## Posts tagged ‘Fibonacci’

### 144 Quadrillion Reasons to Never Attend a Baseball Game with Me

Baseball fans do lots of dumb things. Wear a foam finger. Act like they’re smart because they correctly chose the attendance from the four choices on the big board. Throw trash at the lower levels. Streak the field. Root for the Yankees. (And while we’re at it, the 1980’s called. They want “The Wave” back.)

Yet few things are as dumb as the following game, which some friends and I play at Nationals Park.

- To start, everyone puts in a $1 ante.
- The first gambler then adds another $1 to the cup. If the player batting gets a hit, that gambler gets all the money in the cup. If the player batting does not get a hit, that gambler passes the cup to the second person.
- The second gambler then adds $2 to the cup. Again, that gambler gets the money if the player batting gets a hit, or passes the cup to the next gambler if the player batting does not get a hit.
- This continues, with the next gambler adding double what the previous gambler added, until there’s a hit.

Because of exponential growth, it doesn’t take long for this game to exceed most people’s comfort level. At a recent game, we had gone through five batters without a hit when the cup was passed to Dave. He gladly added $32, but then passed the cup to Joe when the batter struck out.

Joe was reluctant to add $64, but like the rest of us, he was certain that the paucity of hits couldn’t continue. It did, however, and he passed the cup to me.

Now that the cup was requesting a three-digit donation, this was getting serious. Most of my friends don’t carry $128 to a baseball game, and even fewer of them are willing to put it in a betting cup. Dave looked at me. “You in?” he asked.

“Sure am,” I replied as I put $100 in the cup. “You’ll have to trust that I’m good for the other $28,” I said. “That’s the last of my cash.”

It occurred to me that **52.9% of the money now in the cup had come from my wallet**. That made it extra hard to pass the cup to Adam when the batter struck out.

But Adam just looked at it. “Don’t even think about handing that to me,” he said. “I’m out.”

So the cup went back to Dave, begging for $256 more. “If you’re in, I’m in,” he said to me. “Trust me that I’m good for it?” he asked. I nodded my head to let him know that I did.

When Dave handed the cup to Joe after the next batter flew out, it was **Dave’s money that now accounted for 56.3% of the pot**. Joe just passed the cup directly to me. “I’m out, too,” he said.

Judgment is the better part of valor. But no one has ever accused me of being valiant. *If I put in $512 and there’s a hit*, I thought, *I’ll win $377. But if not, I’ll be passing a cup to Dave that has $649 of my money.*

“I’m in,” I said, surprising even myself. Dave looked at me with equal parts disbelief and dismay… amazed that I had pulled the trigger, unsure what he would do if the cup came to him, stunned that we had gotten to this point.

He’d need to deal with the question sooner than he thought. The batter hit a routine ground ball to the second baseman on the first pitch. As the ball popped in the first baseman’s mitt, my gulp was audible. Folks in our section eyed Dave suspiciously as he cheered an out for the home team. I passed the cup to him and asked, “To you for $1,024, sir. Are you willing to sacrifice your kids’ college fund for this?”

Dave looked at the cup. He looked at me. He looked back at the cup. After several long seconds, he looked at me again. “Split it?” he asked.

I paused. “What?”

“Yeah… what?” said Joe.

Dave explained. “You and I split it. Joe and Adam are out.”

“No!” protested Joe. “That’s not fair! You set us up!”

I ignored Joe, because I realized I now had an exit strategy. “Are you suggesting 50/50?” I asked Dave.

“That’s what I was thinking.”

“But I put two-thirds of the money in that cup.”

“Okay, sure,” Dave said. “We’ll split it 2:1.”

So in the end, I won $33, Dave won $51, and Adam and Joe were disconsolate over what had just transpired.

There had been a streak of 10 batters without a hit (one of the batters had walked, but that doesn’t count as a hit, so our silly game continued), and the pot had grown to $1,027. It would have climbed to $2,051 if Dave had continued. That might only be a drop in the bucket for the Koch brothers, but it’s a sizable amount for a math educator.

The hitless streak made us wonder what game in baseball history would have made us dig deepest into our pockets. The all-time winner would have been a game on May 2, 1917, between the Cincinnati Reds and the Chicago Cubs, when **Fred Toney and Hippo Vaughn each threw no-hitters through nine innings**. In the top of the 10th, Vaughn got one out before giving up the first hit of the game… so that’s at least 55 batters without a hit. It’s hard to find play-by-play data for games from 1917, but the box score shows that there were 2 walks in the game. If both of them happened before the first hit, that’d increase the pot to a staggering 2^{57} + 3 = $144,115,188,075,855,874. That’s **144 quadrillion**, for those of you who, like me, go cross-eyed trying to read such large numbers.

