A Paltry Geometric Dilemma, Part 2

Via Internet Time-Wasters III at Making Light, here’s the World’s Hardest Easy Geometry Problem:

From the site:

Using only elementary geometry, determine angle x. Provide a step-by-step proof.

You may only use elementary geometry, such as the fact that the angles of a triangle add up to 180 degrees and the basic congruent triangle rules (side-angle-side, etc.). You may not use more advanced trigonomery, such as the law of sines, the law of cosines, etc.

Small hints are available on the site, but they’re very small. Think of this as something to keep you busy while you’re digesting all that turkey.

(The first Paltry Geometric Dilemma.)

Addition and Multiplication

Monday Math Madness #12 is up at Blinkdagger, and features Marvin the Martian (who turned 60 years old this past July 24. Happy Birthday Marvin! And wasn’t it cool of NASA and friends to use Marvin in the patch for the Mars Exploration Rovers?)

At any rate, this week’s puzzle is particularly challenging. One person picks two whole numbers between 2 and 99, tells the sum to a second person and the product to a third person. The second person tells the third person they  [Person #3] can’t possibly know the original numbers, and the third person realizes that that is enough information to figure it out.  With that revelation, the second person is able to figure it out.  Your job is to find the numbers.

Seriously, that’s all the information that you get, though it’s phrased perhaps a little more clearly at Blinkdagger. And at the moment I have little idea how to solve it, but I’m working on it. It did, however, remind me of one of my favorite problems that I occasionally given to non-majors in a “distribution requirement” math class. The problem involves a census taker who asks a parent the ages of the three children who live in the house. The ages (whole numbers) multiply to seventy-two, and add to the house number. The census-taker looks at the house number and says, “That’s not enough information.” The parent agrees, and comments that the oldest child has a pet rabbit, and that’s enough to solve the problem.

Like I said, I love this problem, but my students are often a little overwhelmed when I assign it. This led to one of my favorite ever teaching exchanges, which went something like this:

Student: Does it matter that it’s a rabbit?
Me: Not in particular. It could be a dog. Or a cow.

The student thought for a while, then:

Student: I got it! “Rabbit” in French is lapin, which has 5 letters. “Dog” in French is chien, which has 5 letters. And “Cow” in French is vache, which also has 5 letters. Am I on the right path?

One the one hand, I loved the student’s enthusiasm (which was not unusual for this student) and also the willingness to try new ways of thinking. And this student was no slouch mathematically, and was a joy to have in class. On the other hand, it really gave me insight into what word problems must seem like to a non-mathematician, if translating the words into a foreign language and then counting the letters seemed like a reasonable course of action. In problem solving, “easy”, “hard”, and “obvious” are in the eye of the beholder, not necessarily the eye of the author of the problem. [Which isn't me -- I've seen versions of this problem in several places.]

And in good news, my student did go on to solve the problem correctly.

Saturday Smorgasbord

Now is the time on Sprockets 360 when we plagiarize share some of the fine things going on around the Internet.

Are you looking for some fun math problems to do? You can try out the bi-weekly Monday Math Madness. MMM #10 at Blinkdagger is a probability problem with a twist: if you know the probability of a group winning the lottery over 25 years, what is the probability of someone in the group winning the lottery at least once in a 5-year period? Solutions are due by Monday night/Tuesday morning at midnight. MMM #11 will appear the following Monday on Wild About Math.

There’s also a new regular math problem being posted. Walking Randomly has started posting an Integral of the Week. Integral #1 is . (Incidentally, if you google “Walking Randomly” then the program suggests that you mean “Working Randomly”, which feels rather like my summer.)

Finally, why was 6 afraid of 7? Because 7 8 9. (Thank you, thank you, I’ll be here all week.) The Barenaked Ladies said it much better on this YouTube video, which I actually found here at Wiskundermeisjes, a site created by Ionica Smeets and Jeanine Daems. (This isn’t the first time I’ve stolen borrowed from them, either — I originally found the idea for Godzilla’s Sierpinski Cookies from Evil Mad Scientist Laboratories on their site. Indeed, their site really makes me wish I could speak Dutch.)

Crossnumbers

The San Francisco Chronicle carries Sudoku by the comics, but it turns out that it has an additional Sudoku-like puzzle hidden in the classifieds. And by Sudoku-like, I mean not really at all like Sudoku except that it involves digits being put into boxes. Actually, it’s more like Kakuro now that I think about it, except you can have the same digit appear more than once in a row or column. So, in fact, it’s not really like either one.

