Science Buddies: "Ask an Expert"

Hi. I'm doing a project on the fib. sequence, but since some of my math skills are weak, and I do not know the exact project I want to (just that I want to do something related to the seq), my teacher reccommended that I do some math problems over the break, so I can get some higher level math practice and more creative.

However, on a list of 15 practice qs, I haven't been able to do more than 2 in the two weeks I've been working on it (a lot, too...)

The two I am most having trouble with follow along with what I have so far.

1. PROVE (ahh) that (x+1)(x^2+1)(x^3+1) is less than or equal to 4(x^6+1) for all real numbers x.
So far, I've only been able to, basically, simplify it. I got to the following (and after that, I just seem to be going in circles): x^4+x^3+2x^2+x <= 3x^6. I know it isn't much. It took a page of math, and it is the seventh one I got, so I'm not sure if that is right. I keep finding mistakes in my work, but so far, that one seems right. That's all I have for this...also, just my rationale that the left side isn't "growing" as quickly, but I know that is not a "proof," so..yeah.

2. What is the largest number of points that can be placed in (or on the boundary of) a 2x2 square so that the distance between each pair of pts is at least one? Justify your answer.
I got the answer 9 - with a pt at each vertex. I tried to find other ways and couldn't. I can't justify it really...I just did it experimentally, so I'm not sure it is right either.

Thanks for helping. I am completely exhausted. I seem to have been going in circles and just doing nothing while trying so hard for days now.

Yay. I found a way to do number one...but still not number two.

Hi brightdevil1!

Re problem 2: you got the right answer! I don’t have a formal proof here, maybe someone can help. As means of justification, If you look at http://www.maa.org/features/mathchat/ma ... _2_99.html it’s clear that nine stones fit. The best solution for 10 points, has to place stones closer than 1 unit apart on the webpage I referenced.

I don't know the exact coordinates of the points, but it's clear that with 10 points, the points in the middle are squeezed closer together.

Yay. Thank you so much!

Finding quick ways to prove problems like #2 involve realizing that they are fundamentally a combination of geometry, boundary conditions, packing density, statistics, and modeling. In other words, different mathmatical approaches can be used to construct proofs and ones familiarity with the different disciplines will lead to slightly different proof approaches.

Being able to visualize and think about spacial or gometric problems is an acquired skill that often starts early in life with building blocks, tinker toys, and puzzles. If you aren't good at it and you want to get better, go play with some toys and think about geometry and math as you do it.

Hi Brightdevil1,

In proving that 9 is the maximum number of stones, three items are required:

1. prove that 9 stones is possible:this is done by placing 8 stones at the vertices and half vertices and 1 stone at the center.

x x x
x x x
x x x


2. prove that the only 9 stone solution is the one listed in 1) above. Proving this is more difficult. Basically a solution must find points distance 1 apart and have 9 points. In reaching 9 stones, the vertices get populated as “farâ€