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, January 15: Martin Luther King Jr. Day, no classes.
• Wednesday, January 17: First day! The first few minutes of class were devoted to me attempting to learn names. Next, I summarized what to expect from the course, toured the the course webpage, and summarized a few items on the syllabus. With the time we had left, we discussed Problems 1 and 2 from the Problem Collection. AM2, CN, and JS2 volunteered to discuss their approaches to Problem 1. Since we were short on time, I blurted out the answer to Problem 2 in the last minute of class.
• Friday, January 19: I think we had a great second day. After fielding a few questions about the syllabus and reminding students about the day-to-day structure, we divided the class up into 6 small groups, each tasked with discussing one of the homework problems. We had JG/JS2, JB/DW/AM2, AW/PH presented their proposed solutions to Problems 3, 4, and 5, respectively.

## Week 2

• Monday, January 22: Another great day! We kicked off with KP presenting Problem 6(a), which involved pebbles on a 5 by 5 grid. Next, we divided the class up into small groups, each tasked with discussing one of the remaining homework problems. After a few minutes, we had KF, HM, JC, JS1 and JG present Problems 6(b), 6(c), 7, 8, and 9, respectively. We spent the last few minutes of class discussing the next few problems.
• Wednesday, January 24: I had a lot of fun today. We kicked off by briefly discussing the upcoming quiz. Next, we had GS present his method of attack on Problem 10. This was followed by AM2 discussing an alternative approach. Next up was Problem 11, which was a disguised version of the Sleeping Beauty Problem. This problem is famous for not having an agreed upon solution. The two most commonly argued answers are 1/2 and 1/3. Roughly a third of the class believed the answer was 1/3 and the rest were in the 1/2 camp. We had DW argue the 1/2 answer and then DC and TM argued the 1/3 position. Problem 12 is meant to be a modified version of Problem 11. TB used a clever sleeping strategy to argue his point of view while PH avoided the attempt to relay extra information. Interestingly, both approaches arrived at the same expected dollar amount in the end.
• Friday, January 26: The students took Quiz 1.

## Week 3

• Monday, January 29: Lots of student presentations today! We had KB, SG, BK, KP, MN, JR, KR, MA, DW, and JS2 present Problems 14(a), 14(b), 14(c), 15, 16(a), 16(b), 16(c), 16(d), 16(e), and 17, respectively. We didn't have time for Problem 13, so we will tackle that one next time.
• Wednesday, January 31: After handing back the quiz, we discussed Problems B1, B2, and B3 from Quiz 1. AM1 presented an elegant solution to Problem 13, which was left over from last time. Then CN, TM, AM2, and JS1 presented Problems 18(a), 18(b), 18(c), and 19, respectively. We didn't have time to discuss Problem 20, so that one will have to wait until Friday.
• Friday, February 2: Dr. Hagood covered for me while I was out of town. My understanding is that JS2, BH, DW, and PH presented Problems 20, 21, 22, and 23, respectively.

## Week 4

• Monday, February 5: We spent the first half of class reviewing Problems 20-23. I drew pictures for the visual proofs for Problems 20-22 and then we discussed two different approaches to Problem 23. Next, we had KF and AM2 present Problem 24. We didn't have enough time to discuss Problems 25 and 26, so we will tackle those next time.
• Wednesday, February 7: First, JG presented an elegant solution to Problem 25 (40 cookies on a plate). Next, AW gave a nice and quick solution to Problem 26 (Sylver Coinage). This was followed by an attempt at Problem 27 (maximal intersections of lines) by SG. KP volunteered an alternate approach to Problem 27, which generalizes to an arbitrary number of lines. We had a quick discussion about the connection to triangular numbers and the formula for $n+1$ choose 2. MN used one of my puzzles as a prop to argue a solution to Problem 28 (mouse eating a $3\times 3\times 3$ block of cheese). Lastly, JC presented a slick solution to Problem 29 (prisoners with dots on the back of their heads), but realized that it didn't generalize to having an odd number of prisoners. GS offered up a modification to the solution presented that could handle any number of prisoners.
• Friday, February 9: The students took Quiz 2.

