We’ll use this page to keep track of what has happened each day in class. It won’t contain any of the nitty-gritty details, but will instead serve to summarize what has transpired each day.

Week 1

  • Monday, August 28: First day! The first few minutes of class were devoted to me attempting to learn names. Next, I summarized the first few homework assignments, which was followed by a whirlwind tour of the course webpage. After this, the students engaged in an exercise aimed at understanding "productive struggle". This exercise is the second half of the Setting the Stage activity. The remaining portion of class was devoted to highlighting some key items on the syllabus.
  • Wednesday, August 30: The first few minutes of class were devoted to learning names and making sure there weren't any questions about the syllabus. Next, we had a quick discuss about writing code to solve some of the problems. For the record, I'm totally fine with students occasionally taking this approach. However, it is important to reflect on the meaning of the output. We also discussed trying to avoid "legal battles" when interpreting the wording of problems. This was followed by a quick discussion of Problem 1 involving taxi fares. We then split the class up into 6 small groups, where each group was responsible for one of Problems 3, 4, 5, 6(a), 6(b), and 6(c). After a few minutes we came back together to discuss the problems. Our very first presentation was given by OW. She presented Problem 3 (1000 doors in a hallway) and did an excellent job! With the time we had left had EB and HR present Problem 5 (sticks in a bag) and Problem 4 (triangles with toothpicks), respectively. Both of these presentations were also great. I'm very happy with how things went today. Unfortunately, we didn't get to Problem 6. The plan is tackle the 3 parts of this problem during the next class.
  • Friday, September 1: Another busy but productive day. After I lead a discussion about Problem 6(a) (Variation 1 of pebbles on a chessboard), we started tackling Problems 6(c), 6(c), 7, and 8, all of which deal with pebbles on a chessboard. DC and MF presented solutions to Problems 6(b) and 6(c). I took the lead on discussing Problems 7 and 8 and along the way, MG provided some key insights. Next, JG presented a proposed solution to Problem 9 and KC verified the solution was optimal.

Week 2

  • Monday, September 4: Labor Day, no class!
  • Wednesday, September 6: I was surprised how much we accomplished today. After a quick discussion of the upcoming quiz, we split the class up into 8 small groups, where each group was tasked with discussing their proposed solution to one of Problems 10-13. When all was said and done, we had AT, MF, HR, and QS present solutions to Problems 10, 11, 12, and 13, respectively. Our discussion of Problem 13 (ants on a log) was a bit rushed, so I hope to return to that one on Monday (since we have a quiz on Friday).
  • Friday, September 8: The students took Quiz 1.

Week 3

  • Monday, September 11: The week is off to a good start. After a few quick announcement, we split the class up into 5 small groups. After a few minutes, we had YS, OW, VM, AG, and KR present solutions for Problems 14, 15(algebraic), 15(visual), 16, and 17, respectively (but not in that order as KR actually went first). I meant to revisit Problem 11, but I completely forgot, so we'll have to do that at another time. We had some time left at the end, which we devoted to discussing the upcoming problems.
  • Wednesday, September 13: While I was out today, Monika Keindl covered for me. My understanding is that JK, KS/DO, VM, and MG presented their proposed solutions to Problems 18, 19,20, and 21, respectively. There was also some preliminary discussion of Problem 22.
  • Friday, September 15: Nandor Sieben covered for me today. It appears that EB, KW/KR, VM, and HR presented Problems 22, 23, 24, and 25, respectively.

Week 4

  • Monday, September 18: After some discussion about Problems 22 and 23, we started discussing Problem 27, which is a lot more involved than it looks at first glance. As a class, we discussed the cases of 2, 3, 4, and 5 orbs. We also got most of the way through discussing the cases involving 6 orbs. Along the way, we heard from QS, EB, TN, JR, AT, MG, HR, and KW. The goal is for the students to wrap up the cases involving 6, 7, 8, 9, and 10 orbs.
  • Wednesday, September 20: After a few announcements, the students formed 8 small groups with 3 students each. Each group was tasked with discussing 2 of the 4 problems that were currently outstanding. After 10 minutes or so, we started having students share out. We had MQ, JL/JK, JR, and SS present solutions or ideas for Problems 26, 27, 28, and 29, respectively. Problems 26 and 27 are still outstanding. VM conjectured a solution to Problem 26 and students were asked to prove this for Monday.
  • Friday, September 22: The students took Quiz 2.

