## AWOKK, Day 3: KenKen Times

Today is Day 3 in MJ4MF’s **A Week of KenKen** series. In case you missed the fun we’ve had previously…

- Day 1: Introduction
- Day 2: The KENtathlon

Yesterday, I introduced you to the **KENtathlon**.

While completing a KENtathlon, my goal is to complete a 6 × 6 puzzle in less than 2 minutes; a 5 × 5 puzzle in less than 1 minute; and a 4 × 4 puzzle in less than 20 seconds. Even though the sum of those times for all three puzzles is 3 minutes, 20 seconds, my goal is a combined time of 3 minutes. It’s good to have goals.

Puzzle Size |
Goal Time |
Personal Best |

4 × 4 | 0:20 | 0:12 |

5 × 5 | 1:00 | 0:27 |

6 × 6 | 2:00 | 1:29 |

KENtathlon | 3:00 | 2:32 |

I don’t always perform well enough to meet those goals. And when I don’t, I repeat the same size puzzle again… and again… and again… for as many attempts as it takes to complete each puzzle in the allotted time. And when I’ve met the time goal for each puzzle individually, if the combined time isn’t satisfactory, then I start the whole thing over.

To say that I’m slightly obsessive would be like saying that the Pope is a little bit Catholic.

As you may have noticed in the table, I once finished a 4 × 4 puzzle in 12 seconds. The key word there is **once**. The stars were in alignment that day — it was an easy puzzle, and the dexterity of my thumbs and fingers was at an all-time high. Though I’ve attained 13 a handful of times, I’ve never replicated that 12-second feat.

That said, I regularly complete 4 × 4 puzzles in 14 or 15 seconds. With that being the case, you have to wonder if the 20-second goal is really a challenge. And what about the goal times for 5 × 5 and 6 × 6 puzzles?

Admittedly, my time goals are arbitrary, though not random. When I chose those goals, I had completed enough KenKen puzzles that I intuitively knew what felt right. Still, it wasn’t based on hard data… and if you’ve read this blog long enough, you know that that bothered me. A lot.

But what’s a boy to do?

I suppose a well-adjusted human might do nothing, think it’s not worth the trouble, and just let the whole thing go. But an obsessive numbers guy? Well, he’d painstakingly solve 132 KenKen puzzles, collect data on the amount of time each one took to complete, meticulously record the data in an Excel spreadsheet, and perform a thorough analysis. You may think that undertaking such a project is ludicrous; but to me, it was absolutely essential.

The graph below shows the results. The circular dots represent my median time for each puzzle size, and the square dots represent the upper and lower quartiles. For instance, the median time for 6 × 6 puzzles was 217 seconds, while the interquartile range for 6 × 6 puzzles extended from 163 to 284 seconds.

What this reveals is that my intuition wasn’t perfect, but not bad.

- I completed
**49% of 4 × 4 puzzles**in less than the goal time of 20 seconds. - I completed
**58% of 5 × 5 puzzles**in less than the goal time of 1 minute. - But, I completed
**only 14% of 6 × 6 puzzles**in less than the goal time of 2 minutes.

Further analysis revealed that I completed 40% of the 6×6 puzzles under 3 minutes, and that seems a bit more reasonable, so **my new goal time for 6 × 6 puzzles is 3 minutes**.

Now, I know you thought this analysis was completely unnecessary, but the proof is in the pudding. The results were invaluable. By considering the data, interpreting the results, and revising my goal time for 6 × 6 puzzles, the probability that I can now complete each size puzzle in the allotted time on the first or second try has increased from 16% to 39%. Or said another way, Remy’s morning walks now last an average of just 15 minutes, whereas some of them used to take an hour-and-a-half.

## AWOKK, Day 2: The KENtathlon

You’ve come back! Great! Welcome to Day 2 of MJ4MF’s **A Week of KenKen** series. In case you missed the fun we had yesterday…

- Day 1: Introduction

As the early riser in our family, I’m responsible for taking our dog Remy on his morning walk. Given his advanced age (13) and breed (golden retriever), our veterinarian says that I should walk him 15-20 minutes every morning.

