## Posts tagged ‘math’

### Stick Figure Math

I’ll never forget the first time I saw the pattern

1, 2, 4, 8, 16, __

and was dumbfounded to learn that the missing value was **31**, *not 32*, because the pattern was *not* meant to represent the powers of 2, but rather, the number of pieces into which a circle is divided if *n* points on its circumference are joined by chords. Known as Moser’s circle problem, it represents the inherent danger in making assumptions from a limited set of data.

Last night, my sons told me about the following problem, which they encountered on a recent math competition:

*What number should replace the question mark?*

Well, what say you? What number do you think should appear in the middle stick figure’s head?

Hold on, let me give you a hint. This problem appeared on a multiple-choice test, and these were the answer choices:

- 3
- 6
- 9
- 12

Now that you know one of those four numbers is *supposed* to be correct, does that change your answer? If you thought about it in the same way that the test designers intended it, then seeing the choices probably didn’t change your answer. But if you didn’t think about it that way and you put a little more effort into it, and you came up with something a bit more complicated — like I did — well, then, the answer choices may have thrown you for a loop, too, and made you slap your head and say, “WTF?”

For me, it was Moser’s circle problem all over again.

So, here’s where I need your help: **I’d like to identify various patterns that could make any of those answers seem reasonable.**

In addition, I’d also love to find a few other patterns that could make some answers other than the four given choices seem reasonable.

For instance, if the numbers in the limbs are *a*, *b*, *c*, and *d*, like this…

then the formula 8*a* – 4*d* gives 8 for the first and third figures’ heads and yields 8 × 6 – 4 × 9 = **12** as the answer, which happens to be one of the four answer choices.

Oh, wait… you’d don’t like that I didn’t use all four variables? Okay, that’s fair. So how about this instead: ‑3*a* + *b* + *c* + 2*d*, which also gives ‑3 × 6 + 7 + 5 + 2 × 9 = **12**.

Willing to help? **Post your pattern(s) in the comments.**

[**UPDATE (3/9/18):** I sent a note to the contest organizers about this problem, and I got the following response this afternoon: “Thanks for your overall evaluation comments on [our] problems, and specifically for your input on the Stick Figure Problem. After careful consideration, we decided to give credit to every student for this question. Therefore, scores will be adjusted automatically.”]

### 2017 KenKen International Championship

If you like puzzles and ping pong, then Pleasantville, NY, was the place to be on December 17.

More than 200 Kenthusiasts — people who love KenKen puzzles — descended on Will Shortz’s Westchester Table Tennis Center for the 2017 KenKen International Championship (or the KKIC, for short). Participants followed 1.5 hours of solving KenKen puzzles with a pizza party and several hours of table tennis.

The competition consisted of three rounds, with the three puzzles in each round slightly larger and more difficult than those from the previous round. Consequently, competitors were given 15, 18, and 20 minutes to complete the puzzles in the first, second, and third rounds, respectively.

Competitors earned 1,000 points for each completely correct puzzle, and 0 points for an incomplete or incorrect puzzle. In addition, a bonus of 5 points was earned for every 10 seconds in which a puzzle was turned in before time was called. So, let’s say you got two of the three puzzles correct and handed in your answers with 30 seconds remaining in the round; then, your score for that round would be

The leader after the written portion was John Gilling, a data scientist from Brooklyn, whose total score was 10,195. And if you’ve been paying attention, then you know what that means — Gilling earned 9,000 points for completing all of the puzzles correctly, so his time bonus was 1,195 points… which is the amount you’d earn for turning in the puzzles 2,390 seconds (combined) before time was called. The implication? Gilling solved all 9 puzzles from the written rounds — which contained a mix of puzzles from size 5 × 5 to 8 × 8 — in just over 13 minutes.

Wow.

As a result, Gilling, the defending champion, earned a spot in the Championship Round against Tess Mandell, a math teacher from Boston; Ellie Grueskin, a high school senior at The Hackley School; and Michael Holman, a technology consultant. In the final round, each of them attempted a challenging 9 × 9 puzzle, which was displayed on an easel for the crowd to see. Solving a challenging 9 × 9 is tough enough; having to do it as 200 kenthusiasts follow your every move is even tougher.

So, how’d they do? See for yourself…

When the dust settled, Gilling had successfully defended his title. For his efforts, he received a check for $500. But more importantly, he retained bragging rights for one more year.

If you think you’ve got what it takes to compete with the best KenKen solvers, try your hand at the 9 × 9 puzzle that was used in the final round. In the video above, you saw how fast Gilling solved it to win the gold. But even the slowest of the four final-round participants finished in under 15 minutes.