Week 5

  • Monday, September 25: We started with a discussion of Problem 27 (Martian Artifacts). We determined that we had a list of upper bounds on the number of tests required for $n$ orbs, but we left open the issue of optimality. We believe that our list is accurate, but we have not verified this. Next, we split the class up into 6 small groups, each tasked with discussing one of Problems 26, 30, and 31. After several minutes, we had HH share his group's proposed solution to Problem 31, which involved some brute-force checking of cases. We also had MQ and QS present their groups' progress on Problem 30 (plate of cookies). We didn't have time to discuss Viola's conjecture to Problem 26. Hopefully, we can revisit this one on Wednesday.
  • Wednesday, September 27: To start, I walked us through one potential approach to solving Problem 32 (light up squares in an $n\times n$ grid). Next, we had VM present a proof of "Viola's Conjecture" concerning Problem 26 (Star Base hyperdrive). The was followed by a quick presentation of Problem 34 (signed permutations) by HH. With the few minutes we had left, OW attacked Problem 33, although we discovered that her solution was for a problem with different constraints. We will revisit Problem 33 next time.
  • Friday, September 29: We kicked off with JG and MQ presenting solutions to Problem 33 (Turnaround). We believe that 6 moves is optimal, but we did not prove this. VM discussed her computer code that hunted for an optimal solution. Next, we discussed some potential notation for Problem 35 (soul swapping) that utilized some diagrams. MG shared her solution to part (a) of Problem 35. This was followed by a presentation of part (b) by SS. VM presented a solution for part (c) that utilized the answer for part (a) and then JG shared a nice solution using only one of Wonder Woman or Superman. With the time we had left, I discussed a solution to the Futurama mind swapping problem, which essentially solves parts (d) and (e) of Problem 35. We will address Problems 36 and 37 on Monday.

Week 6

  • Monday, October 2: At the beginning of class, JG volunteered to show us a solution to Problem 35(d) that utilized a single new person instead of two. She conjectures that her approach generalizes. Next, we discussed Problems B.1 and B.2 from Quiz 2. After this, we had students pair-off to discuss Problems 36-39. We had JK/OW, KR, and JR present their proposed solutions to Problem 36, 38, and 39, respectively. We ran out of time and didn't get to discuss Problem 37 (fastest 3 horses). We'll discuss this one next time.
  • Wednesday, October 4: Today was super productive. After a short "pep talk" about putting more effort into homework and letting me know when students plan to miss class, we had a short discussion about Problem 35 (soul swapping problem). In particular, I clarified the types of scramblings that our proposed solution works on. We also spent some time wrapping up Problems 38 and 39. Then AG, YS, DO, and KW presented proposed solutions Problems 37, 41, 42, and 43, respectively. We will have CW present Problem 40 on Monday.
  • Friday, October 4: The students took Quiz 3.

Week 7

  • Monday, October 9: We had CW, EB, MQ, VM, SS, MG, and JR present Problems 40, 44(a), 44(b), 44(c), 45, 46(ab), and 46(e), respectively. Along the way, we tackled Problem 46(d) as a class. With the time we had remaining at the end, we started discussing Problem 47. AT came to the board and got us started.
  • Wednesday, October 11: EB quickly cranked out a nice coloring for Problem 47 and then QS, OW, and DC used this coloring to tackled Problems 48(a), 48(b), and 48(c), respectively. We cranked through this quickly and then spend the rest of our time discussing Problem 49, which asks which rectangles we can tile using L-shaped trominoes. After discussing some necessary conditions, we claimed that the 3 conditions were also sufficient and then started trying to prove this. We got most of the way through a case analysis and left one subcase for the students to resolve for homework.
  • Friday, October 13: After handing back Quiz 3, I told students the correct answer and then sketched the argument. I asked students to wrap up the argument for Monday. Next, QS volunteered to present the last remaining case on Problem 49 (L-shaped trominoes). This was followed up by a discussion of Problem 50 (product of 3 integers satisfying 3 conditions), which TN got us started on. KS started to present Problem 51 (12 coins), but then realized that he made an extra assumption about the problem. His modified version of the problems was worthwhile, so we spent some time thinking about it. Problem 51 is still open.