The idea is that there is a 4×4 grid with one digit in each box. The sum of each row is given on the right (shown below in bold), the sum of each column is given below the column (shown here in bold), and the sum of the diagonals is also given. In addition, some of the entries are already given.

Cross Numbers

18
3	1			12
6				19
			4	18
		1	4	13
23	7	17	15	11

In theory, these could be solved logically, but it turns out that (at least for the puzzles in the Chronicle) there are multiple solutions, and the practice of making a guess, checking how far off you are, and then adjusting the guess works pretty well for these.

Aside: My brother-in-law Scott, who brought these to our attention (thanks Scott!) said he’d looked around on the internet for info on this puzzle but couldn’t really find anything. This may be due to the fact that “Cross Numbers” seems to refer to more than one game: I also found a game that was more like Crossword puzzles, with teach Across or Down clue leading to a number, one digit per square. You can see an example here.

Some Math for the Fourth of July

In browsing around for Math and Independence Day, I ran across the following puzzle on Ask Dr. Math. The connection with holidays is indirect, but the puzzle itself was fun to do and took me a while to figure out.

Five female-male couples (the Aston, the Barlers, the Cauchys, the Dicks, the Egglers) get together five times a year (Halloween, Independence Day, New Year’s Eve, St. Patricks Day, Valentine’s Day) for a party. Each couple hosts one of the parties at their home (on Palmer, Quinton, Rawlins, Stoddard, Talbot) and serves a different drink (coffee, fruit juice, iced tea, lemonade, punch). The wives’ first names are Freda, Ginny, Helene, Ilene, and Julia. The husbands’ first names are Kermit, Leon, Morton, Norbert, and Orville. The party decorations, food, and costumes are appropriate for each party.

Read the clues below and match everything up.

1. Mrs. Barler, Julia, and Ilene all shop at the same supermarket.
2. Everyone dressed in green for the party on Quinton.
3. Norbert and his wife picked up the Egglers and the couple that served lemonade on their way to the Aston’s party.
4. Helene and Freda both ask Mrs. Dick for the recipe for the frosting on the heart-shaped cake she served at her party.
5. The couple on Stoddard decorated with pumpkins and served iced tea at their party.
6. Orville told Mr. Eggler he should color the drink green at the Eggler’s party, but Mr. Eggler said his wife thinks it would look unappetizing.
7. Ginny’s husband made the coffee for their party while Ginny was on the phone with Mrs. Cauchy.
8. Leon, Norbert, and Mr. Barler are all on the school board.
9. The couple on Palmer always look forward to going to the New Year’s Eve and Independence Day parties because they like to make noise and stay up late.
10. Orville and his wife had a hard time finding confetti and noisemakers for their last party because they waited too long to go shopping.
11. The Rawlins couple were thinking about moving out of the city, but Orville and the Barlers talked them out of it.
12. Neither Helene nor Kermit lives on Stoddard.
13. The Saint Patrick’s Day party is not held at either of Kermit’s or Helene’s houses.
14. The Cauchys and the couple on Talbot sometimes have dinner together at a restaurant.
15. Helene does not live on Rawlins, and neither does Ilene.
16. Ginny is not married to Leon.
17. Fruit juice is not served at Orville’s party.

For some hints, see the original post on Ask Dr. Math.

(This puzzle reminds me of Who Owns the Fish?, which I got by email years ago and have used successfully in classes many times.)

Monday Math Madness Still Going Strong!

Monday Math Madness #9 is over at Wild About Math this week — solutions are due this coming Monday night/Tuesday morning at midnight. This week’s puzzle is the following:

Consider all of the 6-digit numbers that one can construct using each of the digits between 1 and 6 inclusively exactly one time each. 123456 is such a number as is 346125. 112345 is not such a number since 1 is repeated and 6 is not used.

How many of these 6-digit numbers are divisible by 8?

While you may use a computer program to verify your answer, show how to solve the problem without use of a computer.

You can find directions for submitting the solution at MMM #9. Problems are posted every other Monday either by Sol on Wild About Math or by Quan and Daniel at Blinkdagger; solutions are due a week later, and a winner is announced the following Friday. The problems have been fun to play around with, and isn’t that the point after all? (And as a bonus, they offer free prizes to the winners!)