But the length of Remy’s morning walk is completely independent of any professional recommendations. Instead, it depends entirely on the amount of time it takes me to solve a 6 × 6, a 5 × 5, and a 4 × 4 KenKen puzzle.

I call it the **KENtathlon**.

Here are three puzzles for you to solve as a KENtathlon. The order in which you solve them doesn’t matter. I don’t even care if you take a break between puzzles. The only thing I would recommend is that you complete each puzzle in one sitting with no break — but, hell, this is for fun, so I don’t much care if you ignore that suggestion, too.

**Post your combined time for all three puzzles in the comments.**

If you’d like to compete against other KENthusiasts, you can register for the **KenKen International Championship**, to be held in New York City on December 18. (Unfortunately, registration isn’t open yet… check www.kenken.com in the coming months for details.)

## A Week of KenKen, Day 1: Introduction

Welcome to **A Week of KenKen** (AWOKK). Every day this week, the MJ4MF blog will feature a new post about the number puzzle that Sudoku wishes it could be. That’s right — seven days, nothing but KenKen.

Here’s a list of the posts that you’ll see in coming days:

- Day 1: Intro (that’s
*today*!) - Day 2: The KENtathlon
- Day 3: KenKen Times
- Day 4: My KenKen Puzzles
- Day 5: Harold Reiter’s Puzzles
- Day 6: KenKen Glossary
- Day 7: KenKen Puzzle for 2016
- Day 8: KenKen in the Classroom

If the Beatles got nothin’ but love, babe, eight days a week, then I can certainly have a week with eight days of KenKen. Deal with it.

Today is an introduction, for those of you unfamiliar with KenKen. Here are the rules of the puzzle:

- For an
*n*×*n*grid, fill each row and column with the numbers 1 through*n*. A number may not be repeated in any row or column. - Each heavily outlined set of cells, called a
*cage*, contains a mathematical clue that consists of a number and an arithmetic operation: +, –, ×, or ÷. The numbers in that cage must combine (in any order) to produce the target number using the mathematical operation indicated. - Cages with just one cell should be filled with the target number.
- A number may be repeated within a cage, provided it’s not in the same row or column.

*The New York Times* crossword puzzle editor and *Weekend Edition* puzzlemaster Will Shortz explains KenKen in this short video:

Ready to try for yourself? Here’s a simple puzzle, which is dubbed an “easy” puzzle from the KenKen website:

Too easy? Here’s a slightly more interesting one that I created:

Did that whet your appetite for more? If you were a kid who could’ve held out for several minutes to get two marshmallows, then check back tomorrow for the next installment. But if you were a kid who just couldn’t wait and would’ve gobbled that single marshmallow immediately, then here’s your instant KenKen gratification:

- kenken.com
- Illuminations KeKen app (provided by NCTM, iPhone only)
- MathDoku Pro (my personal favorite, but Android only)

Till tomorrow, happy solving!

## Constant Change

I’m also old, cranky, and cynical. Whatever.

My frustration is not the my-flight-was-delayed-three-times-then-eventually-cancelled-and-there-are-no-more-flights-to-Cleveland-till-tomorrow-morning type. It’s not even the can’t-believe-my-boss-is-making-me-go-to-Cleveland kind of frustration. More like the why-aren’t-there-the-same-number-of-hot-dogs-and-buns-in-a-pack variety. So it’s a First World problem, to be sure, but still annoying. I’ll explain more in a moment.

But first, how ’bout a math problem to get us started?

If you make a purchase and pay with cash, what’s the probability that you’ll receive a nickel as part of your change?

Sure, if you want to get all crazy about this, then we can take all the fun out of this problem by stating the following assumptions:

- You only pay with paper currency. If you paid with coins, then the distribution of coins you’d receive as change would likely vary quite a bit.
- You never use 50¢ coins. Honestly, they’re just too obscure.
- Transaction amounts are uniformly distributed, so that you’re just as likely to receive 21¢ as 78¢ or any other amount.
- Cashiers don’t round because they dislike pennies. So, if you’re supposed to get 99¢ change, the cashier doesn’t hand you a dollar and say, “Don’t worry about it.” Instead, you actually get 99¢ change.