Week 8

  • Monday, October 16: Today wasn't quite as productive as I'd hoped. I got us started on Problem B.3 from Quiz 3 and then QS finished it off for us. Next, we had JL share his thoughts on Problem 51 (12 coins). His solution was almost perfect, but then he realized that the last case would require 4 steps instead of 3. I proposed an alternative approach and then AT showed us how to finish one of the cases. JK provided an alternative (and potentially simpler) approach to the same case. We ended with me quickly summarizing what to do on the remaining cases.
  • Wednesday, October 18: We spent the first half of class discussing the solution to Problem 51 (12 coins). Next we tackled the multiple parts of Problem 53 ($n$ coins in $k$ weighings). We had MG, EB, JG, OW, and HR share solutions to parts (a), (d), (e), (f), and (h), respectively of Problem 53. Along the way, I presented parts (b) and (c). Today was the first time that I was able to put all the pieces together for part (b). I was pretty excited about this. We also implicitly answered part (g).
  • Friday, October 20: The students took Quiz 4.

Week 9

  • Monday, October 23: I think we were all a bit sluggish today. I gave students a few minutes to chat about today's problems and then we had KR present his solution to Problem 55. Next, CW presented Problem 54 (All Different). We got started on Problem 56 (Checker Mate), but we didn't quite nail it down.
  • Wednesday, October 25: First, we discussed B.2 from Quiz 4 and then potential generalizations for Problem 55. This was followed by presentations by HR, KS, AG, MG, and VM of Problems 58(a), 58(b), 59(a), 59(b), and 59(c), respectively. With the few minutes we had left, we quickly discussed Problem 57 group of 6 people). Problem 56 (Checker Mate) is still outstanding.
  • Friday, October 27: I had a lot of fun today, but the energy level in the room was low. Our first order of business was revisiting Problem 57 (group of 6 people). In particular, we reviewed our argument and discussed how many cases would need to be checked in order to attack the problem via brute-force. Next, JK quickly put Problem 60 (balls in incorrectly labeled bins) to rest. No one seemed to have progress on Problem 61, so we discussed it as a class. Lastly, AG had some good ideas about Problem 62 (Quilt), but his analysis overlooked one of the hypotheses in the problem. Together we solved this one during the last few minutes of class.

Week 10

  • Monday, October 30: I divided the class up into 5 small groups, 4 of which were responsible for working on various cases of Problem 63 (set of 7 integers) and 1 group was responsible for working on Problem 64 (puzzle pieces). We had JL, TN, MQ, and EB present the 4 cases of Problem 63. Next, we had CW, JK, and MF present potential solutions for Problem 64. It wasn't clear whether we solved Problem 64 or not, so we will have to return to that one on Wednesday.
  • Wednesday, November 1: To start, our visitor Oliver presented an elegant algebraic proof to Problem 64 (puzzle pieces). Next, we had DC and JR discuss their approaches to Problem 67 (paper folding problem). This was followed by discussion of Problem 66 ($1/3 = 1/A+1/B+1/C$). We had some insight from KR, MG, JG, and VM along the way. With the few minutes left, we had KC get us started on Problem 65 (Klingon senate).
  • Friday, November 3: The students took Quiz 5.

