Entrance Slip Sept 21 - Locker Problem

-

I really enjoyed working on this problem! I was reminded, however, about the importance of not leaving my work to the last minute, as I caught myself working on it late Sunday evening and didn't want to give up. I was able to solve it, and I've included my thought process below. I must admit that my justification may not be up to the spec of a number theory proof.

1. I first thought there might be some clever incorporation of binary numbers given the on/off character of the problem. I realized that any locker that had its state changed an odd number of times (including the first pass) would end up closed, and any one that had its state changed an even number of times would end up open. I then realized that I needed to think about divisibility but wasn't sure how to proceed.

2. Eventually, I decided to break down the problem into a smaller locker sample. I started with 10 lockers, and then scaled it up to 25 as I noticed a pattern emerge.

  • I used a "c" to represent a closed locker and "o" to represent an open locker
  • I simply followed the "rules of the game" to draw out a diagram as seen below:
  • I noticed that the only lockers that remained closed were the perfect squares in the defined range. I decided to extend the diagram up to the 25th locker to confirm the pattern (and it was confirmed).
3. I decided to think a bit about the special divisibility properties of perfect squares and realized the following:
  • All natural numbers have factors in pairs (including prime numbers—itself and "1")
  • The only exception is natural numbers that are perfect squares (since one of the pairs of factors is in fact a duplicate). Thus, the perfect squares have an odd number of distinct factors.
4. Using this "lemma", I realized that only the lockers numbered by the perfect squares between 1 and 1000 would have their state changed an odd number of times. That is because a locker would only be touched a number of times equal to its total number of distinct factors.

Conclusion: The only lockers remaining closed would be the set of perfect squares of the natural numbers between 1 and 1000 inclusive.

Comments

  1. Susan Gerofsky28 September 2020 at 11:24

    Lovely, Adam! I'm so glad that you did get caught up in the puzzle -- what a happy state to be in (even if it was a bit late in the evening...) I like your thought about the possibility of a binary model. Your solution is very good!

    ReplyDelete

Post a Comment

Popular posts from this blog

Dishes Problem

-
Image
To solve this problem without algebra, I used proportional reasoning. Specifically, I re-interpreted the givens as the average fractional amounts of each dish that one guest would eat, and then constructed a ratio of those values. By extension, this ratio also represents the relative proportions of each type of dish for the whole dinner party. Since there must be an integer number of dishes of each type, we simply scale up the lowest whole number expression of that ratio until the total number of dishes is equal to 65. The number of any one of the types of dishes can then be used to determine the number of guests. I think that culturally diverse examples with rich imagery have the potential of engaging a broader swath of students, and there is little to know disadvantage of using such examples so long as the teacher makes sure to not appropriate other cultures or reduce examples to a sort of tokenism. 
Read more

Mathematical Understanding and Multiple Representations

-
What convinces (or doesn't convince) you in the authors' argument? I would have to say that the authors’ arguments were quite convincing to me personally, not only because of the points made and studies cited, but also because I am predisposed to agree based on my personal experience and observation. The main conclusions—that students must practice using multiple representations, that representation is a fundamentally social activity, and that instruction must use varying techniques—are in general agreement with what I have personally realized in my time as a student and my limited time as a teacher. Furthermore, the way that the authors used not only examples, but also real case studies, to support their points was effective in convincing me of their position. For example, Tchoshanov’s pilot study where three groups of students were taught trigonometry with controlled levels of representation (one group with just an analytic approach, one with just a visual approach, and one w...
Read more

Homework Writing - Sept 23

-
  1. Writing about your favourite and least favourite math teachers -- and why. Favourite Teacher: I am quite lucky to have had many great math teachers over the course of my academic career, and perhaps this is one of the reasons why I am attracted to this particular occupation. If I must choose just one, I will focus on my grade 12 Functions teacher. I choose to write about him because he was one of the first math teachers that truly challenged me as a student, but simultaneously provided the support for me to succeed (and he sincerely wanted me to succeed). It was much more difficult to receive high marks in his class compared to the sections run by his colleagues, and so I, along with many other students, was initially upset by what seemed to be an injustice to us who were competing for university entrances and academic awards. However, as time went on, I quickly realized that this teacher pushed his students so that they would understand concepts more deeply, learn to be adapt...
Read more