But stating assumptions is a form of mathematical douchebaggery, isn’t it? (As an aside, check out the definition of douchey that’s returned when you do a search. Sexist, anyone?) I prefer problems with no assumptions stated; let folks make their own assumptions to devise a model. If you and I get different answers because of different assumptions, no worries. Maybe we both learn something in the process.

Anyway, where was I? Oh, yeah…

Understanding the solution to that problem is a precursor to the issue that’s causing me frustration. I’ll give the solution in a minute, so pause here if you want to solve it on your own, but let me now allow the proverbial cat out of its bag and tell you why I’m frustrated.

At our local grocery store, there’s a **coin counting machine** that will count your change, sort it, and spit out a receipt that you can take to the customer service desk to exchange for paper currency. Walk in with a jar full of change, walk out with a fistful of fifties. Pretty nifty, right? Except the machine charges a ridiculous 8.9% fee to perform this service. No, thank you.

My bank used to have a similar coin counting machine, and if you deposited the amount counted by the machine into your account, there was no fee. The problem is that everyone was doing this to avoid the grocery store fee, so the machine broke often. The bank finally decided the machine wasn’t worth the maintenance fees and got rid of it. Strike two.

Which brings me to my current dilemma. One Saturday morning every month, we now spend 30 minutes counting coins and allocating them to appropriate wrappers. Which is fine. The problem, however, is that we run out of quarter and penny wrappers way faster than we run out of nickel or dime wrappers. Which brings me to the real question for the day:

Since pennies, nickels, dimes, and quarters are not uniformly distributed as change,

why the hell does every package of coin wrappers contain the same number for each coin type?

The Royal Sovereign Assorted Coin Preformed Wrappers is the best-selling collection of coin wrappers on Amazon, and it provides 54 wrappers for each coin type. They also offer a 360‑pack with 90 wrappers for each coin type; Minitube offers a 100‑pack with 25 wrappers for each coin type; and Coin-Tainer offers a 36‑pack with 9 wrappers for each coin type. But what **no one** offers, so far as I can tell, is **a collection of coin wrappers with a distribution that more closely resembles the distribution of coins that are received as change**.

Whew! It feels good to finally raise this issue for public consideration.

So, the question that I really wanted to ask you…

Given the distribution of quarters, dimes, nickels, and pennies that are received in change, and given the number of coins needed to fill a coin wrapper — 40 quarters, 50 dimes, 40 nickels, and 50 pennies — how many of each wrapper should be sold in a bundled collection?

To answer this question, I determined the number of coins of each type required for every amount of change from 1¢ to 99¢. The totals yield the following graph:

The number of pennies is nearly five times the number of nickels. And there are nearly twice as many quarters as dimes.

But I realize that’s a **theoretical** result that may not match what happens in practice, since this assumes that the amounts of change from 1¢ to 99¢ are uniformly distributed (they aren’t) and that cashiers don’t round down to avoid dealing with pennies (they do). In fact, when I made a purchase of $2.59 yesterday, instead of getting one penny, one nickel, one dime, and one quarter as change, the cashier gave me one penny, **three** nickels, and one quarter, in what was clearly a blatant attempt to skew my data.

So for an **experimental** result, I counted the pennies, nickels, dimes, and quarters in our home change jar. The results were similar:

The ratio of pennies to nickels is closer to three, but the ratio of quarters to dimes is still roughly two.

Using a hybrid of the theoretical and experimental results, and accounting for the fact that only 40 quarters and nickels are needed to fill a wrapper whereas 50 pennies and dimes are needed, it seems that an appropriate ratio of coin wrappers would be:

**quarters : dimes : nickels : pennies :: 17 : 8 : 6 : 19**

Okay, admittedly, that’s a weird ratio. Maybe something like **3:2:1:4**, to keep it simple. Or even 2:1:1:2. All I know is that 1:1:1:1 is completely insane, and this nonsense has got to stop.

Hello, Royal Sovereign, Minitube, and Coin-Tainer? Are you listening? I’ve completed this analysis for you, free of charge. Now **do the right thing**, and adjust the ratio of coin wrappers in a package accordingly. Thank you.