## Week 5

• Monday, February 12: After a quick rambling about growth mindset and productive failure, we jumped into student presentations. We had BK, JB, KR, and KB present Problems 30 (prisoners with flashlight), 31 (color vertices in grid), 32 (color points in plane), and 33 (nuggetable numbers).
• Wednesday, February 14: After handing back Quiz 2, we discussed Problems B1, B2, and B3. Next, we had JR, DW, and AW present Problems 34 (Star Base), 35 (more Star Base), and 36 (Zoltar). We made a conjecture about Problem 35, but didn't wrap up the details. We'll revisit this problem on Friday.
• Friday, February 16: Most of the class meeting was spent discussing Problem 35 (Star Base). AM2 proved one implication of the conjecture about Problem 35 and AM1 attempted to prove the reverse implication. With the time we had left, BH and TB quickly presented proofs for Theorems D and E from Problem 37.

## Week 6

• Monday, February 19: Lots of Circle-Dot today! We divided the class up into 6 small groups, each tasked with proving one of the remaining theorems about Circle-Dot. We had MY, BH/JS1, MN/TM, KF, GS, and PH, present Theorems F, G, H, I, J, and K, respectively from Problem 37. We also had AM2 share her conjecture and some justification about which sequences of circles and dots are attainable in Circle-Dot. We left it as an open question.
• Wednesday, February 21: At the beginning of class, JS2 offered to revisit Circle-Dot. His initial conjecture was that all sequences of circles and dots were obtainable, but he realized there was a flaw in his argument. Note that this doesn't imply anything one way or the other. All we know is that his approach failed. Our discussion of Circle-Dot was followed by a discussion of axioms, axiomatic systems, theorems, incompleteness, and decidability. Next, KP and AW nicely handled Problem 38 (number of ways to make 110 from 14 distinct numbers). Then HM presented an easy to follow algebraic proof of Problem 39 (yet another identity involving triangular numbers). With the few minutes we had left, I hinted at a visual proof for Problem 38. We will tackle Problem 40 on Monday next week.
• Friday, February 23: The students took Quiz 3.

## Week 7

• Monday, February 26: The first thing we did was revisit a visual proof for Problem 39. Next, we had JC, GS/TB, KP, and SG present Problems 40, 41, 42, and 43. Along the way, I discussed pancake sorting, as well as the connection between genome rearrangements and reversal sorting. We had a couple minutes left at the end to discuss Problems 44 and 45, which are due on Wednesday.
• Wednesday, February 28: Despite the weather, most students were in attendance. We had KF, JS2, and AM2 present Problems 44-46, respectively. Problem 44 went smoothly, but as expected Problems 45 and 46 were left slightly unresolved. Certainly, a solution to Problem 46 will take care of Problem 45.
• Friday, March 2: Dr. Hagood covered for me while I was out of town. My understanding is that SG, DW, KP, KR, and JS2 presented Problems 47(abc), 47(def), 48, 49, and 50, respectively.

## Week 8

• Monday, March 5: After finally handing back Quiz 3, I spent a few minutes discussing Problem B3 (Circle-Dot stuff). After talking about the quiz, we revisited Problem 46 (light up squares in $n\times n$ grid), which we never officially settled last week. Next, we had PH and MA present Problems 49 and 51. We revisited Problem 49 because it wasn't officially settled. We will tackle Problems 53 and 54 on Wednesday.
• Wednesday, March 7: I started class by kicking around ideas about Problem 57, but we never ended up formally discussing it. We will come back to this one next week. We had JR, JB, AM1, BK, MN, and AW present Problems 52, 53, 54, 55(a), 55(b), and 56, respectively.
• Friday, March 9: The students took Quiz 4.

## Week 9

• Monday, March 12: DW, HM, PH, and KB presented Problems 57, 58(a), 58(bc), and 59, respectively. We will get to Problem 60 next time.
• Wednesday, March 14: The first few minutes of class were devoted to wrapping up Problem 57. Next, we had TM, KR, DW, and BK present Problems 60, 61, 62, and 63, respectively.
• Friday, March 16: Good turn out today, considering it is the day before spring break. We had CN, JC, AM, and BK present Problems 64, 65, 66/67, and 68, respectively. We didn't nail down all the details about Problem 67, so we will have to briefly revisit that one after the break.

