## Archive for July, 2011

### Coming Up Through the Ranks

Through some special features at Amazon Author Central, I am able to know the daily sales rank of Math Jokes 4 Mathy Folks. My sales rank at the end of each of the last three days was 44,404, 96,990, and 35,355, respectively. I thought that was interesting — three days in a row when the sales rank was a five-digit number in which one digit occurred at least three times. What’s the likelihood of that? Stated more formally:

Assuming that the sales rank of MJ4MF is always a five-digit number, what is the probability that three consecutive days’ sales ranks will contain a digit that occurs in the sales rank at least three times?

The sales rank of MJ4MF has never been a five-digit number in which the same digit is repeated five times. (Bummer!) The probability of that occurrence, though, is even less likely than the situation described above — though I won’t tell you exactly how much less likely, so as not to spoil your fun!

### Math Vacation

Not much activity at MJ4MF recently. The family was on vacation for the past 10 days, and I’m happy to report that I was able to avoid my computer for the entire trip. It was hard, though, as I am slightly addicted…

Is it better to have a wife or a mistress? Both. Your wife thinks you’re with your mistress, your mistress thinks you’re with your wife, and the whole time you can be alone with your computer.

The following is a true story from our vacation. At one of our stops, my twin sons and I entered a toilet stall that was huge. It’s typical for my sons to urinate in the same toilet at the same time, but because this particular stall was so huge, I decided to join them. (The family that sprays together stays together, right?) As we were taking care of business, Eli said, “Three people making pee-pee!”

Upon hearing this, Alex asked, “That’s hexa-pee, right, daddy?”

God love ’em.

And finally, a math joke from an activity book that Eli was doing on the plane.

Six is about to enter a party, but notices over his shoulder that Five is coming up the sidewalk. So Six holds the door open for Five and says, “After you.”

### If This is What Hades is Like, We Better Change Our Ways

It was so hot in Virginia today, I saw two fire hydrants fighting over a dog.

Truth be known, I’m travelling through California right now, so I don’t actually know how hot it was in Virginia today. But I got an email that said my wine o’ the month shipment would be delayed until the weather cools to a point where the wine won’t be in danger of spoiling. Zoiks! In California this week, it’s been rather pleasant — even chilly at night, with temps in the low 50’s.

A recent conversation with the Northern California friends we’re visiting turned to temperature conversions. John said that on international trips, he amuses himself by converting Celsius temps to Fahrenheit. I informed him that the rule “times 2 plus 30” gives an accurate estimate for temps in the typical range — it isn’t really necessary to remember the exact rule. I also shared a poem about Celsius temps that he and his wife hadn’t heard before:

30 is hot,

20 is nice,

10 is cold,

and 0 is ice.

Finally, a stupid math and weather joke.

What is the name of a person who only adds when it’s hot outside?

Summer.

### Skill-Testing Questions and Captcha

All lotteries have three major components:

- There is a value associated with the prize;
- The organization running the sweepstakes benefits financially; and,
- The winner is chosen at random.

To avoid being an illegal private lottery, at least one of the three components must be removed. Canadian sweepstakes law requires that the third component (winners chosen at random) be removed. That is, the sponsoring organization cannot use pure luck alone to determine who wins. There must be some element of skill involved.

Hence the disclaimer on lotteries open to residents of Canada, such as the following from the Bizrate sweepstakes:

If a Canadian resident wins a prize, that person must also answer correctly within a 5 minute time period a mathematical skill-testing question (STQ)…

Apparently, Canadian courts have determined that a mathematical expression containing at least three binary operators is sufficient to qualify as an STQ. Hence, a person whose name is chosen at random might have to determine the value of the following before being awarded the prize:

8 × 6 – 5 + 9

The expression above is an actual STQ that was used in a Tim Horton’s contest a few years ago. (A woman with a learning disability gave the incorrect answer of 51. When she appealed, she was given a second chance, and they gave her the same question. Again, she answered 51. Amid much protest, Tim Horton’s eventually relented and awarded the prize to the woman anyway, though it’s unclear to me how they got around the STQ requirement.)

Similar STQ’s are now being used as captchas on web sites. I was presented with the following math question when submitting a comment to a site the other day:

I really thought they missed a golden opportunity, though, so I submitted a second comment with the following image attached:

### Smooth Operators

I recently learned about a cool problem that involves the four binary operations.

“My life is all arithmetic,” the young businesswoman explained. “I try to add to my income, subtract from my weight, divide my time, and avoid multiplying.”

A fifth-grader teacher, who is spending a week at NCTM Headquarters for the Illuminations Summer Institute, shared the problem with me. He uses it to practice basic operations and to develop an understanding of place value with his students.

Choose any positive integer as the starting value, and choose a different positive integer as the ending value. Then, perform any of the following moves on the starting number:

- Add 1.
- Subtract 1.
- Multiply by 10.
- Divide by 10.

