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A base-ic problem
Consider numbers written in base b, where 1< b<10 (and b is an integer).
Find an easy-to-check criterion to test whether the number expressed in base b is even or odd. Justify your answer. (In case of different criteria that work, one that is easier to check will be preferred)
Submit your solutions by email to jgalovich, or bring to my office, PE 261. Deadline is Monday, 9/23/09, at noon.
Thanks to all who entered the Monty Hall contest! We got quite a few proposed solutions, but the earliest correct one was from Sean Landman, so congratulations to Sean, this week’s winner!
Here is Sean’s solution:
Let’s make a(nother) deal….
Alice and Bob face three doors marked 1, 2, 3. Behind the doors are placed, randomly, a car, a key, and a goat. The couple wins the car if Bob finds the car and Alice finds the key.
First Bob (with Alice removed from the scene) will open a door; if the car is not behind it he can open a second door.
If he fails to find the car, they lose.
If he does find the car, then all doors are closed and Alice gets to open a door in the hope of finding the key and, if not, trying again with a second door. Alice and Bob do not communicate except to make a plan beforehand. What is their best strategy?
Send your solutions by email to jgalovich or put by my office door, PE 261. Deadline is Monday, 9/14/09.
Congratulations to Wengzhi Wang who had the earliest submitted correct entry to last week’s contest. Here is Wenzhi’s solution:
Because en and ne are both larger than 1, we can compare eπ and πe by ln(eπ) and ln(πe). Since ln(eπ)=π and ln(πe)=e*ln(π),then ln(eπ) / ln(πe)=n/(e*ln(π)).
Now consider the function f(x)= x/(e*ln(x)). Taking the derivative,
f'(x)=[e*ln(x)-e] / (e*ln(x))2=[ln(x)-1]/ e*[ln(x)]2
Notice that when x=π, ln(x)>1 ,so f'(π)>0. We conclude that f is increasing and so f(π)>f(e)=1. That is,
ln(eπ) / ln(πe)>1, so ln(eπ) > ln(πe)>0. Since g(x)=ln(x) is an increasing function,
we know that eπ> πe.
How big is your π?
Determine, without using any technology, which of the following numbers is larger:
e^pi or pi^e
Prove that your answer is correct!
Please send your solutions electronically to me at jgalovich, or drop off at my office, PE 261.
Contest ends at noon on Friday, September 4.
First correct solution gets to choose from my prizes box!
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