## Archive for May, 2011

### Observations

Some things I’ve noticed…

- Algebra is
*x*sighting. - Rational people are partial to fractions.
- Geometricians like angles… to a degree.
- Vectors can be ‘arrowing.
- Calculus teachers can go on and on about sequences.
- Translations are shifty.
- Complex numbers are unreal.
- Most people’s feelings about integers are positive.
- On average, people are mean.

### Finger Multiplication

A *digital computer* is one who adds on his fingers, and this definition reminds me of a quote by Tom Lehrer.

Base eight is just like base ten, really… if you’re missing two fingers!

Many students know the trick for multiplying by 9 on their fingers, which I mentioned in yesterday’s post. However, most folks don’t know the following finger trick for multiplying two numbers that are greater than 5 but less than or equal to 10.

- On each hand, raise a number of fingers to indicate the difference between each factor and 5. For instance, 8 – 5 = 3 and 7 – 5 = 2, so to multiply 8 × 7, raise three fingers on the left hand and two fingers on the right hand, as shown below.

- Count the total number of fingers that are raised on both hands, and append a 0. As shown above, there are 3 + 2 = 5 fingers raised, so this gives an intermediate result of 50.
- Count the number of fingers that are folded on each hand, and find their product. Again using the example above, there are two fingers folded on the left and three fingers folded on the right, and 3 × 2 = 6.
- Add the results of Steps 2 and 3. For our example, 50 + 6 = 56, so 8 × 7 = 56.

We can use algebra to understand why this trick works.

Let *L* = the number of fingers raised on the left hand, and let *R* = the number of fingers raised on the right hand. Then the numbers we are multiplying are (5 + *L*) and (5 + *R*)*.*

The total number of fingers raised is *L* + *R*. Consequently, the result from Step 2 is equal to 10(*L* + *R*).

The number of fingers not raised on the left hand is 5 – *L*, and the number of fingers not raised on the left hand is 5 – *R*. Their product, which is required in Step 3, is (5 – *L*)(5 – *R*) = 25 – 5(*L* + *R*) + *LR*.

Finally, combining them in Step 4 gives

10(*L* + *R*) + 25 – 5(*L* + *R*) + *LR* = 25 + 5(* L* +

*) +*

*R**= (5 +*

*LR**L*)(5 +

*R*),

which is the product we initially wanted to compute.

I recently learned one other method of multiplication. It does not require fingers, and I’m not even sure it’s useful, but it is interesting.

To find the product of two numbers, *a* and *b*, do the following:

Plot the points A(0, *a*), B(*b*, 0), and C(0, 1).

Draw segment CB, and then construct segment AD parallel to CB.

As a result, point D(*d*, 0) is a point on the horizontal axis such that *a* × *b* = *d*.

For instance, to compute 6 × 3, plot A(0, 6), B(3, 0), and C(0, 1). Then draw a line through A that is parallel to CB. This line will intersect the horizontal axis at the point D(18, 0), so 6 × 3 = 18.

### Most Eggs-Cellent Math Jokes

You may have received the following advice about dividing fractions without thinking about it:

Ours is not to reason why; just invert and multiply.

Similarly, don’t waste your time trying to figure out why I’m posting a bunch of jokes about chickens and eggs. I can’t explain it. Just enjoy them, and please don’t analyze me.

How do you teach math to a chicken?

Show it lots of egg samples!Why do chickens hate school?

They don’t like eggs-aminations!Who tells the best math jokes on the farm?

Comedi-hens!How can you drop an egg six feet without breaking it?

Drop it from seven feet!Why did the chicken go to school?

To get an egg-ucation!Why do chicken coops have only two doors?

Because if they had four doors, they’d be sedans!

And a joke about the smartest chicken I know…

A chicken walks into a bar. “I’d like a burger and a beer,” he says to the bartender.

“Oh, my God!” the bartender says. “You can talk!”

“Well, look at that,” the chicken replies. “Your ears work!”

“But, you’re a chicken!” the bartender says.

“Ah, I see your eyes work, too,” the chicken says. “Now, can I have my burger and beer?”

“Certainly,” the bartender says. “Sorry about that. It’s just not every day that I see a talking chicken. What are you doing around here?”

“I’m working at the university,” the chicken says. He goes on to explain that he’s helping a professor with research on representation theory and integrable systems, but the bartender clearly has no idea what he’s talking about. So, the chicken enjoys his burger and beer and leaves.

A little while later, the owner of the circus comes into the bar. The bartender says, “You’re the owner of the circus, right? Well, have I got an act for you! I know this chicken who talks, reads, and drinks beer!”

“Sounds great!” says the circus owner. “Have him give me a call.”

The next day, the chicken returns to the bar. The bartender explains that he thinks he can get the chicken a great job at the circus.