## Week 10

• Monday, March 26: After reviewing Problem 68, we had KR, DW, JR, MA, PH, and AM present aspects of Problem 67. With the time we had left, we discussed the next batch of problems, making progress on most of them. Along the way, we had AW present Problem 71.
• Wednesday, March 28: We kicked off by reviewing Problem 64 (Quilt), which we discovered we had made a mistake on previously. Next, we had JS1, JS2, and HM present their proposed solutions to Problem 69. This was followed by presentations for parts (b) and (c) of Problem 70 by JB and AM, respectively. Next, we had AW summarize his approach to Problem 71, which he had previously shown us on Monday. Then TB did an excellent job of describing the solution to Problem 72. We had some time at the end of class to discuss the next batch of problems.
• Friday, March 30: The students took Quiz 5.

## Week 11

• Monday, April 2: We had BH, TM, GS, and KF present Problems 73, 74, 75, and 76, respectively. With the time we had left at the end, we briefly discussed Problems 77 and 78.
• Wednesday, April 4: We had AW, JG, KP, AM, and KF present B2 from Quiz 5, B3 from Quiz 5, Problem 77, Problem 78, and Problem 79, respectively.
• Friday, April 6: Sarah Watson covered for me while I was out. SG, KB, and JS2 presented Problems 80, 81, and 82 respectively.

## Week 12

• Monday, April 9: We spent the majority of the time reviewing the key ideas from Problems 80 and 82. We left one aspect of Problem 82 open with the intention that students would resolve it by Wednesday. We also had AW summarize his approach to Problem 81. With the time we had left, CN presented a solution to Problem 84, which is one of my all-time favorite problems. Problems 83 and 85 are still outstanding.
• Wednesday, April 11: After revisiting Problem 81, we had KF, DW, and AM/PH presented Problems 82, 83, and 85, respectively.
• Friday, April 13: The students took Quiz 6.

## Week 13

• Monday, April 16: We spent some time reviewing Problem 85 (12 coins) and then started discussing Problem 87. MA and TB discussed their thoughts on parts (a) and (b) of Problem 87 and then we spend the rest of class discussing my potentially sketchy approach to parts (b) and (c).
• Wednesday, April 18: AM shared her thoughts on part (b) of Problem 87 and then I addressed part (d). Next, we had KP, TM, MN, JC, and KR, DW present the remaining parts of Problem 87. This was followed by presentations of the cases for 2, 3, 4, and 5 orbs by DW, KF, and JS2.
• Friday, April 20: We kicked off with JS1 presenting Problem B.2 from Quiz 6 and then I discussed Problem B.3. After I fumbled with the case of 5 orbs for Problem 88, we decided to put it aside and tackle Problem 89. We had KF, BH, MN, and JS1 present parts (a), (b), (c), and (d), respectively, of Problem 89.

## Week 14

• Monday, April 23: We spent some time at the beginning of class discussing the Futurama Theorem. Then we had TB, JB, AM, JR, and GS present Problems 88($n=6$), 88($n=7$), 88($n=8$), 91, and 90, respectively. Problem 90 is related to Google PageRank and we will return to this on Wednesday.
• Wednesday, April 25: I spent the first few minutes discussing Google PageRank in the context of Problem 90. Then we had KB and AM share their proposed solutions to Problem 92 (Good Teacher). This one was left unresolved. Next, SG presented an elegant solution to Problem 93. With the few minutes we had left, we briefly discussed induction.
• Friday, April 27: After a short wrap-up discussion of Problem 92, we had AW and BH present Problems 94 and 95, respectively.

## Week 15

• Monday, April 30: The students took Quiz 6.
• Wednesday, May 2: We had KP, JS1, and JC/PH present Problems 96, 97, and 98, respectively.
• Friday, May 4: Last day of classes! I had a fun semester with this group of students. We had BK, CN, TB, JS2, JS1, and GS present Problems 99, 100, 101, 102(b), 103, and 104, respectively.

# Dana C. Ernst

Mathematics & Teaching

Northern Arizona University
Flagstaff, AZ
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