Again, wow.

Finally, I’d be failing as a father if I didn’t mention that my sons Alex and Eli competed in the Delta (age 10 and under) division. Though bested by Aritro Chatterjee, a brilliant young man who earned a trip to the 2017 KKIC by winning the UAE KenKen Championship, Eli took the silver, and Alex brought home the bronze. They’re shown in the photos below with Bob Fuhrer, the president of Nextoy, LLC, the KenKen company and host of the KKIC.

#proudpapa

For more KenKen puzzles, check out www.kenken.com, or see my series of posts, A Week of KenKen.

### Number Challenge from Will Shortz and NPR

Typically, the NPR Sunday Puzzle involves a word-based challenge, but this week’s challenge was a number puzzle.

[This challenge] comes from Zack Guido, who’s the author of the book

Of Course! The Greatest Collection of Riddles & Brain Teasers for Expanding Your Mind. Write down the equation

65 – 43 = 21You’ll notice that this is not correct. 65 minus 43 equals 22, not 21. The object is to

move exactly two of the digits to create a correct equation. There is no trick in the puzzle’s wording. In the answer, the minus and equal signs do not move.

Seemed like an appropriate one to share with the MJ4MF audience. Enjoy!

### 8-15-17

Today is a glorious day!

The date is 8/15/17, which is mathematically significant because those three numbers represent a Pythagorean triple:

But August 15 has also been historically important:

- It’s the birthday of some famous people, including Jennifer Lawrence, Kerri Walsh-Jennings, Napoleon Bonaparte, Julia Child, and Ben Affleck, as well as some not-so-famous people, including one of my sisters, one of my aunts, one of my uncles, and my maternal grandfather.
- The Mayflower departed from Southampton, England, on August 15, 1620.
- The Panama Canal opened to traffic on August 15, 1914.
*The Wizard of Oz*premiered at Grauman’s Chinese Theater on August 15, 1939, and exactly 40 years later,*Apocalypse Now*was released.- In 1945, the Japanese surrendered on August 15.

But as of today, August 15 has one more reason to brag: It’s the official publication date of a bestseller-to-be…

Like its predecessor, this second volume of math humor contains over 400 jokes. Faithful readers of this blog may have seen a few of them before, but most are new. And if you own a copy of the original * Math Jokes 4 Mathy Folks*, well, fear not — you won’t see any repeats.

What kind of amazing material will you find on the pages of * More Jokes 4 Mathy Folks*? There are jokes about school…

An excited son says, “I got 100% in math class today!”

“That’s great!” his mom replies. “On what?”

The son says, “50% on my homework, and 50% on my quiz!”

There are jokes about mathematical professions…

An actuary, an underwriter, and an insurance salesperson are riding in a car. The salesperson has his foot on the gas, the underwriter has her foot on the brake, and the actuary is looking out the back window telling them where to go.

There are Tom Swifties…

“13/6 is a fraction,” said Tom improperly.

And, of course, there are pure math jokes to amuse your inner geek…

You know you’re a mathematician if you’ve ever wondered how Euler pronounced Euclid.

Hungry for more? Sorry, you’ll have to buy a copy to sate that craving.

To purchase a copy for yourself or for the math geeks in your life, visit **Amazon**, where * MoreJ4MF* is already getting rave reviews:

For quantity discounts, visit **Robert D. Reed Publishers**.

### Just Sayin’

Heidi Lang is one of the amazing teachers at Thomas Jefferson Elementary School. When she’s not challenging my sons with interesting puzzles and problems, she’s entertaining them with jokes that make them think. On her classroom door is a sign titled *Just Sayin’*, under which hangs a variety of puns. Here’s one of them:

Last night, I was wondering why I couldn’t see the sun. Then it dawned on me.

That reminds me of one of my favorite jokes:

I wondered why the baseball kept getting larger. Then it hit me.

Occasionally, one of her puns has a mathematical twist:

Did you know they won’t be making yardsticks any longer?

And this is one of her mathematical puns, though I’ve modified it a bit:

When he picked up a 20‑pound rock and threw it 5,280 feet, well, that was a real milestone.

