Off on a Tangent 
A Fortnightly Electronic
Newsletter from the Hope College Department of Mathematics 
December 2, 2016  Vol. 15, No. 6 
http://www.math.hope.edu/newsletter.html 
Acoustical engineer will speak at next week's colloquium 
Title: Understanding what we Hear  
Speaker: Peter Laux, Ph.D.  
Time: Tuesday, December 6 at 11 am  
Place: VanderWerf 102 
Student's work comes crashing to the floor 
In a recent lesson on semiregular
polyhedra in the History of Mathematics class, the students spent time
trying to discover different semiregular polyhedra using some
construction toys.
Daiga Cers was very enthusiastic about constructing a truncated
icosahedron (soccer ball), but she was even more enthusiastic about
deconstructing as shown in the video made by fellow student Alli
VanderStoep.

Problem
Solvers of the Fortnight 

In our last Problem of the Fortnight we
presented the following conversation between Math Man and Vectoria.
"Hey, Double M! How's it going?" "Oh, hi, Vectoria. Pretty well, I suppose. Daria and Philip helped me out with my last conundrum, which was great  especially since you weren't of any assistance." "Oh, c'mon, M's. I was running late and had to jet. Besides. I gave you some hints." "Well, I suppose you did. But they didn't help me any. . . . And now I'm stuck again." "Well, what'cha got?" "Oh, it's a math problem." "Uh, yeah. I could've guessed that. What is the problem?" "Well, the problem is that I can't solve the problem!" "Uh, right. You already told me that, though, Math Man." "No, I didn't. I said I was stuck." "Same diff. . . . Let's try this again. Hey, M&M, what's the math problem you're working on?" "Oh, it's a neat one, but I can't seem to get it. The problem is: Find a formula for the function f(x), knowing that 0 < x < 1, that f(x) is positive for such x, and that [f(x)]^{2}
 x[f(x)]^{2}  x^{3}[f(x)] = x^{4}."
"Hey, that is a cool one! How in the world could somebody find the formula for a function knowing so little about it?" "I know! That's my predicament exactly," says Math Man, scratching his forehead just under his helmet. After a while, Vectoria says, "Sorry, M's. Don't think I can help you on this one. It's a stumper! I gotta get going. See ya around." "Uh, okay. yeah, see ya, Vectoria. Have a nice day" Just as she's about to head out the door, Vectoria exclaims, "Oh!" "What? Did you solve it?" "No, but I think I have an idea for how to go about it." "What is it?" "Aw, M's. I don't want to spoil it for you. And besides, it's not that hard. Any of the math students you see roaming these halls can help you. See ya later! Happy thinking!" Vectoria says as she disappears out the door. Absorbed in thought, Math Man distractedly says, "Um . . . yeah . . . See you later. . . ." Your job was to help Math Man by finding a formula for the function and explaining your reasoning carefully. Congratulations to Fiona Batamuliza, Josiah Brouwer, Branden Derstine, Ania Dlugosz, Cassie Harders, Richard Edwards, Maya FrostBrophy, Claire Hallock, Katlyn Hettinger, Erik Johnsen, Marissa Karadsheh, Jordan Lahr, David Lakanen, Philip LaPorte, Kachikiwu Nwike, Thea Patterson, John Peterson, Zheng Qu, Jada Royer, Heidi Schaetzl, Sara Seckler, Daria Solomon, Sean Traynor, Anna Wormmeester, and Jincheng Yang  all of whom correctly solved the Problem of the Fortnight in the last issue of the newsletter. 
Problem
of the Fortnight 
"Hi, M Squared! Whatcha
looking at?" "Oh, hi, Vectoria. This?" says Math Man, turning the paper he was gazing at toward Vectoria so she can see. "This is a square." "Oh, so that's what those things are called!" jokes Vectoria. "No, I mean, what's so interesting about it?" "Well, I was trying to figure out its area, but I'm stumped," says Math Man, adding in a forlorn sigh, "as usual." "That's easy, M's. The area of a square is the length of one of its sides squared." "Very funny." "All right. What's the problem?" "Well, the problem is that I can't get any of these problems!" says Math Man, erupting in frustration. "It's as if every thought I have about these Problems of the Fortnight is wrong." "Well, if every instinct you have is wrong, then the opposite would have to be right, right?" "Yes," says Math Man pensively, "I suppose so. But, I don't know. I've been staring at this figure for a long time, and I'm getting nowhere. Maybe I'll just go watch some TV." "No, don't give up M's. Let's try this again. Hi, I'm Vectoria. What's that math problem you're pondering, Double M?" "It's this. Point M is placed inside a square ABCD. Letting MB = x and MD = y and MC = z, what's the area of the square?" says Math Man, holding the paper toward Vectoria. "But, wait! There's more," says Vectoria. "The problem reads: If x^{2 }+ y^{2} = 2z^{2}, what's the area of the square in terms of x, y, and z?" "I didn't think that detail was important. But, maybe I should . . . think . . ." says Math Man, his voice trailing off as he lapses into deep thought. " . . . the opposite," says Vectoria, finishing his sentence. Help Math Man find the area of the square in terms of x, y, and z. Write your solution on a square piece of paper, and be sure to explain your reasoning carefully. Drop it in the Problem of the Fortnight slot outside Professor Mark Pearson's office (Vander Werf 212) by 3:00 p.m. on Friday, December 9. As always, be sure to include your name and the name(s) of your math professor(s)  e.g. Cy deLength, Professor A. Rhea  on your solution. Good luck and have fun! 
Off
on a Tangent 