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

  • Wednesday, January 18: First day! After attempting to learn names, we took a tour of the course webpage and then discussed an overview of the course and the syllabus. Next, groups of 2-3 pondered Problem 1 from the problem collection for a short bit before we had a brief full-class discussion about it. We wrapped up with a quick discussion of Problem 2. Comically, I accidentally ended class 10 minutes early. I won't do that again.
  • Friday, January 20: Great first day of student presentations. I'm extremely happy with how things went. After answering a few quick questions, we divided the class up into 6 small groups, each tasked with discussing one of the homework problems. Groups spent about 20 minutes kicking around ideas related to their respective problem and then we had representatives from each group share out. Both LL and LB each presented an approach to Problem 1 and then SS discussed Problem 3. KG chimed in on Problem 3 along the way. We wrapped up with BR presenting his take on Problem 4(a).

Week 2

  • Monday, January 23: Despite the weather we had great attendance and class went really well. After reviewing a few key ideas, we split the class up into four small groups to work on Problems 5, 4(b)/6, 7, and 8, respectively. I left the groups chat for about 5 minutes and then we started discussing proposed solutions. We had MH, JM, and RD present Problems 5, 4(b)/6, and 7, respectively. Along the way, several other people chimed in. We ran out of time to discuss Problem 8. I hope we can squeeze that one in next time.
  • Wednesday, January 25: Today was a good day and we got a lot accomplished. After summarizing some key ideas and the status Problems 1-7, we moved onto Problems 8-11. SW shared her solution to Problem 8 and then MS summarized a slick solution that involved less analysis. For Problem 9, we had AT1 and AC present their solutions to Problem 9. Next, BG provided a nice take on Problem 10. I spent the last few minutes of class giving some hints on Problem 11.
  • Friday, January 27: Despite being a tremendous amount of pain, I think things went well today. We were really productive. KG's explanation (with comments from LL) for Problem 7 finally put that one to rest. Next, we had RG discuss a conjecture for Problem 12, which connects with triangular numbers. Then BS, JK, and NP presented Problem 13, a visual approach to Problem 14, and an algebraic proof for Problem 14. We wrapped up with a general discussion of Problem 11, which included guidance from RD and MS.

Week 3

  • Monday, January 30: The week is off to a good start. We kicked off by having SS and MR present visual and algebraic proofs, respectively, for Problem 15. After allowed students to chat in small groups for a few minutes, JS2 presented a slick solution to Problem 16. Next, we had students chat in small groups about Problem 17. AS discussed a case analysis for Problem 17 and then we distilled the argument down to a single case. We wrapped up by discussing Erdos and the next few homework problems.
  • Wednesday, February 1: The students took Quiz 2. We will get back to working on problems on Friday.
  • Friday, February 3: While I was out of town, Monika Keindl covered for me. Monika handed out a half sheet of paper that contained 4 problems. Students spent the majority of the class meeting working on these problems in small groups. Along the way, BG and RD presented solutions to the first and second problems, respectively. The intention was for students to finish working on the remaining two problems over the weekend.

Week 4

  • Monday, February 6: After checking in to see how things went on Friday while I was out of town, we discussed solutions to the last 3 problems on Quiz 1. Next, we recapped the solutions to the first and second problems from Friday and then discussed the third and fourth problems. We had the class split up into two groups to act out the solution to the third problem involving the frogs. NP suggested there is a nice formula to count the number of steps in the general case. We wrapped up by discussing the fourth problem involving $x-y=85$ and $\sqrt{x}+\sqrt{y}=17$. MR presented a method for finding a solution, but we left it open as to whether there are any other solutions. We still need to discuss Problems 18-20, which we will do on Wednesday.
  • Wednesday, February 8: We kicked off with a discussion of the problem involving $x-y=85$ and $\sqrt{x}+\sqrt{y}=17$. After discussing a few different approaches, we settled on there being a unique solution. However, after class, MR pointed out to me that what we said only guarantees that there are at most 2 solutions. Perhaps we will have time to address this one on Friday. Next, AC presented Problem 18, which generated good discussion about rigorous arguments compared to data having a pattern. This was followed by a nice argument for Problem 19 by KP. We finished up with AT1 giving a slick approach to Problem 21. We briefly discussed why we couldn't improve upon our answer in Problem 21.
  • Friday, February 10: I really enjoyed today's class. We started with allowing students to chat in small groups about Problems 21 and 22. We had LB and SW present Problem 21 and then MH, JM, and RD discussed Problem 22. Next, AT2, AS, and JJ tackled Problem 23. We wrapped up with NP presenting Problem 24.

