Rounding Up – Way Up

Ever heard of Dudeney numbers?  Neither had I, until yesterday, when I discovered them completely by accident while reading (Wikipedia, what else?) about narcissistic numbers.  A Dudeney number (named after famous English mathematician and puzzle author Henry Dudeney) is a number that is the cube of the sum of its digits.  For example,

There are only six Dudeney numbers.  Neat numbers, but I was a little disappointed by that.  What to do next?

Generalize, of course!  Generalized Dudeney numbers (discussed here, but the link appears to be dead, so I used Google’s cached version) are numbers that are some power of the sum of their digits:

The largest number on the above site is , which has 147253 digits.  The site links to Wolfram Alpha to confirm this.  Here’s where it gets weird:

How many digits is that?  About ?  About a million?  What kind of rounding is that?  It gets worse.  Try a number with just 100,002 digits (despite what Alpha says).  I think Alpha is a great tool, and I’ve had (far too much) fun playing with it, but I’m a tad disappointed (that’s twice in one post).  So, hey, get on that, Wolfram.

Luggage Math Mistake No More!

Remember that post on August 1 about the math mistake?

The mistake here is that while individual dimensions were correctly converted to centimeters (by multiplying by 2.54), the Capacity was incorrectly converted, since the 4720 cubic inches were multiplied by 2.54 (and not the cube of 2.54) to get 11988.8 cubic centimeters.

I found this error a month or so before posting it so I know it was around for a while, but I just discovered that the error has been fixed.  Behold, the new stats!

In this case, the somehow-determined capacity of 4720 cubic inches is multiplied by (2.54)³, or approximately 16.387 cubic cm per cubic inch, to get 77,346.9 cubic centimeters.  But one liter is, conveniently [although not exactly coincidentally], exactly 1000 cubic centimeters, so this translates to 77.3469 liters, which does round to 77.3 liters.  The stats for other pieces of luggage were similarly updated.

I’m so happy!  I’d like to think that the People In Charge actually read this blog, or at least the email I sent them about it, but alas I have no evidence of this.  Still, it’s nice to see the mistake corrected.

Billions and Billions of Burgers

There’s a new restaurant in Manhattan called 4Food which is geared towards Social Networkers, according to this September 16 review on CNNMoney.com:

Customers who step into the restaurant are met by staffers ready to take orders on their iPads, a 240-foot screen featuring live Twitter feeds and Foursquare check-ins, and a menu that offers more than 96 billion customizable burger options.

This paragraph goes on to provide a link to this article in which Stacy Cowley actually explains the mathematics.  It turns out that there are some items where you can only pick 1 [1 bun among 5 choices; 1 scoop of something I don’t normally think of as being on a burger —  mac n cheese and sushi are two of the options — among 18 choices, where one of the choices is “No thanks.”; and finally 1 burger among 8 choices] but for the 4  add-ons like lettuce, the 12 condiments, the 7 cheeses, and the 4 meat slices, you can have as many different kinds as you want.  This means the total number of combinations becomes

which is 96,636,764,160.

The article contains more specific information about each of the choices, and the many many comments include additional information, like that someone in theory should be able to order no bun or no meat, although the comments mostly seem to be arguing whether or not the math is correct (no math mistakes that I see!).  Incidentally, if you allow the option not to have a bun or not to have the patty at all, which seem reasonable options to include, the 5 in the equation changes to a 6 and the 8 to a 9, giving 130,459,631,616 combinations. And that’s a lot of choice.

It’s a New Newsletter!

We finally published the Winter Spring SUMMER(ish) edition of our department newsletter!  This issue is named the Taniyama Times after Yutaka Taniyama (谷山豊 ) of the Taniyama-Shimura conjecture (proved by Andrew Wiles, and giving Fermat’s Last theorem as a wee little corollary).

This issue will admittedly hold less interest for non-alumni than most of our issues, since it’s primarily about where the Class of 2009 has been spending the past year.  Still, it does contain the following fun problems to work on!

Problem 4.2.1: (2006 AMC10) If x◊y =x³—y, what is h◊(h◊h)?

Problem 4.2.2: Which fits better: a square peg in a round hole or a round peg in a square hole?

Problem 4.2.3: The figure below shows the first three circles in an infinite sequence. What is the total area of the circles? What is the total circumference?

Answers are welcome in the comments, and you might just be acknowledged in the next newsletter!  [If we remember, which is sort of a risk since I’m right now remembering that we forgot to check that when we put together this issue.]