Wow, that was a long rant. Sorry. If you’ve made it this far, you deserve some comic relief.

How many mathematicians does it take to change a light bulb?

Just one. She gives it to a physicist, thus reducing it to a previously solved problem.If you do not change direction, you may end up where you are heading. – Lao Tzu

The only thing that is constant is change. – Heraclitus

Turn and face the strange ch-ch-ch-changes. – David Bowie

A Buddhist monk walks into a Zen pizza parlor and says, “Make me one with everything.” The owner obliged, and when the pizza was delivered, the monk paid with a $20 bill. The owner put the money in his pocket and began to walk away. “Hey, where’s my change?” asked the monk. “Sorry,” said the owner, “change must come from within.”

As for the “probability of a nickel” problem that started this post, here’s my solution.

For change amounts from 1¢ to 25¢, there are ten values (5‑9 and 15‑19) for which you’ll receive a nickel as part of your change.

This pattern then repeats, such that for change amounts from **25 n + 1** to

**25**, where

*n*+ 25*n*is the number of quarters to be returned, you’ll receive a nickel when the amount of change is

**25**, where

*n*+*k**k*∈ {5, 6, 7, 8, 9, 15, 16, 17, 18, 19}. For 0 ≤

*n*< 4, there are 40 different amounts of change that will contain a nickel, so the probability of getting a nickel as part of your change is 40/100, or 40%.

## Math of the Rundetaarn

As we were exiting the Rundetaarn (“Round Tower”) in Copenhagen, Denmark, I noticed a man wearing a shirt with the following quotation:

Find what you love, and let it kill you.

The only problem is that the shirt attributed the quotation to poet Charles Bukowski, when apparently it should have been attributed to humorist Kinky Friedman. For what it’s worth, my favorite Friedman quote is, “I just want Texas to be number one in something other than executions, toll roads, and property taxes.” But this ain’t a post about Kinky Friedman, or even Charles Bukowski. So, allow me to pull off the sidewalk and get back on the boulevard.

Whoever said it, the quotation hit me as drastically appropriate. I suspect that **math will someday kill me**… likely as I cross the street while playing KenKen on my phone, oblivious to an oncoming truck. As I exited the Rundetaarn, I was thinking about all the math that I had seen inside — much of which, I suspect, would not have been seen by many of the other tourists.

The Rundetaarn, completed in 1642, is known for the 7.5-turn helical ramp that visitors can walk to the top of the tower and, coincidentally, to one helluva view of the city. That leads to Question #1.

Along the outer wall of the tower, the winding corridor has a length of 210 meters, climbing 3.74 meters per turn.

What is the (inside) diameter of the tower?

Above Trinitatis Church is a gift shop that is accessible from the Rundetaarn’s spiral corridor. The following clock was hanging on the wall in that little shop:

I have no idea who the bust is, but the clock leads to Question #2.

What

sequence of geometric transformationswere required to convert a regular clock into this clock?

And to Question #3.

Do the hands on this clock spin

clockwiseorcounterclockwise?

And to Question #3a.

What is the “error” on the clock?

A privy accessible from the spiral corridor in the Rundetaarn has been preserved like a museum exhibit. Sadly, I have no picture of it to share, but a sign next to the privy implied that the feces deposited by a friar would fall 12 meters into the pit below.

That leads to Question #4.

What is the

terminal velocity of a depositwhen it reaches the bottom of the pit? (Or should that be “turd-minal velocity”?)

The first respondent to correctly answer all of these questions will earn inalienable bragging rights for perpetuity.

## How Wide and How Deep?

In 2002, William Schmidt described the U.S. math curriculum as “a mile wide, an inch deep,” and it’s been bugging the sh*t out of me ever since.

I mean, I get what he and his co-authors were saying: The curriculum contains too many topics, so they can’t be covered with sufficient depth.

But if a mile is **too wide** and an inch is **too shallow**, then what dimensions *would be* appropriate?

One-inch wide and a mile deep would be problematic, too. That’d be like spending an entire year teaching kids to count to 10.