Why do you dislike the number 144?

Because it’s gross.Why do you dislike the number 144 quadrillion?

Because it’sverygross!

Perhaps most interesting to me, though, was **the percent of the cup contributed by the person currently holding the cup**. A tabulation of the possibilities is shown below. This scenario assumes four gamblers with no one dropping out. The rows highlighted in yellow (rounds 4*n* – 3) indicate when the first gambler is holding the cup. Likewise, the second gambler’s turns occur in rows 4*n* – 2, the third gambler’s turns occur in rows 4*n* – 1, and the fourth gambler’s turns occur in rows 4*n*.

Round |
Contribution |
Gambler Total |
Cup Value |
Percent |

1 | 1 | 2 | 5 | 40.00% |

2 | 2 | 3 | 7 | 42.86% |

3 | 4 | 5 | 11 | 45.45% |

4 | 8 | 9 | 19 | 51.43% |

5 | 16 | 18 | 35 | 52.24% |

6 | 32 | 35 | 67 | 52.67% |

7 | 64 | 69 | 131 | 52.90% |

8 | 128 | 137 | 259 | 53.20% |

9 | 256 | 274 | 515 | 53.26% |

10 | 512 | 547 | 1,027 | 53.29% |

11 | 1,024 | 1,093 | 2,051 | 53.31% |

12 | 2,048 | 2,185 | 4,099 | 53.33% |

13 | 4,096 | 4,370 | 8,195 | 53.33% |

14 | 8,192 | 8,739 | 16,387 | 53.33% |

15 | 16,384 | 17,477 | 32,771 | 53.33% |

16 | 32,768 | 34,593 | 65,539 | 53.33% |

17 | 65,536 | 69,906 | 131,075 | 53.33% |

Here’s how to read the table:

- The
**Contribution**is the amount that a gambler adds to the cup in that round. - The
**Gambler Total**is the combined amount contributed by that gambler so far. For instance, the “Gambler Total” in Row 13 is $4,370, because the gambler has contributed 4,096 + 256 + 16 + 1 on his previous four turns, plus the $1 ante at the beginning of the game. - The
**Cup Value**is the amount in the cup, which includes the ante plus all previous contributions. - The
**Percent**then shows the percent of the money in the cup that was contributed by the gambler holding the cup.

In Round *n*, the Cup Value is given by the wonderfully simplistic formula

because the sum of the first *n* powers of 2 is 2^{n} – 1, and the initial ante contributes $4 more.

More difficult is the formula for the combined contribution shown in the Gambler Total column. For Gambler 1, the formula is:

The Gambler Total formula for gamblers 2, 3, and 4 are similarly complex. (You can have fun figuring those out on your own.) But the purpose of determining those formulas was to investigate the percent of the Cup Value represented by the Gambler Total. And as you can see in the final column, **the percent of the cup contributed by the gambler holding the cup tends toward 53.33%**.

So here’s the way to think about this. If you pass the cup and the next gambler wins, 53.33% of the money in the cup is his; **26.66% of the money in the cup was yours**; and, the remaining 20.00% came from the other two gamblers. Which is to say, **you’ve contributed 26.66/46.66 = 57.14% of the winner’s profit**.

Ouch.

Taking all this into account, and realizing that I could bankrupt my family if there were ever a streak of just 20 batters without a hit, some additional rules became necessary:

- A person can choose to leave the game at any time. Play continues with those remaining. Obviously, if only one gambler remains, he or she wins.
- Anyone can declare a “cap,” wherein the amount added to the cup continues at the current rate without increasing. For instance, if I added $32 to the cup on my turn and called a “cap,” then every person thereafter would simply need to add $32 on their turn, instead of doubling.

This latter rule seems like a good idea; it transfers the game from exponential growth to linear growth. Still, the pot grows quite rapidly, as shown below:

After 10 consecutive batters without a hit, the pot would still grow to $160, even if a “cap” had been implemented at the $32 mark.

We’ve also switched to **the Fibonacci sequence**, because that grows less quickly over time.