Continue to perform moves until you’ve reached the ending number.

For example, you can get from 8 to 71 with the following sequence of moves:

- Subtract 1: 8 – 1 =
**7** - Multiply by 10: 7 × 10 =
**70** - Add 1: 70 + 1 =
**71**

For any given starting and ending number, what is the fewest number of moves required?

In general, can you find an algorithm to predict the minimum number of moves for any given starting and ending numbers?

The teacher who shared the problem with me said that students have fun with 777 as the starting number and 888 as the ending number. It is a fine problem for kids to explore, but it proved to be a red herring for me — it led me to make false assumptions for determining a general solution.

Enjoy.

### USA Ultimate Grand Masters National Championships

Completely off-topic, but I’ve got to share. In the bucket list that I mentioned in a previous post, I claimed that I wanted to play Ultimate Frisbee until I was well into my 60’s. What I failed to include on that list, but one item that’s been subconsciously on my bucket list for a long time, is that I wanted to win a National Championship in something. I’ve played Ultimate Frisbee for almost two decades, and though I’m pretty good — I’ve played at the National Championships eight times, the World Championships once, and I’ve won numerous local leagues and small tournaments — I never thought I’d win a national title for anything athletic.

Well, hear ye, hear ye! This past weekend, my teammates from Scrapple (a Philadelphia-based team with two other guys and myself from DC) and I descended on Lebanon, OH, and we won the USA Ultimate Grand Masters National Championships! En route to a perfect 6-0 record, we beat teams from Milwaukee, California, Boston, Chicago, North Carolina and Colorado.

Cross another item off the list!

Here’s a Frisbee joke that involves relativity (or is it a relativity joke that involves a Frisbee?):

I wondered why the Frisbee kept getting larger and larger… then it hit me.

### 5 Math Strategy Games to Practice Basic Skills

The summer is a great time for kids to hike, bike, swim… and forget everything that they learned during the school year.

The son returned to school after summer break. At the end of the first day, his mother received a call from the teacher about his poor behavior. “Now, just one minute,” said the mother. “He had poor behavior all summer, yet I never called you once!”

In *Outliers*, Malcolm Gladwell purports that poor kids lose ground to affluent kids during summer break. Their experiences and academic progress during the school year are similar, he contends, but their out-of-school experiences during the summer are very different. Though minor at first, the cumulative effect of those summer losses becomes noticeable as children get older.

The following are five games/puzzles that can be used with young kids to prevent summer losses and, possibly, even elicit some summer gains. Each has the characteristics that I love about a good game for young kids: It requires students to use and practice basic skills, but there is a higher purpose for doing so.

**1. KenKen**

This is a game that’s kind of like SuDoku, but a million times better. If you don’t know the game, check it out at www.kenken.com. My sons noticed me playing it one afternoon and asked what it was. I explained, and they asked if they could do it with me. We now solve three or four games every afternoon. I used to help them a lot, but now they pretty much know all of their math facts up through 7 × 7. How do you not love a game that helps four-years-olds learn the times table?

**2. Wormhole**

This is a puzzle, not a game, and you can learn all about it at Math Pickle. The general idea is that you start with a sequence of numbers in a flower-like pattern. You then multiply two adjacent numbers, subtract 1, and divide by the number below. The cool and surprising part is that every intermediate result is an integer, so there are no ugly decimals for kids to deal with. And by the twelfth ring of petals, every result is 0. Happens every time.

**3. Squares of Differences**

The good folks at Math For Love reminded me of this great problem, and Josh Zucker discussed it at length on the NYTimes Numberplay blog. Draw a square, and put a positive integer at each vertex. Then at the midpoint of each side, write the difference of the numbers at the two adjacent vertices. Now connect the midpoints to form a rotated square inside the original square, and repeat. It seems that if you continue this process long enough, you’ll eventually get all 0’s. But does that always happen?

By the time kids test this conjecture with three or four attempts, they’ve done a hundred subtraction problems without even realizing it.

**4. Decimal Maze**

The Decimal Maze (PDF) comes from the lesson Too Big or Too Small on Illuminations. Trying to obtain the maximum value while traversing a maze with decimal operations, students learn about the effects of multiplying and dividing by decimals that are greater or less than 1. The activity is good for upper elementary and middle school students, but I’ve used modified versions with very young kids. For instance, a modified maze for kids in first grade uses single-digit positive integers while limiting the operations to just addition and subtraction; for older kids, a maze could include fractions or powers instead of decimals.

**5. Dollar Nim**

As I mentioned in a previous post, my wife created a great game that I call Dollar Nim. The idea is simple. Imagine you have 100¢, and on your turn you can remove 1¢, 5¢, 10¢, or 25¢. Players alternate turns; the player to reduce the amount to 0¢ is the winner. The optimal strategy is not obvious, and kids practice a whole lot of subtraction, especially as it relates to making change.