Week 5

  • Monday, February 13: We accomplished more today than I expected. We kicked off with small groups discussing potential solutions to Problem 25. We had JS1 share a solution he encountered when chatting with his group. Next, we had JS2 share his proposed solution to Problem 26, but we discovered it wasn't optimal because RB proposed a better solution. I suggested folks attempt to prove that RB's solution is the best we can do. After having AT2 share his proposed approach to Problem 27, we had BG present slick solutions to both Problems 27 and 28.
  • Wednesday, February 15: The students took Quiz 2. I love watching the students collaborate for the few minutes I allow them to chat.
  • Friday, February 17: I was energetic today, but I might have been the only one. We started with a brief follow-up discussion on Problems 27 and 28. NP told us how we could take the solutions from Problems 27 and 28 and combine them into a single strategy. Next, KG provided an argument about why 17 minutes is the best we can do on Problem 26. AT2 put Problem 33 to rest with easy; another example of triangular numbers. RD and AT1 tackled Problem 34 for us. I think everyone should be able to handle any problem like this one in the future. We spend the last 20 minutes of class discussing Problem 35. LL and RB presented their proposed solutions, but it turns out that we can do better. We'll revisit Problem 35 next week.

Week 6

  • Monday, February 20: Unfortunately, I spent almost the entire class talking. I spent a good chunk of time going over Quiz 2 and then wrote down a complete solution to Problem 35 involving 12 coins. There was a few minutes at the end for students to think about what to do with 11 coins.
  • Wednesday, February 22: Great day! BS presented Problem 37, which illustrated the importance of working backwards. Next MS walked us through Problem 38. I followed this up with a discussion of how to explicitly count the number of friendships on each team. This lead to a brief discussion about how to generalize the problem. KG attempted Problem 39, but his approach was thwarted by an inability to resolve one inequality. KF followed this up with an elegant solution involving the triangle inequality. We wrapped up class with a short discussion of Problem 40, which turns out to not work as expected.
  • Friday, February 24: The students took Quiz 3.

Week 7

  • Monday, February 27: I had surgery on my back this morning and I will be out for 2 weeks. In my absence, Dr. Wilson is covering for me. My understanding is that MS presented an elegant solution to Problem 41 and then there was some general discussion of Problems 42 and 43. These problems and likely a couple more will be tackled on Wednesday.
  • Wednesday, March 1: The word on the street is that after some discussion of Problem 42 (Bin Laden), KG and RB presented the cases for $n=5$ and $n=7$ for Problem 43 (Martian Artifacts), respectively which was followed up with some general discussion of the cases involving $n=2k$ versus $n=2k-1$. Next, RD, followed by MS, presented their proposed solutions to Problem 44 (Aslant).
  • Friday, March 3: Rumor has it that the following transpired: KG tackled the case with $n=9$ for Problem 43, RD led a discussion for Problem 45, and then MS and NP presented for Problem 46.

Week 8

  • Monday, March 6: My understanding is that KP presented Problem 44 (Aslant), MS finished off Problem 43 (Martian Artifacts), RB and AC tackled Problem 48 (Magic!), and RD presented Problem 49 (Two Deep).
  • Wednesday, March 8: Today MS presented Problem 46 (the one with the $3\times 3$ and $2\times 2$ squares), RD presented Problem 45 (Double Tangent), SS and BS presented Problem 52 (One Over), and KG presented Problem 51 (The Good Teacher), but there was an algebra mistake in Problem 51, so we haven't found a solution to that one yet.
  • Friday, March 10: The students took Quiz 4.

