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Problem of the Week (POW)

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:

There are six (3 factorial) possibilities for how the goat, key, and car are arranged in the doors. Bob will find the car 2/3 of the time since he gets to select two doors if need be, so the other 1/3 of the time they lose for sure.

Assuming they go by the ideal plan listed above, Alice will find the key 75% of the time in her two picks. This is because the door which was not part of Bob's plan is guaranteed to have either the goat or the key. If it had the car, Alice wouldn't be playing because Bob wouldn't have picked it since it wasn't part of his plan. Thus, Alice has a 50% chance of finding the key with that door, and then if that fails it is a 50/50 guess between the other two (one would have the key and the other would have the car that Bob found). Note that it doesn't matter which order Alice selects doors in since she has two picks if she needs them, thus any plan in which she involves the door which Bob did not include would give them equivalent chances. Thus, 66.6% (Bob's success rate) * 75% (Alice's success rate) = 50% success rate with this strategy.

If Bob and Alice did not do this strategy, in other words, neither of their plans include one of these doors, their success rate is always 33.3%. This is because when Alice picks, one of her doors is guaranteed to have the car (since she is selecting from the same two that Bob did), and the other one thus has a 50% chance of having the key. Thus, 66.6% (Bob's success rate) * 50% (Alice's success rate) = 33.3% success rate for the couple for every strategy in which neither of them consider of these doors.

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 eⁿand n^e are both larger than 1, we can compare e^πand π^eby ln(e^π) and ln(π^e). Since ln(e^π)=π and ln(π^e)=e*ln(π),then ln(e^π) / ln(π^e)=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(e^π) / ln(π^e)>1, so ln(e^π) > ln(π^e)>0. Since g(x)=ln(x) is an increasing function,

we know that e^π> π^e.

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!