More generally, any one-pile nim game is great for the purpose of having kids practice subtraction without realizing it.

I hope you find some free time this summer to enjoy these games. I’ll leave you with a joke/truth about summer school.

I never understood the concept of summer school. The teacher’s going to go up there and go, “OK, class. You know that subject you couldn’t grasp in nine months? Well, we’re going to whip it out in six weeks.” – Todd Barry

### Real-Life Double Negatives

“Can a Jewish person eat a ham calzone?” I asked my Jewish friend.

She gave me a look that said, “Oh, goodness, I can’t wait to hear this one…”

I continued. “It seems to me that things cancel out. You’re not allowed to eat ham, but you also aren’t allowed to mix meat and cheese, so this is a double negative.”

“Nice try,” she replied. “Just means you get to ride the express elevator to hell.”

The error with my logic, of course, is that sins are additive, not multiplicative. This is actually quite lucky, if you think about it. Otherwise, you’d have to spend your entire life being remarkably fastidious in keeping track of your sins: an even number of sins, you go to heaven; an odd number of sins, you end up a little lower. The graph looks something like this:

I’m not the first to tell a joke involving Jews and double negatives. Stanislaw Ulam did it with this joke from his autobiography:

Two Jews are riding on a train through Russia. One asks the other, “Where are you going?” The second replies, “To Kiev.” Whereupon the first says, “You liar, you tell me you are going to Kiev so I would think you are going to Odessa. But I know you are going to Kiev, so why do you lie?”

### Twice the Sum of Its Digits

As my sons and I were doing a 6 × 6 KenKen puzzle today, we needed two numbers with a product of 18. “What two numbers multiply together to give 18?” I asked the boys.

Eli answered, “4 and 4½.”

“Um, yeah,” I responded. “How do you know that?”

“Well,” he said exuberantly, “4 × 4 is 16, and 4 × 5 is 20, so 4 × 4½ is 18.”

By the look on my wife’s face, I could tell that she was as shocked as I was. “Holy sh*t,” I said. “He just did linear interpolation!”

Why is it that the more accuracy you demand from an interpolation function, the more expensive it becomes to compute?

That’s the Law of Spline Demand.

As it turns out, our paths crossed the number 18 several times today. While brushing their teeth, Eli explained that he had to floss 18 times. “There are 4 spaces between my fingers, so there are 9 spaces between my teeth. So I have to floss 18 spaces on the top and bottom.”

The number 18 walks into a bar. “I’d like a beer,” he says.

“Sorry, I can’t serve you,” says the bartender.

“Why not?” asks 18.

“Because you’re under 21.”

After each night’s bedtime story, either my wife or I count to the boys before they go to sleep. Tonight, Eli asked mommy to count to 198 by the Phibby (pronounced *fee*‘-*bee*) numbers, which is Eli’s word for the multiples of 11. Nadine said, “That seems like a lot of counting,” to which Eli responded, “Not really — 198 is only the 18th Phibby number.”

**[Footnote]** As I was doing “research” for this post, I did a Google search for “joke interpolation.” I was directed to *Joke Retrieval: Recognizing the Same Joke Told Differently*, an academic paper that codifies jokes independent of context, characters, and location; instead, jokes are compared by punch line, and professions/countries are viewed as interchangeable. The result is a model that attempts to identify two different versions as the same joke. Several variations of the following joke appeared within that paper:

An engineer driving westbound collides with a mathematician driving eastbound on the same highway. Their cars are completely demolished, yet neither driver has even a scratch. They each crawl from the wreckage, and they begin to marvel at what just transpired. “This is a miracle!” says the engineer. “Can you believe that neither of us got hurt?”

“I know!” says the mathematician. “And look! This bottle of whiskey in my back seat is still intact. Such an amazing occurrence calls for a celebration,” he says, as he unscrews the cap and hands the bottle to the engineer.

The engineer swigs half the bottle, then hands it back to the mathematician. The mathematician puts the cap back on and sets the bottle on the ground.

“Aren’t you having any?” asks the engineer.

“Nah,” says the mathematician. “I think I’ll wait till after the police arrive.”

### Stuck in the Middle With You — July 2

Today is an average day, exactly halfway between the beginning and end of the year.

Benoit Mandelbrot, the father of fractal geometry, often said he was born in Poland and educated in France — making him German, on average.

*Average* is something upon which hens lay their eggs. For instance, “My hens lay four eggs a week on average.”

When she told me that I was only average, she was just being mean.

A statistician with his head in the freezer and his feet in the oven will say that, on average, he feels fine.

Three statisticians go hunting. When they see a duck flying overhead, two of them take a shot. The first fires six inches over the duck; the second fires six inches under the duck; and, the third excitedly exclaims, “We got it! We got it!”