Week 11

  • Monday, November 6: I was out sick today and class ran without me. According to a report from MQ, here is what transpired in my absence. EB's job was to keep everything running smoothly. JR presented Problem 68 and showed that we can arrange the ambassadors. I'd like to revisit this one on Wednesday (along with Problem 65) to make sure we have the right approach. OW presented Problem 69 and went with a "half-life" approach, reasoning that if the man took the pills after a certain amount of time, the dosage would be small enough to not kill him. MG followed up and said that if all four pills were cut in half, and the man took one of each half, the total dosage would add up to one pill of each. Next, JG worked through the algebraic proof for Problem 70 and using the triangular number formula, plugged it in and rearranged as needed. Lastly, VM showed us the visual proof for Problem 70 by clever rearrangement. SS and KW captured some pictures of what got put on the boards.
  • Wednesday, November 8: Even though we didn't accomplish everything we set out to do today, we got a lot done. We spent over half the class period reviewing the key ideas of Problems 65 and 68. I left a few of the details for Problem 68 to the students to take care of for homework. With the time we had left, we heard from DO, KS, and MG about Problem 72. We didn't have time to discuss Problem 72, so we will have to squeeze that one in next time.
  • Friday, November 10: Veteran's Day! No class.

Week 12

  • Monday, November 13: We finally wrapped up Problem 68 (ambassadors at table), which I spent the first minutes of class going over. Next, we had TN discuss her approach to Problem 71 (paper folding area). This was followed up by a very elegant solution to the problem by KR. Then MQ presented his excellent solution to Problem 74 (100 prisoners and a light switch). After that we had AG discuss his solution to Problem 73, but we realized that we made a mistake. We will return to this one on Wednesday. With the few minutes we had left, I quickly summarized the technique of induction.
  • Wednesday, November 15: We had VM/MQ, SS, and JL present solutions to Problems 73, 75, and 76, respectively. With the time we had left, we had a brief discussion of the Four Color Theorem and got started on two of the next problems.
  • Friday, November 17: The students took Quiz 6.

Week 13

  • Monday, November 20: We had KS, EB, and CW presented Problems 77, 78, and 79, respectively. I also discussed Google Page Rank after Problem 79.
  • Wednesday, November 22: First, we quickly summarized Problem 80 (which turned out to more or less be identical to Problem 79). After that I walked us through Problem 81(a), which was followed up by a presentation of Problem 81(b) by DC. After showing an alternative approach to Problem 81(b), we modified it to get a solution to Problem 81(c). Next, MQ presented a short and slick solution to Problem 82. We we spent the last few minutes going over the problems due for the next homework assignment.
  • Friday, November 26: No classes, Thanksgiving Holiday!

Week 14

  • Monday, November 27: We accomplished quite a bit today. After discussing Problem B.1 from Quiz 6, we started discussing the problems that were due today. We had AT, YS/JL/TN, and JK/QS/MQ present Problems 83, 84, and 85, respectively. We had time at the end to get a jump start on Problems 86-88.
  • Wednesday, November 29: DC, HR, and MG each made contributions to Problem 86 (chameleons) and then I led a discussion to wrap it all up. Next, MF and VM presented two distinct solutions to Problem 88 (cut to make square). This was followed by discussion of Problem 87 (Good Teacher). JG showed us a solution that involved a repeated root and then EB shared her proposed method of attack. VM summarized her approach to the problem and provided a solution that did not involve a repeated root. We spent the last few minutes getting a start on Problem 89 (colored number line).
  • Friday, December 1: Students took Quiz 7.

Week 15

  • Monday, December 4: We had JG, JR, HR/MQ, and EB present Problems 90, 91, 92, and 89(a). We didn't have time to discuss part (b) of Problem 89. Hopefully, we can do that on Wednesday.
  • Wednesday, December 6: We began by revisiting Problems 89 and 92. Next, we had MF present his solution to Problem 93, which TN also chimed in on. This was followed by a discussion of Problem 94, which was mostly lead by me. With the time we had left, JG presented Problem 95.
  • Friday, December 8: Last day! I'll miss this group of students. After discussing the final exam briefly, we had YS, JL, EB, and JR present Problems 96(a), 96(b), 96(c), and 97, respectively. Next, we reviewed a few problems and key ideas. In particular, we discussed parity arguments, in general.

Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
  Google Scholar
  Impact Story

Current Courses

  MAT 411: Abstract Algebra
  MAT 690: Genome Combinatorics

About This Site

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Land Acknowledgement

  Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.