I so enjoy reading Ms. Lang’s *Just Sayin’* puns that I decided to create some of my own. I suspect I’ll be able to hear you groan…

- He put 3 feet of bouillon in the stockyard.
- When the NFL coach went to the bank, he got his quarterback.
- She put 16 ounces of poodle in the dog pound.
- The accountant thought the pennies were guilty. But how many mills are innocent?
- His wife felt bad when she hit him in the ass with 2⅓ gallons of water, so she gave him a peck on the cheek.
- Does she know that there are 12 eggs in a carton? Sadly, she dozen.
- When his daughter missed the first 1/180 of the circle, he gave her the third degree.
- She caught a fish that weighed 4 ounces and measured 475 nm on the visible spectrum. It was a blue gill.
- When Rod goes to the lake, he uses a stick that is 16.5 feet long. He calls it his fishing rod.
- What is a New York minute times a New York minute? Times Square.
- I wanted to dance after drinking 31 gallons of Budweiser, so I asked the band to play the beer barrel polka.
- The algebra teacher was surprised by the mass when she tried to weigh the ball:
*b*ounces.

And because this post would feel incomplete without it, here’s probably the most famous joke of this ilk:

- In London, a pound of hamburger weighs about a pound.

### Our Library’s Summer Math Contest

Every summer, our local library runs a contest called *The Great Big Brain Game*. Young patrons who solve all of the weekly puzzles receive a prize. The second puzzle for Summer 2017 looked like a typical math competition problem:

Last weekend, the weather was perfect, so you decided to go to Cherry Hill Park. When you got there, you saw that half of Falls Church was at the park, too! In addition to all the people on the playground, there were a total of 13 kids riding bicycles and tricycles. If the total number of wheels was 30, how many tricycles were there?

First, some comments about the problem.

- I dislike using “you” in math problems. I believe it’s a turn-off to students who can’t see themselves in the situation described. There are enough reasons that kids don’t like math. Why give them another reason to shut down by telling them that they went somewhere they didn’t want to go or that they did something they didn’t want to do?
- Word problems are not real-world just because they use a local context, and this one is no exception. This problem attempts to show an application for a system of linear equations, but true real-world problems don’t have all the information neatly packaged like this.
- Wouldn’t the person posing this problem already have access to the information they seek? That is, if she counted the number of kids riding bikes and the total number of wheels, couldn’t she have just counted the number of bicycles and tricycles instead? It has always struck me as strange when the (implied) narrator of a math problem wants you to figure out something they already know.

All that said, this was meant to be a fun puzzle for a summer contest, and I don’t mean to scold the library. I don’t know that I’d use this puzzle in a classroom — at least, not presented exactly like this — but I love that kids in my town have an opportunity to do some math in June, July, and August.

Now, I’ll offer some comments on the solution. In particular, the solution provided by the library was different than the method used by one of my sons. Here’s what the library did:

Imagine that all 13 kids were on bicycles with 2 wheels. That would be a total of 26 wheels. But since 30 wheels are needed, there are 4 extra wheels. If you add each of those extra wheels to a bicycle, that’ll create 4 tricycles, leaving 9 bicycles. So, there must have been 4 tricycles at Cherry Hill Park.

And here’s what my son did:

If you can’t see what he wrote, he created a system of two equations and then solved it:

2a + 3b = 30

a + b = 13a + 2b = 17

13 – b + 2b = 17

b = 42a + 12 = 30

2a = 18

a = 9

That’s all well and good. In fact, it’s perfect if you want to assess my son’s ability to translate a problem and solve a system of equations. But I have to admit, I was a little disappointed. What bums me out is that he went straight to a symbolic algorithm instead of considering alternatives.

I think I know the reason for this. This past year, my son was in a pull-out math program, in which he studied math with someone other than his regular classroom teacher. In this special class, the teacher focused on preparing him to take Algebra II in sixth grade when he enters middle school. Consequently, students in the pull-out class spent the past year learning basic algebra. My fear is that they focused almost exclusively on symbolic manipulation and, as my former boss liked to say, “Algebra teachers are too symbol-minded.”

A key trait of effective problem solvers is flexibility. That type of flexibility comes from solving many problems and filling your toolbox with a variety of strategies. My worry — and this isn’t just a concern for my son, but for every math student in the country — is that students learn algorithms at the expense of more useful problem-solving heuristics. What happens when my son is presented with a problem that can’t be translated into a system of linear equations? Will he know what to do when he doesn’t know what to do?

The previous pull-out teacher said that when she presented my sons with problems that they didn’t know how to solve, their eyes would light up. They liked the challenge of doing something they hadn’t done before. I’m hopeful that this enthusiasm isn’t lost as they proceed to higher levels of mathematics.

