Solving Problems From The Back

Most of you have heard about "working backwards", where you start at the end and work your way through equations to figure out what the starting conditions are. When it comes problem solving, there is a similar approach of working backwards that allows you to figure out how to find a possible path towards your end goal.

I am a huge champion / believer in "working from the front and from the back to see how they meet in the middle". (Yes I need a much shorter name for that, any suggestions?) Let me explain that with an interesting problem that I recently came across:

India Selection Test
Find all such polynomial f(x)f(x) with integer coefficients such that for all integers nn, we have gcd(f(n),f(2n))=1 \gcd(f(n),f(2^n))=1.


At first glance, it is hard to see how one can start to prove this statement. Who am I kidding, at second or third glances, there is no clear path forward. This is when it helps to "work from the end result", and see what hints (breadcrumbs) are left for us to follow.

We will use the well known results that
1) For any fZ[x] f \in \mathbb{Z} [x], abf(a)f(b) a-b \mid f(a) - f(b).
2) Fermat's Little Theorem (Euler's Theorem)
Exercise 1: Prove these results.

Hint: Read Polynomial Sprint: Useful Lemma.

Ponder this, and then move on to the next note in this set.

Note by Calvin Lin
5 years, 3 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...