At a recent Nationals-Cubs game, four of us played this game with the amended rules. Everyone anted $1 to start, and then the amounts added to the cup were:

1, 2, **3**, 5, 8, 13, **21** (*cap*), 21, 21, 21, **21**

Dave was third, and the amounts he added to the cup are bold in the sequence above. Both Tanner Roark and Tsuyoshi Wada started strong, and the first 10 batters were retired without a hit or a walk. But the 11th batter of the game got a hit in the bottom of the second inning. Dave had contributed $46 of the $141 in the cup, so he won $95 on that beautiful swing by Wilson Ramos of the Nationals. (It wouldn’t have been “beautiful” if it had been made by a Cub or if it had cost me more than $27.)

The adjustment of using the Fibonacci sequence was a good one. The seventh donation to the cup was a reasonable $21, compared to a $64 contribution using the doubling scheme. In addition, instituting the “cap” permitted folks to continue playing long after they otherwise would have dropped out.

We continued playing through the end of the game. Which brings me to one final rule:

- At the end of the game, the gambler holding the cup does not win. Instead, the money in the cup goes to the next gambler.

This only makes sense. You wouldn’t want a gambler winning if he was holding the cup when an out was recorded.

Last Friday’s game ended with an astonishing 18 hits. There were no other streaks of 10+ batters without a hit, so none of the wins were as large as Dave’s first. On the other hand, I won more than I lost the rest of the game, and I was only down about $10 by game’s end.

Still, I’m concerned that I’ll go broke by the end of the season. One solution, of course, would be to stop playing, but that would require willpower and a higher intellect. Instead, I’m thinking that perhaps I should try to earn some extra money. If you’d like to contribute to this worthy cause, **I’m available for tutoring, stand-up comedy, and blog post writing**.

And if you have a need for analyzing the dumb games that boys play while watching a baseball game, well, there’s data to suggest that I’m pretty good at that, too.

—

*I’d like to thank Marjan Hong for her help in analyzing this game. I’d also like to thank Dave Barnes for teaching me this game, for wasting countless hours discussing the rules and amendments, and for taking my money.*

### Mathiest Week of 2013

Can you hear it? That’s the sound of the awesomeness approaching.

It starts this Wednesday.

5/8/13

The month, date and year are consecutive terms in the Fibonacci sequence.

It continues on Thursday.

5/9/13

The numbers form an arithmetic sequence.

And then there’s Sunday.

5/12/13

That’s a Pythagorean triple.

Arithmetic sequence dates are a dime a dozen. In fact, there are six of them in 2013 alone. Pythagorean dates and Fibonacci dates are far more rare. There are only eight Pythagorean dates and six Fibonacci dates in the entire 21st century. To have all of them occur within a six-day span is incredible.

**How will you celebrate?**

### Out of Sequence

If you came here looking for a good math joke and were thinking, “My God, I sure hope he didn’t post another story about his sons,” well, now would be a good time to close your browser.

Still with me? Good. Because I’ve just got to tell you about two things that have happened recently with Alex and Eli.

As we were walking through the neighborhood today, I was pointing to things that I thought the boys would enjoy. “Look,” I said as I pointed to the sky. Alex looked up. “An airplane!” he shouted.

I pointed to an automobile on the right. “A red car!” Eli exclaimed.

Then I pointed to a house on our left. Alex announced, “A square Fibonacci number!”

I admit, I didn’t see that coming. I had pointed to a house on our street, and there was a cat on the front porch that I wanted the boys to see. Instead, Alex saw the house number — 144 — and identified it as part of two different sequences. (I don’t think he recognized the lovely coincidence that 144 is the 12th term in both sequences, though I can’t be sure.)

My boys’ love of sequences is a result of teeth brushing. (No, really.) To make sure they brush their teeth long enough, I count while they brush — 15 seconds on the left side, 15 seconds on the right side, 15 seconds for the front teeth, and finally 15 seconds for “all around,” during which they’re supposed to brush the inside parts of their teeth. As you might well imagine, though, I was getting bored counting 1, 2, 3, …, 15 four times every night. I started to mix it up.

- I counted 1‑15 for the left side, then 16‑30 for the right, 31‑45 for the front, and 46‑60 for all around. But that got boring rather quickly, too.
- I switched to counting respectively by 1, 2, 3, and 4; that is, I’d count 1‑15 for the left, 17‑45 (by 2’s) for the right, 48‑90 (by 3’s) for the front, and 94‑150 (by 4’s) for all around. That was unsettling, though, because the boys started to think that 94, 98, 102, …, 150 were multiples of 4. While they recognized that counting by 2’s from 17‑45 gave the odd numbers, they weren’t able to discern that 94‑150 by 4’s analogously gave numbers
*n*≡ 2 mod 4. Nor did I expect them to — they’re only 3½. - I therefore switched it up again and counted by 1’s, 2’s, 3’s, and 6’s, which meant that “all around” was now 96, 102, 108, …, 150, which provided the more satisfying pattern of multiples of 6.