### Russell, Robertson, and Ratios

In the NBA, a **triple-double** happens when a player has a double-digit total in three of the five categories (points, assists, rebounds, steals, and blocks) in a game. Triple-doubles are very rare; on average, one has been recorded only once every 27 games since 2003. So far this season, there have been 111 triple-doubles throughout the entire NBA — and **Russell Westbrook** has 41 of them.

In 1961-62,** Oscar Robertson** set a record that Westbrook is about to break. That year, Robertson recorded 41 triple-doubles in 80 games. Westbrook recorded his 41st triple-double of the current season in just 78 games. When two fractions have the same numerator, the one with the smaller denominator is larger. Consequently, 41/78 > 41/80, so Westbrook’s accomplishment exceeds Robertson.

But ratios can be used to make the point even more dramatically. In the early 1960’s, pro basketball games were played at a faster pace than they are today. In 1961‑62, the average game featured 126.2 possessions, meaning that Robertson typically had more than 60 tries to grab a rebound, make an assist, or score some points. By comparison, there have been an average of just 96.4 possessions per game during the current NBA season, meaning that Westbrook generally has fewer than 50 attempts per game to improve his stat line. So another ratio — the comparison of points, rebounds, and assists to number of possessions — also leans in Westbrook’s favor.

Who knew that either of these guys were such fans of math?

At Discovery Education, we’ve been having a lot of fun writing basketball problems based on real NBA data. Check out a few problems at http://www.discoveryeducation.com/nbamath, and get a glimpse of the NBA Analysis Tool within Math Techbook^{TM} by signing up for a free 60-day trial at http://www.discoveryeducation.com/math.

#mathslamdunk

### College Basketball Round-Up

The sheepdog returned to the farmhouse and told the shepherd, “All 200 sheep have been returned to their pens.”

“200?” asked the shepherd. “But we only have 196 sheep.”

The dog replied, “Well, yeah, but you know I like to round up.”

Rounding up has been a topic of conversation in college basketball this week.

Marcus Keene, a guard for the Central Michigan Chippewas, scored 959 points in 32 games this season, giving him a points-per-game (PPG) average of 30.0.

Sort of.

Technically, his average is 29.96875, just shy of the highly coveted 30 points-per-game mark that’s only been attained by a few dozen players in NCAA history. Since 1981, only 8 players have reached 30 PPG, most recently Long Island’s Charles Jones in 1996‑97.

But the controversy swirled this week because Keene didn’t actually average more than 30 points per game. He was one point shy. His lofty accomplishment was nothing more than smoke-and-mirrors due to round-off error, or so the critics say.

Per-game statistics are used to compare players with one another, because totals can’t be compared for players who have played a different number of games. And let’s face it, no one wants to get into the habit of comparing per-game stats to seven decimal places. The NCAA reports all per-game statistics to the nearest tenth, and the truth is that Keene’s PPG average would be reported as 30.0, 30.00, 30.000, and 30.0000 if rounded to tenths, hundredths, thousandths, and ten-thousandths, respectively.

It’s been a good year for math and basketball. Anthony Davis can have an asterisk for his record-setting 52 points in the NBA All-Star Game because no one played defense; and now Marcus Keene can have an asterisk for his 30.0 points-per-game average.

In related news, it was reported that 53% of men say that they will watch the NCAA Division I Men’s Basketball Championship (aka, “March Madness”). And just to prove the men are the dumber sex, 61% of them admitted that they’ll watch while at work. Simple math says that 32.3% of men will watch the tourney at work. Which means that if you’re a man with two friends who don’t like basketball, then you’ll be the one killing office productivity next Thursday.

### 2 Good 2 Be True

I was eating a bowl of shepherd’s pie at the Irish pub in our neighborhood. A man walks up to my table and asks, “What’s your favorite number?”

“Uh, 153,” I respond.

“And 153 × 2 is 306,” he says, then hurriedly scurries away.

He approaches another table, asks another patron for her favorite number, and again multiplies it by 2. He does this over and over, popping from table to table, annoying customer after customer. Eventually, the manager notices this eccentric behavior and approaches the man.

“Sir,” says the manager, “You can’t keep interrupting people’s dinners by asking them for a number and then multiplying by 2.”

“What can I say,” he responds. “I love Dublin!”

A little while later, the gentleman at the table next to me says to his companion, “I know a sure-fire way to double your money.”

This piqued my interest, so I leaned over to eavesdrop on his advice.

“Fold it in half,” he said.

Dismayed that you’ve read this far and have only heard two terrible jokes? Well, buck up, because your fortune is about to change. I can’t help you double your money, but I can help you get twice as much for it.

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Folks, this is a linear relationship that you’d be foolish to ignore!