Week 9

  • Monday, March 20: After checking in and making a few announcements, we divided the class up into 11 small groups, where each groups was tasked with coming to consensus on one of multiple parts of the homework problems. We had LB, JJ, SS, JK, SW, JM, MR, LL, BS, AT2, and RB present Problems 54(a), 54(b), 55(a), 55(b), 55(c), 55(d), 55(e), 55(f), 55(g), 55(h), and 55(i), respectively.
  • Wednesday, March 22: We finally saw solutions to Problem 50 (Checkermate) and Problem 53 (Quilt), which were presented by RD and BG, respectively. Next, AC tackled Problem 56(a). The solutions had an error, but the essence of it was correct. After this, I explained how we could model the soul-swapping via "string diagrams". This was followed by a presentation of solutions of parts (b) and (c) for Problem 56 by KG.
  • Friday, March 24: After handing back Quiz 4, we had a brief discussion of Problem B1 from the quiz. Next, MR provided a solution to Problem 56(d), which involved the swapping of souls of 4 individuals. We spent a few minutes reframing the solution in terms of the one I hinted at the previous class and then discussed how to handle any soul swapping bonanza, which settled Problem 56(e). This was followed with NP providing a solution to the 2 real versus 2 counterfeit coins in Problem 57 (Zoltar). MS then presented an elegant solution to the 7 versus 7 case, which can easily be generalized to handle any $n$ versus $n$ case (for $n\geq 2$). We wrapped up with BS presenting a solution to Problem 58 (chicken nuggets) and JM explained the code that he wrote to solve the problem.

Week 10

  • Monday, March 27: We finally saw a solution to Problem 50 (Good Teacher), which NP presented for us. Next, we split the class up into 6 small groups, each tasked with discussing one of Problems 59-61. After a few minutes, JS1 presented a solution to Problem 59 (Modified Sylver Coinage). Next, we heard from JK, AT2, and LB about Problem 60 (Cookies). Lastly, AT1 discussed the beautiful solution to Problem 61 (Prisoners and a light switch).
  • Wednesday, March 29: The students took Quiz 5.
  • Friday, March 31: Today, RB presented a wonderful solution to Problem 62. It appeared that no one had made much progress on on Problems 63 and 64, so we spent the rest of class kicking around some ideas about how to approach these problems.

Week 11

  • Monday, April 3: We started with a discussion about how to frame Problem 65 (7 positive integers and their gcd's mod 3) in terms of a graph with 7 vertices and edges that have been colored using 3 different colors. The goal is the justify that there is always a triangle of the same color. After discussing Problem 65, we split the class up into 3 small groups, which we tasked with discussing Problems 63-65, respectively. We had KG share a solution to Problem 63, which was followed by a really nice solution to Problem 65 by LL. We wrapped up with MR and RD discussing most of the details of Problem 64. There is one remaining detail to resolve before we will have complete solution to Problem 64.
  • Wednesday, April 5: We finally nailed down Problem 64 (Federation ambassadors around table). After I got us up to speed with what we knew about the problem, MS and KG filled in the rest of the details. Next, we had AC present an algebraic proof for Problem 66. This was followed by attempts at a visual proof by SW, NP, and MR for the same problem. We were pressed for time at the end, but JM was able to share his approach to Problem 67 before we departed.
  • Friday, April 7: Since Problem 67 was discussed so quickly on Monday, we revisited that problem first. During our discussion, we heard from MR, BS, MS, and RD. Next, JK presented a quick solution to Problem 69 (mellow yellow and high fructose thorn scallop). This was followed by presentations by AC, KG, MS, and MR for Problem 70 (ten people in a circle). Problem 68 (rectangle vs pentagon) was the last one we discussed. MH shared his progress and then KG started to share his approach using the triangle inequality, but we ran out of time. We will revisit this one on Monday.

Week 12

  • Monday, April 10: The first few minutes of class were devoted to wrapping up Problem 68 (rectangle vs pentagon), which I presented. Next, SS led us on a discussion of Problem 71 (4 by 4 grid with lights) that involved some case analysis. We attempted to generalize potential arguments to handle the 5 by 5 case and the 8 by 8 case. We all believe the answer to Problem 72 (8 by 8 grid with lights) is that it is impossible to start with fewer than 8 lights, but the problem remains open.
  • Wednesday, April 12: The students took Quiz 6.
  • Friday, April 14: We spent the first few minutes of class discussing my proposed solution to Problem 72 (8 by 8 grid with lights). Next, we quickly dispensed with Problem 74 (liars and truth tellers) and then moved onto Problem 73, which AT2 presented. Then SW led a discussion of Problem 75 (lions on an island). I spent the last few minutes of class discussing induction.