Two dimensions and beyond (or at least between)

I fear that this will be a shaggy dog post. One of my students posed an interesting question the other day, and I found myself surprised by the answer. So here’s the quick version of today’s post: What’s the dimension of a tetrix? A tetrix is also known as a Sierpinski tetrahedron, and indeed it is like a three-dimensional version of the Sierpinski triangle, except that, being a fractal, its dimension is less than three. There’s a picture below (which I think of as upside-down), and also a cool java applet version here which you can spin around.

Moving the 6

George Mach posed this problem to one of his classes (out at Cal Poly SLO in California), and my dad passed it along for posting here. The answer may be surprising, in the sense that it’s not something that can be easily guessed. Here’s the question:

Find a whole number ending in 6 which is doubled if you move the 6 from the end to the beginning. (e.g. the number 316 almost works because 631 is close to the double of 316, but not quite)

I’ll post the answer here tomorrow….

Pi Day Sudoku

In honor of Pi Day, Brainfreeze Puzzles (“we turn coffee into puzzles”) created a Pi Day Sudoku on a 12×12 grid.

The rules are a little different from standard Sudoku, in part because the blocks are jigsaw pieces rather than 3×3, and in part because the first 12 digits of π are used instead of the standard 1-9. Each row, each column, and each colored block (“jigsaw region”) contains the first 12 digits of pi

3 1 4 1 5 9 2 6 5 3 5 8

in some order. In particular, there are two 1s, one 2, two 3s, one 4, three 5s, one 6, no 7s one 8, and one 9.

As a bonus, on Brainfreeze’s Pi Day site there are instructions for how to (possibly) win an autographed book by completing the puzzle. Woo hoo!
The contest is over: you can find the solution here.

“Double Down” Dumbs It Down

Double Down is a quiz show for New York State high school students that airs on PBS (WCNY to be specific). I happened to catch a rerun last night, and one of the categories was “Math”. Here are a couple of the questions:

1. A polygon with 5 sides is called what?
2. A polygon with 8 sides is called what?

These are high school students, remember. My 20-month old daughter knows what an octagon is. Can we give these kids some credit?!

Birthday Math

Today is Shawna’s birthday, and I was reminded, as I am every time someone celebrates a birthday, of a problem my high school physics teacher posed to us when asked about his age:

Last year my age was a perfect square. Next year my age will be a perfect cube.

In fact, his age was the only solution to that problem.

I wondered if I could come up with such a description of my own age (Shawna’s too, but I’m not going to share that one). I wanted the description to be unique in some sense, and the best I could come up with was a minimal solution:

My age is prime, the sum of two consecutive composite integers, won’t be prime again for six years (sexy primes!), and is the smallest such age.

Does anyone else have a cool way of describing their age?  (Note the implication about my own description.)

Why Doesn’t This Work?

A common example of integration by parts used in many Calculus II classes has students compute

by integrating by parts twice, then rearranging terms to arrive at a solution. This technique is handy for many functions whose derivatives eventually repeat, that is, functions satisfying

for some integer n and some constant c. (Question: Is there a name for such functions? I feel like I should know this.)

Yes, I’m a Nerd

On a recent visit to my mom’s house (also the house in which I grew up), I was going through some of my old things (that is, I was asked to take them with me or throw them out) and I came across some math competitions I had taken in high school. (See, here’s the part where you say, “Wow, you’re a nerd,” and I reply, “See the title of this post.”) Specifically, I found two years of the American High School Mathematics Examination (AHSME, pronounced ahz-mee), and one of the American Invitational Mathematics Examination (AIME, pronounced ay-mee). My initial reaction was to recycle them, but something made me peek at the questions, and I ended up bringing them home, mostly to find out if I had any idea what I was doing back then.

Something to contemplate in the kitchen today

Consider the following advice for cooking the perfect Thanksgiving turkey:

The final temperature of the bird, after “resting” for 15 to 20 minutes, should be at least but not much more than 165 degrees Fahrenheit. The temperature will continue to rise 5 to 10 degrees after the turkey is removed from the oven. [theperfectturkey.com]

Does this mean that turkeys violate Newton’s Law of Heating and Cooling?

Your Brain Games for Today

Here are a couple problems to work on during your lunch break:

1. Let a and b be real numbers such that a+b=1. What is the relationship between a²+b and a+b²? Does this change if we allow a and b to be complex numbers?
2. Let n be a 1-digit whole number. How many 6-digit numbers contain at least one n? Generalize this to k-digit numbers.