“The circus?” asks the chicken. “You mean the place with the big tent, animals, lion tamers and trapeze artists?”

“Yeah!” says the bartender. “The owner would love to hire you!”

“Why?” asks the chicken. “What use would he have for an algebraist?”

### Dissections are Fun, No Matter How You Slice It

Some folks have referred to me as “a real cut‑up.” Of course, those folks are generally septuagenarians who still use words coined in the mid‑1800’s. But I take it as a compliment, perhaps because I’m a fan of dissection problems.

Speaking of dissection, here’s one of my favorite quotes about jokes, which comes from E. B. White:

Humor can be dissected as a frog can — but the thing dies in the process, and the innards are discouraging to any but the pure scientific mind.

Like jokes and amphibians, geometric figures can be dissected, too. The following are three of my favorite dissection problems.

My all-time favorite is the Haberdasher Problem, originally proposed by H. E. Dudeney in 1902. It’s my favorite because of its simplicity — it can be described with just a few words, and it asks the would‑be solver to turn the simplest of all polygons, an equilateral triangle, into a square:

With three cuts, dissect an equilateral triangle so that the pieces can be rearranged to form a square.

The second isn’t really a problem to be solved. Instead, it’s a dissection that offers a surprising result. (At least, I was surprised the first time I saw it.)

Remove a tetrahedron of edge length 1 unit from each vertex of a larger tetrahedron with edge length 2 units. In total, remove four tetrahedra, one from each vertex. The image below shows one of the tetrahedra being removed. What is the shape of the solid that remains?

Finally, here’s an original dissection problem.

Form a pentagon by placing an isosceles triangle with height 1 unit on top of a square with side length 2 units. Then find a point

Palong the perimeter of the pentagon such that whenPis connected to two vertices, the three pieces into which the pentagon is divided can be rearranged to form a square. Find the area of the largest piece.

As usual, these problems are presented without answers or solutions. More fun can be had if you solve them yourself.

### Let Me Translate For You

As any student of algebra will tell you, sometimes translating words to equations is harder than the actual math. For instance, the following limerick looks like a real bear:

A dozen, a gross, and a score,

Plus three times the square root of four,

Divided by seven,

Plus five times eleven,

Equals nine squared and not a bit more.

It’s a mouthful to read, but once you translate it to the equation below, you recognize that it is perfectly true.

This poem was written by Leigh Mercer, a panhandling palindromist who earned money by drawing sidewalk caricatures. He is also the one who gave us the famous palidrome, “A man, a plan, a canal — Panama!” Although the poem above is often attributed to math textbook author John Saxon, it originally appeared in *Games Magazine* long before Saxon ever put it in a textbook*.*

What can be more difficult than translating language to an equation is doing the opposite — taking an equation and figuring out the text from which it was derived. For instance, the equation 21*x* = 63 could correspond to the problem, “Walter Melon has 21 bags, each with the same number of apples, and all together he has 63 apples. How many apples are in each bag?” Or, it could correspond to, “How many dice do you have if there are a total of 63 pips?”

The following are some equations that were generated from quotations by famous people. Can you determine the original quotes and identify the authors?

- Humor = Tragedy + Time
- Success = Work + Play + Keeping Your Mouth Shut
- Necessities ≠ Wants
- lim (time → infinity)
*Love*=_{take}*Love*_{make} - Hesitation / Risk = Age
- Man = What He Has Done + What He Can Do + 0
- Results ∝ Effort
- Action = –Reaction
- Goals – Doubts = Reality
*V*_{life}∝ Courage- History = Σ(Things That Could Have Been Avoided)

Original Quotes

- “Humor is tragedy plus time.” – Mark Twain
- “If
*A*equals success, then the formula is*A*equals*X*plus*Y*and*Z*, with*X*being work,*Y*play, and*Z*keeping your mouth shut.” – Albert Einstein - “Our necessities never equal our wants.” – Benjamin Franklin
- “In the end, the love you take is equal to the love you make.” – Paul McCartney
- “Hesitation increases in relation to risk in equal proportion to age.” – Ernest Hemingway
- “A man is the sum of his actions, of what he has done, of what he can do — nothing else.” – John Galsworthy
- “The results you achieve will be in direct proportion to the effort you apply.” – Denis Waitley
- “Action and reaction are equal and opposite.” – Gertrude Stein
- “Your goals, minus your doubts, equal your reality.” – Ralph Marston
- “Life shrinks or expands in proportion to one’s courage.” – Anais Nin
- “History is the sum total of all things that could have been avoided.” – Konrad Adenauer

### Mathy Names

Walking into the store was a large woman wearing flip flops, shorts that were too tight, and a shirt that revealed an inappropriate amount of her midriff. She approached the customer service desk, and she told the clerk that she would like to return the pink lawn flamingoes she had recently purchased. When asked the reason for the return, she responded, “My husband thinks they don’t look good next to the plastic deer and gnome on the lawn.” The clerk gave her a form to fill out. I watched over her shoulder as she scribbled her name:

Lois Carmen Denominator

That incident reminded me of my first teaching assignment. If name implies destiny, then I was primed for a spectacular year!