I suppose we could opt for a square curriculum instead. A curriculum that is a mile wide and an inch deep has an area of 5,280 × 1/12 = 440 square feet, so the conversion would look something like this, with the thin line representing a mile by an inch and the square representing 21 feet by 21 feet:

Sorry, it’s not to scale because of space limitations.

The square curriculum doesn’t feel quite right, either. The only way I know to make this problem tractable is to look at data.

In the late 1990’s, I was a standards weenie. I was fascinated by the variety from state to state. Because I didn’t have a girlfriend (and ostensibly didn’t want one, either), I would read state standards documents **for fun**. At the time Schmidt coined his phrase, Florida had more than 80 standards in each grade, and Utah subjected students to over 130 standards each year. As I recall, the average state had more than 100 standards at each grade level.

Today, Common Core represents a significant reduction in the number of standards. There are approximately 30 standards per grade for K‑8, and closer to 40 standards per course in high school.

Which means that if the curriculum used to be a mile wide, then the current curriculum is closer to ⅓ × 5,280 = 1,760 feet wide.

But if it’s ⅓ as wide, then it needs to be 3 times as deep. Which means the current curriculum is 1,760 feet wide by 3 inches deep, so it looks something like this:

Doesn’t feel like much of an improvement, does it? And the phrase “3 inches deep” doesn’t inspire confidence that the curriculum now has the depth it needs.

So, I give up. I don’t know what the proper dimensions ought to be. I just know that Schmidt’s phrase was hyperbole for dramatic effect, and it worked.

**What do you think are the proper dimensions of the math curriculum?**

Here’s a puzzle about width and depth:

How much dirt is in a hole that measures 4¾ feet × 5¼ feet?

And I know a joke about width, but you need to be able to read CSS:

.yomama { width: 99999999px; }

** **

## 3 Questions to Determine if You’re a Math Geek

Yesterday morning on Cooley and Kevin, a local sports radio show, the hosts and producer each posited three questions that could be used to determine if someone is **a real man**. (The implication being, if you can’t answer all three, then you ain’t.) I didn’t like that many of the questions focused on sports, but I’m not surprised. I was, however, surprised by some of the non-sports questions. What do you think?

**Thom Loverro (guest host):**

- Who wrote
*The Old Man and The Sea*? - What was the name of the bar owned by Humphrey Bogart in
*Casablanca*? - Name three heavyweight boxing champions.

**Kevin Sheehan (regular host):**

- Who was Clark Kent’s alter ego?
- Name one of the two fighters in the “Thrilla in Manila.”
- Who won the first Super Bowl?

**Greg Hough (producer):**

- Name one James Bond movie and the actor who played James Bond in it.
- Who did Rocky beat to win the title?
- With what team did Brett Favre win a Super Bowl?

During the rounds of trivia, Loverro remarked, “If you can name three heavyweight champs but haven’t seen *Casablanca*, then you’re still in puberty.”

This made me wonder:

What three questions would you ask to determine if someone is

a real woman?

One possible question might be, “Name two of the three actresses who tortured their boss in the movie *Nine to Five*.” Then I remembered that women don’t play the same stupid games that men do. And I realized that strolling too far down that path will lead to hate mail or a slap or both. So, let’s move on.

It also made me wonder if there are three questions you could ask to determine if someone is **a real math geek**. Sure, you could use the Math Purity Test, but that’s 63 questions. A 95% reduction in the number of items would be most welcome.

So, here are my three questions:

- What’s the eighth digit (after the decimal point) of π?
- What’s the punch line to, “Why do programmers confuse Halloween and Christmas?”
- Name seven mathematical puzzles that have entered popular culture.

And my honorable mention:

- What’s the airspeed velocity of an unladen swallow?

One of my initial questions was, “Have you ever told a math joke for your own amusement, knowing full well that your audience either wouldn’t understand it or wouldn’t find it funny?” But I tossed that one, because it’s a yes/no question that was personal, not factual. Eventually, which questions were kept and which were discarded came down to one simple rule: If nothing was lost by replacing a question with, “Are you a math dork?” then it should be rejected.

How’d I do? Opinions welcome. **Submit new or revised questions for determining one’s math geekiness in the comments. **