Week 13

  • Monday, April 17: After handing back Quiz 6, I briefly discussed Problems B1, B2, and B3. This was followed by another discussion of induction. Next, AT1 and NP presented two different approaches to Problem 77 (sums of distinct powers of 2). Then BG presented Problem 76 ($n$ lines).
  • Wednesday, April 19: More than half of the class period was spent discussing Google's PageRank, which was inspired by Problem 78 (who is the coolest?). With the time we had left, NP and LL presented both parts of Problem 79 (infinite hotel). We'll come back to Problem 80 on Friday.
  • Friday, April 21: We had JS1 and MR present Problem 80(a) and 80(b), respectively, which was followed by me presenting Problem 80(c). Next, we heard from JK and KP on Problem 81.

Week 14

  • Monday, April 24: We had 4 different solutions to Problem 82. Three of these were from SW, RD, and KG, and the fourth was from me. Next, we saw a solution to Problem 83 by NP. We spent the last few minutes of class reviewing for Wednesday's quiz.
  • Wednesday, April 26: The students took Quiz 7.
  • Friday, April 28: We started by discussing Problem 85 (top player). RD discussed his intuition about the problem, but no one had a rigorous argument. I walked us through an inductive proof (with some minor confusion on one case). With the time we had left, I guided a discussion of the first part of Problem 84. The plan is to revisit this one and Problem 86 on Monday.

Week 15

  • Monday, May 1: First, I attempted to salvage a crappy presentation that I did for Problem 85 (top player), but more or less blew it again. Maybe we will have time to come back to it at some point. Next, MH waltzed through Problem 87(a) and explained a little bit of his thinking about part (c). This was followed by a very elegant solution to both (b) and (c) for Problem 87. JJ walked us through a different approach to Problem 87(b). Ignoring my horrible hint on Problem 88 (all tetronimoes), AC cranked out all 5 possibilities. The best part of class came next. When I asked for a solution to Problem 86 (papal vote), KG rambled off a quick justification, which I promptly dismissed. I said something vague about why his approach wouldn't work and then asked for a different solution. BS suggested that I follow the hint I gave and outlined an approach. We were all in agreement that this worked, but then some students asked again why KG's original proposed solution did not work. I still wasn't convinced and again tried to push us away from it. However, after class some students followed me back to my office and walked me through KG's proposed solution. It took me a few minutes to come around, but it turns out he and the other students were correct. In fact, KG's solution is better than the one I proposed since it is shorter and handles more possibilities. I think I was hung up on my solution and couldn’t see what he and others were saying. The best part of what happened is that no one backed down from me. I think this is really cool. You were convinced by KG’s argument and didn’t just change your mind because I said it didn’t hold. Awesome. With the few minutes we had left at the end of class, KP provided a potential coloring for Problem 84. I hope to wrap this one up on Wednesday.
  • Wednesday, May 3: We got a lot done today! First, JS1 showed up possible solutions to parts (a), (b), (c), and (e) of Problem 89 (tiling with tetrominoes). AT2 provided an alternate solution to part (b). Next, MH showed us his take on Problem 91 (ages of 3 children), which was followed by a more accurate solution presented by LB. Then we spent quite a bit of time discussing Problem 90 (tiling with L-shaped triominoes). After coming up with some necessary conditions, we started hunting for sufficient conditions. BG provided the first key insight, which allowed us to dispense with the case involving at least one even dimension. To get us started on the case involving two odd dimensions, SS showed us how to tile a $5\times 9$ board, which set us up to handle the general case. I took the last few minutes to summarize all possible colorings of the number line for Problem 84.
  • Friday, May 5: A great last day! First, we had RB show us a beautiful coloring of the $10\times 10$ grid for Problem 93. Next, LL showed us potential tilings for part (a) and (b) of Problem 94. Utilizing RB's coloring from Problem 93, RD proved that no tiling exists for part (c) of Problem 94. This was followed by an attempt at unscrambling the signed permutation in Problem 95 by JJ, but he realized at the end that he had a flaw. JK then showed us one potential unscrambling that had the desired number of moves. The last problem to be discussed this semester was Problem 92 (tiling a board with a missing square with L-shaped triominoes). BG got us started, but we discovered that his solution was quite general enough. BS presented an elegant solution to the problem involving induction. With the few minutes we had left at the end, we spent some time reflecting on the semester. I'm pleased with how things went, but have several ideas for how to improve the course.

Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
  Google Scholar
  Impact Story

Current Courses

  MAT 226: Discrete Math
  MAT 690: CGT

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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 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.