- Matt Amatics
- Cal Culator
- Vin Culum
- Cal Culus
- Rose Curve
- Polly Gon
- Al Gorithm
- Polly Hedron
- Al Jabra
- Ella Ment
- Perry Meter
- Polly Nomial
- Hy Perbola
- Lisa Perbound
- M. T. Set
- May Trix
- Al T. Tude
- Norm Ull

Know any others of this ilk? Post them in the comments section.

### Husband of the Year

My wife and I are celebrating our seventh anniversary today. The following joke epitomizes our relationship.

My wife and I were comparing notes the other day. “I have a higher IQ, I did better on my SATs, and I make more money than you,” she pointed out.

“That’s all true,” I said, “but if you’re so much smarter than I am, why weren’t you smart enough to marry up?”

I’m a good but not great husband. (My wife would agree with that assessment.) But it’s not my fault. It’s just the way I’m wired. I spend way too much time thinking about numbers and shapes. While listening to her story about a really frustrating colleague she’s been dealing with, I’ll begin to daydream of triangles. Awakened when she asks, “Can you believe that?” I garner no good will when I respond, “Of course not. I mean, who would have thought that the first two Brocard points are isogonal conjugates?”

Still, I’m not as bad as this guy…

A mathematician drives to his colleague’s office early one morning, and they start discussing a problem that he’s been working on. They talk and talk and talk, making good progress. Around noon, the colleague says, “I’m hungry. How about we continue this conversation over lunch?”

They spend a long lunch discussing the problem further. They then return to the office and continue talking. Before they know it, it’s dark outside. The colleague says, “I’m getting hungry again. Should we continue this conversation over dinner?”

The mathematician thinks for a minute. “No,” he says, “I better not. My wife’s waiting in the car.”

Happy anniversary, Nadine. The first seven have been wonderful, and I’m looking forward to the next seventy.

### Pentagon Flower and Polygonal Jokes

Frisbee golf is a sport enjoyed by many mathy folks. Almost certainly, this is a result of the physics that surround the flight of the disk.

When a ball dreams, it dreams it’s a Frisbee.

— Stancil Johnson

I love Frisbee golf, because I can make the Frisbee bend and curve around trees in a way that I’m not able to do with a real golf ball. But I also love it because Frisbee golf courses are typically located in beautiful natural settings, like my home course at Bluemont Park in Arlington, VA, which meanders through an urban park with trees, flowers, and a creek.

While playing a round with my sons yesterday, I encountered some flowers with five‑fold rotational symmetry. Taken by the perfect pentagonal shape in the middle of the flower, I snapped a pic to share with you:

Speaking of closed plane figures bounded by straight sides, here are pieces of some recently overheard conversations at the Icosagon Cafe:

- “He’s degenerate, but I say, ‘Let digons be digons!'”
- “I don’t have the energy for another attempt. Trigon.”
- “That hot chick walked away. Pentagon.” (Warning: OFFENSIVE; see definition 2 of
*pent*in the Urban Dictionary. ) - “I took an antidote for the wizard’s curse. Hexagon.”
- “I just looked at the calendar. Septagon. Octagon. Nov is here!”
- “Nonagon. They’re all here!”
- “Someone stole my pack of cards. Decagon.”
- “My parrot just died. Polygon.”

Just outside the cafe, a hendecagon was walking by…

### Avoid, On the Whole, Mathematicians

I was recently shown a chapter from Graham Masterson’s book *How to Drive Your Man Wild in Bed*. The chapter gives suggestions on how to choose a mate, and it includes a list of professions to avoid. At the top of the list, Masterson writes:

Avoid, on the whole, mathematicians…

Of course, I think this advice should be ignored. My reaction, of course, is an ego defense, since I’m a mathy folk. But it also seems to me that a heterosexual male horror writer should not be trusted when dispensing advice about pleasing men in bed.

As it turns out, math and sex have a lot in common…

- Explicit discussions of either topic are generally not received well at social events.
- Historically, men have had control, but recent efforts have tried to get women more involved.
- People engage prominently in both activities on college campuses.
- Most people wish they knew more about both subjects.
- Both activities are typically practiced indoors.
- Both involve long and hard problems and can produce interesting topology and geometry.
- Both merit undivided attention, but mathematicians are prone to thinking about one while doing the other.
- Saint Augustine was hostile to both.
- Alan Turing took an unusual approach to both.
- Both typically begin with a lot of hard work and end with a great but brief reward.
- Professionals are viewed with suspicion, and most of them are underpaid.