and always got back to me within 24 hours, offering advice and helping me with any questions I had. And the animation is THE BEST!! Who cannot love that little girl??

Within a few months, it was up! The idea is that teachers can even customize a lesson of their own with the riddle. Here is the actual link to the entire Ted-Ed page that contains the riddle: http://ed.ted.com/lessons/can-you-solve-the-locker-riddle-lisa-winer.

My students LOVE these riddles, the first five of which can be found here. They beg for more. Do you have a good riddle? Consider nominating yourself and writing one for Ted-Ed. 

Figuring out Figurative Numbers--a mini unit

Last week, my Honors Problem Solving Seminar students wrote a collaborative guest blog about their unit on Figurate (or figurative) numbers here.

Here are the handouts. They are mostly from a book I wrote 16 years ago called Investigating Discrete Math. Unfortunately, I did not publish the book. Fortunately, I use it a lot as a resource. 

Most of the students had already seen The Locker Problem, which I blogged about here. This is the first example of a figurate number investigation, though they may not know it at the time.

I gave them the handout below and had them work in pairs, investigating the patterns in oblong and triangular numbers. Interestingly, they can get the oblong number explicit formula and the triangular number recursive formula quickly, but they have a hard time with the oblong recursive and the triangular number explicit. 

What I like about the worksheet is it leads right into the sums of  counting and even numbers, and that the nth triangular number represents the sum of the first n counting numbers, and that the nth oblong number represents the sum of the first n even numbers.

Many of my students needed a hint on finding the explicit formula for triangular numbers. I drew a diagonal through the middle of an oblong and soon they discovered triangular numbers were half the oblong numbers. So oblong number formula is n(n+1) and triangular is that divided by 2.

Since they remembered the locker problem, we reviewed square numbers and showed that the nth square number represents the sum of the first n odd numbers.


I also talked about the story of Gauss whose teacher was trying to keep him out of trouble as a very young boy.  She told him to add the numbers from 1 to 100, thinking it would take him quite a while to do so.  As the story goes, Gauss had the answer within moments.  How did he do it?  Problems 1 and 2 demonstrate two quick ways to do it:

1.         Shown below is the list of numbers he was told to add.

            Notice it you add together the first and last number, you get 101, as shown above.

            a.  Add together the second and the second to the last number.

            b.  If you continue adding (i.e., 3 and 98), you continue getting the same number of 101.  
                 How many pairs of numbers sum to 101?

            c.  Find the answer to the problem using the information above.

            d.  Generalize this for adding together the numbers from 1 to n.

2.  Another interesting way to find the sum of the numbers from 1 to 100 is to list the numbers from 1 to 100 and then underneath it, list the numbers from 100 to 1, as shown below.  Let both sums equal X.

          X =   1 + 2 + 3 + 4 +…+ 97+98+99+100

          X = 100+99+98+97+…+  4  +3 + 2 + 1   

Now add each of the pairs of numbers that are underneath each other; for example, add together the 1 and 100, the 2 and 99, etc.  How many pairs of the same number do you get?  Note that this answer equals 2X.  Why?  Use this new method to solve for X, the sum of the numbers from 1 to 100.

Once we had all of that "figured" out, I gave them the Handshake Problem. Many knew it, but could not remember the formula. I try to encourage students not to remember the formula, but to get it from scratch.

It was pretty cool when they eventually realized that the sequence for the number of handshakes with n people: 0,1,3,6,10,15,... (i.e., for 1 person there are 0 handshakes, for 2 people there is 1 handshake, etc) is one off of the triangular numbers: 1,3,6,10,15, .... Thus, they developed the formula n/2(n-1) by substituting n - 1 into n for the triangular numbers formula. Or, they came up with the formula by realizing that for n people, each person shakes hands with n-1 people, but you need to divide that by 2 since when I shake your hand and you shake my hand, that counts only once not twice.

Next, I gave them the Pentagonal Numbers Investigation.



This was much harder, and again, I encourage you to look at the blog the students wrote here to see their thinking. But one student figured out that pentagonal numbers (by looking at the 3rd figure) is just the sum of the two nth triangular numbers + one (n-1)st triangular number - 1. Pretty cool!!