Then one night, Eli shocked me. He said, “Daddy, I figured out another way to count by 5’s.” I wasn’t really sure what he meant, but I assumed that “counting by 5’s” was a general term he was using to refer to skip counting. I asked him to explain. He said, “Like this: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55.”

Upon hearing this, I thought what any geeky American father would think. “Holy sh*t. Did my three-year-old son really figure out the triangular numbers by himself?”

So I asked him, “What are those numbers, Eli?”

“Well,” he began, “there’s 1,” and then 1 + 2 = 3, and 1 + 2 + 3 = 6, and 1 + 2 + 3 + 4 = 10, and…

Through a few more questions, I realized that he had ascertained these sums using my old TI‑83 calculator. I am proud to share a picture of the boys happily playing with technology:

The boys had been playing with some of my graphing calculators because they liked typing letters to spell words. Little did I know they were entering expressions and learning some math, too. (Take that, all you stodgy opponents of calculators in the classroom!)

After that, there were no holds barred; during teeth brushing, I started busting out all kinds of sequences:

- Triangular numbers: 1, 3, 6, 10, …, where
*T*(*n*) = ½(*n*)*(*n + 1). - Square numbers: 1, 4, 9, 16, …, where
*S*(*n*) =*n*^{2}. - Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, …, where
*F*(*n*) = F(*n*– 1) +*F*(*n*– 2). - Feeby numbers: 11, 22, 33, 44, 55, …, where
*Fb*(*n*) = 11*n*.

(These are just the multiples of 11, but Eli named them the Feeby numbers, I think because it sounded a little like the first part of Fibonacci numbers, and two of the numbers in the list (55, 66) are elements of the Fibonacci sequence.) - Perrin sequence: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, …, where
*P*(*n*) =*P*(*n*– 2) +*P*(*n*– 3).

I’ve been thinking about sharing Moser’s Circle Problem with them and showing them the sequence 1, 2, 4, 8, 16, 31, 57, 99, …. Then again, maybe not.

Okay, so if you’re one of the folks who came for a math joke and tolerated that entire story, then by gosh you deserve a math joke. I don’t have a joke about triangular numbers (note to self: create a joke about triangular numbers), but I do have a joke that involves the word *triangular.*

What is small, green, and triangular?

A small, green triangle.

Yeah, I know… it’s lame.

Let me try to make amends with a cool triangular number problem.

Append the digit 1 to the end of every triangular number. For instance, from 3 you’d get 31, and from 666 you’d get 6,661. Now take a look at all of the divisors of the numbers you’ve created. What are the units digits of the divisors for every number created in this way? Can you prove that this result always holds?

I have a proof, but you’ll have more fun solving it on your own than reading my solution.

### Math Clocks

Jiminy. The folks at Clock Zone make a math class wall clock — and I would like to be the first to publicly chastise buy.com, amazon, and anyone else who is selling it. It contains at least two mathematical errors:

SPOILER ALERT: In my rant below, I identify the errors in the clock. If you’d like to identify them for yourself, don’t read any further.

I say “at least” two errors because there may be more. The obvious errors are for 9 (the expression assumes that the exact value of π is equal to the common approximation 3.14) and for 7 (because the equation is quadratic, *x* = 7 is only one of the answers; the other possible answer is *x* = ‑6).

More generally, I have an issue with any of the algebraic equations that are meant to represent integers. For instance, the equation 50/2 = 100/*x* has solution *x* = 4, but I believe that it is incorrect to say that the equation itself is equal to 4. So perhaps the clock has four errors, if you consider the algebraic equations for 4 and 10 to be erroneous, as I do.

This clock is meant to be a math joke. Edward de Bono in *The Mechanism of the Mind* (1969) suggested that when a familiar connection (such as seeing the numerals 1‑12 on a clock) is disrupted, laughter occurs as a new connection (seeing mathematical expressions instead of numerals) is made. Sadly, math jokes are supposed to make you laugh… yet this clock makes me want to cry.

To ease the pain, I did a little research and uncovered several clocks of famous mathematicians. I present them here for your enjoyment.

Leonardo da Pisa:

Kenneth Appel:

Rene Descartes:

Karl Friedrich Gauss: