## Posts filed under ‘Uncategorized’

### 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

### Big Brother Knows My Sons Are Smarter Than I Am

While pointing and clicking, I stumbled upon an online quiz, **Can We Guess Your Education Level?** Intrigued, I tolerated the 70‑question multiple-choice quiz to see if they could make an accurate prediction. Sure enough, they correctly declared, “It looks like you’re a master with that Master’s Degree.”

**How did they know?**

The optimist in me thinks they use some incredible adaptive engine to figure out exactly what I know and what I don’t, and then they use that information with a correlation of what people at various educational levels know. Sounds plausible, right?

But the pessimist in me was pretty sure they just mined info from my LinkedIn and Facebook profiles, and they likely knew the answer before I responded to a single question.

So, I tested my theory. I took the quiz a second time and deliberately missed a bunch of questions. When I finished, I scored only 21%, and they told me, “It appears that you completed high school, and then graduated from the School of Life.”

Okay, so it is at least based on percent correct. I’m still dubious that it’s rigorous, but at least it isn’t digging through my personal information just to dupe me.

For fun, my 9‑year old son said that he’d like to take the test. And this is when I knew it was complete bullshit — because he scored higher than I did:

Hold on a second. You’re telling me that I spent five glorious years at the Pennsylvania State University earning my undergraduate degree, and then I spent five magnificent years at the University of Maryland earning my master’s degree, and yet my son — who hasn’t spent even five years total in the educational system — was able to outperform me on an academic quiz?

“Hello, is this Penn State? I’d like my money back.”

What really got me, though, is that the math on this quiz — just like every other online quiz, multidisciplinary test, and academic competition — was paltry.

There were seven math-related questions on the test, none of which rose above the level of “basic,” and some were even lower than that. But don’t take my word for it; decide for yourself…

- Speed is defined as…
- What is the name of the result when you add four numbers and then divide the sum by 4?
- What is the definition of
*binary*? - How many events are in a decathlon?
- What is the value of the Roman numeral IX?
- Who wrote
*The Elements*, and what was it about? - The year 1707 is part of which century?

Can we all agree that these are rather easy math questions? It makes me wonder if our discipline is just so abstract or elusive that even the most basic of questions is perceived as difficult by a large portion of the population. If so, what accounts for this perception?

Your thoughts are most welcome.

### More or Less

“That’s Qatar. Its capital is Doha,” I said as we were talking about world capitals and looking at a map of the Middle East. “It’s a very small country.”

“Doesn’t look much smaller than Djibouti,” Alex replied.

And it’s true. On the map, they don’t look much different in size…

Which led me to wonder, which is larger?

And that one little question led to the creation of a game we now call **More or Less**, in which one person names two items, and the other person needs to identify which of the two is larger. It’s a great game for passing time on a car trip or during a long walk.

Here are some of our favorites:

Category |
Option A |
Option B |

Land Area | Qatar | Djibouti |

Percent of U.S. Flag | Red | White |

Distance from St. John’s, NL | Vancouver, BC | Rome, Italy |

Weight | $10 in Quarters | $10 in Dimes |

Population | New York City | London |

Calories | Big Mac | Whopper |

Equatorial Radius | Neptune | Uranus |

Length | Distance from North to South Pole | Great Wall of China |

Official Capacity | Rungrado May First Stadium, North Korea | Michigan Stadium (“The Big House”) |

Caffeine (per 12 fl. oz.) | Coca-Cola | Pepsi-Cola |

Net Worth | Jeff Bezos | Warren Buffett |

Loudness | Squeeze Toy | Vacuum Cleaner |

Stores Worldwide | Dunkin Donuts | Starbucks |

Number by All Teams in a Season | Home Runs in MLB | Goals in NHL |

Top Speed | Slug | Snail |

Heart Beats per Minute | Pig | Human |

Estimated IQ | Newton | Leibniz |

Number of Factors | 144 | 192 |

Value | 3^{16} |
2^{25} |

You should definitely try to figure out whether Option A or B is larger in each row above, before you look at the answers below.

And if you can answer at least 15 of these correctly, you’re **more or less** a genius!

Answers:

- Maps can be deceiving. Although they look similar in size, the area of Djibouti is more than double that of Qatar. Djibouti is 23,200 km
^{2}, Qatar only 11,586 km^{2}. - The U.S. flag is about 41.5% red, 40.9% white.
- Vancouver is 5,117 km from St. John’s, and Rome is only 5,050 km. (Canada is a big country!)
- A dime weighs 2.268 g, a quarter weighs 5.670 g. So 100 dimes and 40 quarters will both weigh 226.8 g.
- London has 8.7 million people. New York has just slightly fewer with 8.6 million.
- A Whopper (no cheese) has 680 calories, whereas a Big Mac (with its two patties, a slice of cheese, and an extra bun in the middle) has only 540 calories. Go figure.
- Uranus is larger with a radius of 25,500 km; the radius of Neptune is 24,700 km.
- Traveling from the North pole to the South pole would be circumnavigating half the Earth, which is about 12,430 miles. But the Great Wall of China is estimated to be 13,170 miles.
- Rungrado holds 114,000, whereas The Big House only holds just under 108,000.
- Pepsi has 58 mg of caffeine, Coke only 54 mg.
- Jeff Bezos is worth $70 billion, Warren Buffett is worth $65 billion. They’re both ridiculously rich, but Bezos was more efficient in acquiring his wealth.
- A vacuum cleaner will reach 75 dB, which is “slightly annoying,” whereas a squeeze toy can reach 90 dB. A vacuum cleaner seems louder and more annoying because the sound persists, whereas most squeeze toys make a noise once, then stop.
- In 2016, there were 25,085 Starbucks but only 12,258 Dunkin’ Donuts, according to Statista.
- There were 6,672 goals scored in the NHL during the 2015-16 season, but just 5,610 home runs hit in the MLB in 2016. There are more than 6,000 hockey goals scored every year, but only two seasons in the past decade have seen more than 5,000 home runs.
- A fast slug can move 0.2 mph, but the poor snail — with that heavy shell on its back — can only muster about 0.02 mph.
- An average human’s heart beats about 60 times per minute; an average pig’s heart, about 70.
- According to
*The Early Mental Traits of Three Hundred Geniuses*by Catharine M. Cox, Leibniz was 183, Newton 168. (Sorry, Isaac!) - The number 144 has 15 factors, the number 192 has only 14.
- 3
^{16}– 2^{25}= 9,492,289.

### Codes, Keypads, and Sequences

When my colleague Chris Meador says, “I’ve thought of a math problem,” rest assured that I’ll spend a good portion of that workday trying to find a solution instead of tackling the items on my to-do list.

Last week, he emailed me the following:

My garage door opener has an exterior keypad that allows me to open the door by entering a 5‑digit number. There is no ENTER key, so the keypad “listens” for the correct code and disregards a false start. How many key presses would it take to test every possible code?

Theoretically, there are 10^{5} possible codes, so entering all of them sequentially would require 5 × 10^{5} key presses. However — because the keypad ignores false starts — some key presses can be saved. For example, typing 123456 will actually test two codes, 12345 and 23456.

Chris continued by asking:

Is it possible to construct an optimal string of key presses of minimal length that tests every possible code?

And with that, my Tuesday was ruined.

I had seen this problem before, or at least a version of it. The top four students at the MathCounts National Competition compete in a special event called the Masters Round, and one year the problem was about something called **D Sequences**. The author used this nickname because such sequences of minimal length are known as *de Bruijn sequences*, after the mathematician Nicolas Govert de Bruijn who proved a conjecture about the number of binary sequences in 1946.

Luckily for Chris, he caught a nasty viral infection last week, which gave him plenty of time to lie in bed thinking about the problem. He emailed me on Monday to inform me of his progress:

I did not manage to prove anything, but I did write a computer program that generates sequences using a pretty straightforward algorithm, and I was able to confirm that solutions are possible for 2‑, 3‑, 4‑, and 5‑digit codes.

That note reminded me that the best way to ensure a happy life is to surround yourself with intelligent people who share similar interests. Chris concluded his email to me with this:

I’d say [that my garage] is pretty secure, since it would take me about 14 hours to punch in all the possible numbers, reading from a list.

Feel free to read more about **de Bruijn sequences** at MathWorld, but you might want to try the following problems first.

- Construct a de Bruijn sequence that contains every two-digit permutation of 0’s and 1’s.
- Construct a de Bruijn sequence that contains every three-character permutation from an alphabet with three characters.
- What is the minimum length of a string of letters that would contain every possible five-letter “word,” that is, every possible permutation of 5 letters, using the Latin alphabet?

Counting things is something that mathematicians, especially those studying combinatorics, do quite often. Yet how they count can be atypical:

When asked how many legs a sheep has, the mathematician replied, “I see two legs in front, two in back, two on the left, and two on the right. That’s eight total, but I counted every leg twice, so the answer is four.”

And there you have it.

### Σ Π :: The Sum and Product Game

This joke, or a close facsimile, has been taking a tour of email servers recently, and it’s now showing up on t-shirts, too:

…and it was delicious!

Appropriate for Pi Day, I suppose, as is the game my sons have been playing…

Eli said to Alex, “18 and 126.”

Alex thought for a second, then replied, “2, 7, and 9.”

“Yes!” Eli exclaimed.

I was confused. “What are you guys doing?” I asked.

“We invented a game,” Eli said. “We give each other **the sum and product of three numbers**, and the other person has to figure out what the numbers are.”

After further inquisition, I learned that it wasn’t just any three numbers but **positive integers** only, that **none can be larger than 15**, and that they must be **distinct**.

Hearing about this game made me immediately think about the famous Ages of Three Children problem:

A woman asks her neighbor the ages of his three children.

“Well,” he says, “the product of their ages is 72.”

“That’s not enough information,” the woman replies.

“The sum of their ages is your house number,” he explains further.

“I still don’t know,” she says.

“I’m sorry,” says the man. “I can’t stay and talk any longer. My eldest child is sick in bed.” He turns to leave.

“Now I know how old they are,” she says.

What are the ages of his children?

You should be able to solve that one on your own. But if you’re not so inclined, you can resort to Wikipedia.

But back to Alex and Eli’s game. It immediately occurred to me that there would likely be some ordered pairs of (sum, product) that wouldn’t correspond to a unique set of numbers. Upon inspection, I found eight of them:

(19, 144)

(20, 90)

(21, 168)

(21, 240)

(23, 360)

(25, 360)

(28, 630)

(30, 840)

My two favorite ordered pairs were:

(24, 240)

(26, 286)

I particularly like the latter one. If you think about it the right way (divisibility rules, anyone?), you’ll solve it in milliseconds.

And the Excel spreadsheet that I created to analyze this game led me to the following problem:

Three distinct positive integers, each less than or equal to 15, are selected at random. What is the most likely product?

Creating that problem was rather satisfying. It was only through looking at the spreadsheet that I would’ve even thought to ask the question. But once I did, I realized that solving it isn’t that tough — there are some likely culprits to be considered, many of which can be eliminated quickly. (The solution is left as an exercise for the reader.)

So, yeah. These are the things that happen in our geeky household. Sure, we bake cookies, play board games, and watch cartoons, but we also listen to the NPR Sunday Puzzle and create math games. You got a problem with that?

### 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.

### Pancake Gift Cards

Ah, gift cards. The perfect present when you want to say, “I was too busy to go to the mall and buy you a gift.”

And if you want to say, “In fact, you’re so unimportant, I bought your gift card while I was eating breakfast,” then head to a pancake house.

That’s right. Our local pancake shop is now offering gift cards — and, boy, do they have a great sale! Check this out…

You smell it coming, don’t you? No, not a stack of pancakes! I’m referring to the math question on the other side of that image.

**Which is the best deal?**

One way to attack this problem is to compare the free amount to the original price. That is, which fraction is greatest: 2/15, 4/25, 7/50, 11/75, or 15/100?

Ack. Too much work.

A better way is to do a piece-wise comparison:

- Which is better, $15 or $25? Clearly the
**$25**. You get twice as much free for only $10 more. - Which is better, $25 or $50? Well, two $25’s will get you $8 free, but for the same price, you’ll get one $50 card and only $7 free. So, the
**$25**card wins again. - Which is better, $25 or $75? Well, three $25’s will get you $12 free, but for the same price, you get one $75 card and only $11 free. Don’t look now, but
**$25**is on a roll. - Which is better, $25 or $100? Well, four $25’s will get you $16 free, but for the same price, you get one $100 card and only $15 free. There you have it,
**$25**is the champ.

Now that that’s out of the way, you can probably anticipate my next question.

**Who the hell came up with this pricing scheme?**

It’s not typical to get a smaller reward when you spend more money. Usually, the more you spend, the more you get free. Then again, it’s a pancake shop. Maybe they did some significant market analysis, recognized that no one could actually spend $100 on pancakes, and since $25 is a more common breakfast total, that’s the one that gets the biggest reward.

Or, maybe they just goofed up the math.

Sorry, I know no jokes about gift cards. But here are a couple about finance.

Why didn’t the mathematician report his stolen credit card?

The thief was spending less then his wife.We didn’t exceed the budget. The allocation simply fell short of expenses.

I’m flat broke, so my financial advisor recommended plastic surgery: cut up all my credit cards.