Finally, I gave them the Pizza Problem.


Students discovered the sequence to be 2,4,7,11,... which most recognized as the triangular numbers + 1, which was pretty cool. I did not do the Pyramid Numbers Investigation, but next year I will, as it is a nice intro to the Method of Finite Differences, which demonstrates how to solve explicit polynomial formulas without looking at patterns.

Here is the Pyramid Number Investigation:



And the Method of Finite Differences plus homework handout. Keep in mind that this was made 16 years ago and so the typeset is a little "old!"

Students really enjoyed that this was out of the normal "math" that they see in their other math classes. We also went to https://www.desmos.com/ and did some regression there as well. There are many ways to solve for the formulas!!

The highlight of this was twofold. One was the collaborative blog they wrote. But the second was when they were the teachers for the Algebra 1 classes, and taught them the classic handshake problem with a twist: The Fist Bump Problem, a la Sam Shah. It was amazing. I will blog about that soon.

~Lisa

Two of my favorite problems: Einstein's Puzzle and the Locker Problem

As school is now in full swing, I am finding that I don't have time to blog as much about the "play" and the "eat." But truthfully, the thing that brought me to blogging is the "math." I was going to skip blogging last week because I didn't have the "perfect" eat play math combination...but I have heard if you have a blog, you should blog at least once a week...so if I have to skip a little eating and playing, so be it!!

This week, I want to blog about two of my favorite problems/investigations. I just wrapped up teaching the first unit in my Algebra 2 Honors class, and I decided to take two days on our special schedule days to allow students to "play." I think they were relieved that they could attempt some difficult "famous" problems and not be graded on them. Here was their homework the night after their test...with a 40 minute time limit:

Einstein's Puzzle.

Students came in very excited to present. One came to the board and got stuck, and then another came to help him finish. We crossed out everything we used until we realized we needed to write all remaining possibilities on the board. Once they realized through process of elimination that there was only one possibility for one of the houses, everything started to fall in place. One student said she solved hers by guessing at one point, going the wrong way, and then finishing by choosing the other method. Explaining to each other in class was hard for the kids to do, but it is early in the year, and they will soon come to realize that explaining to me is not what matters, it's explaining to those who do not yet understand, and getting them to see the light that matters! After all, that's the best part of teaching!!

Once the explanation was done, I handed out the next investigation. Students who did not get Einstein's riddle the night before were extra determined to solve this puzzle. Students went up to the board in pairs to try to solve this puzzle for about the last 15 minutes of class.

The Locker Problem:

On a recent problem from the first day of class (I will have to blog about that another time), the problem solving method was "the method of exhaustion"...writing down every possibility. Obviously that was not the best method here. So the only hint I gave them was to start with a smaller, simpler problem and generalize a pattern. Some asked how many should they start with, and I told them it was up to them.

I walked around, watching students and advising them if they made a mistake, but not telling them much. Finally, a group discovered the answer. I did not want them to give it away to the others, so we talked quietly...but the next, perhaps more important question was WHY  did these locker numbers that remained open. Another group got the pattern, and again, I asked why...then the bell rang. The assignment for the night was, for some, to find the pattern, and for all, to find why the pattern. Again the next day, the students who discovered the answer were excited to share.

SPOILER ALERT: DO NOT READ IF YOU WANT TO SOLVE THE PROBLEM YOURSELF. It turns out that lockers are touched by the number of factors the locker number has...and the pattern that remain open are perfect squares...because perfect squares have an odd number of factors. We talked about why that is...for locker 6, for example, the factors are 1,2,3, and 6...and so they pair up...1x6 and 2x3 = 6. But for locker 4, the factors are 1,2,4. The factor of 2 repeats, which results in an odd number of factors, and this only happens with perfect squares. An extension could be which lockers are only touched twice? (the lockers that are prime.)

Finally, each student was given an index card with a locker number on it and they "acted" as lockers...opening and closing by turning away and toward the front of the classroom so any who did not understand could visually see which lockers remained open.

We ended class with a few more brain teasers...most notably Crossing the river with a goat, a wolf, and a cabbage. Students acted it out, and they were very cute. Their homework was to come in with their favorite (or newly discovered) brain teaser that we will solve throughout the year.