## Posts tagged ‘baseball’

### They’re Moving Second Base

When I first heard that baseball is moving second base, my first thought was, “My, goodness! Isn’t it enough that we’re dealing with a global pandemic, a Russian tyrant invading a neighboring country, a humanitarian crisis in Nicaragua, food insecurity in Somalia, Haiti, and Madagascar, an ever-widening wealth gap, an uptick in calls from unknown numbers, paper cuts, and excessively long lines at the Starbucks drive-thru? I mean, when’s it gonna stop?”

But my second thought was, “This is going to wreak havoc on the secondary textbook publishing industry.” Just look at all the problems that exploit the baseball context:

- Location of the pitcher’s mound (Q2)
- Throw from catcher to second base
- Throw from catcher to second base
- Distance from first base to third base (Q4)
- Runner’s speed in relation to second base

All of those problems are predicated on a consistent distance between bases. Won’t the relocation of second base cause inconsistency?

Well, actually, it won’t.

According to the official rules of baseball, one vertex from first base, third base, and home plate are to be coincident with three vertices of the infield square; but, the *center* of second base is to be coincident with the fourth vertex. With the rule change, second base will be moved so that one of its vertices will be coincident with the fourth vertex of the infield square, finally bringing a state of geometric consistency to the game that I, for one, believe is long overdue. The image above shows the new (white) and old (gray) locations of second base.

The question all fans should be asking isn’t why are they changing the layout of the infield. The more pertinent question is, what took so damn long?

As it turns out, second base doesn’t get all the credit for the previous configuration issues. To the contrary, it was the movement of the other bases that resulted in a problem. In the 1860s, it was generally agreed that all four bases should be positioned with their centers at the vertices of the infield square. And by “generally agreed,” I mean that there was consensus about this, but it wasn’t officially stated in the rules until 1874. Then in 1877, the rules changed so that the back corner of home plate — at the time, home plate was still a square, not a pentagon like today — coincided with the vertex of the infield square, positioning all of home plate in fair territory. A decade later, first base and third base were moved to be entirely within fair territory, too; but most folks didn’t even notice, because that same year (1887) a number of other rules changes garnered more attention:

- Pitchers were limited to just one step when delivering a pitch; previously, they could take a running start
- Batters were prohibited from requesting a high or low ball from the pitcher, as they had been allowed in the past
- The pitcher’s mound was moved back five feet (from 50′ to 55′)
- Five balls were required for a walk, reduced from six
- Four strikes were required for a strikeout, increased from three

With so many drastic rules changes happening simultaneously, it’s hardly a surprise that first and third were repositioned in relative obscurity while second was left floundering in geometric misalignment.

Just so you know, the rules change will only occur in the minor leagues this year. If it pans out, you can bet you’ll see it in the MLB in a year or two.

But why stop there? Here are some other rules changes in sports that should probably be implemented.

**Scoring system in football.** I mean, you can score 1, 2, 3, 6, 7, or 8 points depending on what you do. Isn’t that a little excessive? While we’re at it, let’s change the width of the field, too — who the hell thought 53⅓ yards was an appropriate dimension?

**College basketball uniforms.** Bring back 6, 7, 8, and 9. You may not have known that those digits are not allowed, because each of them requires two hands. Referees indicate the player who committed a foul using their fingers — for instance, holding up two fingers on the right hand and three fingers on the left to indicate that an infraction was committed by number 23 — and the digits 6‑9 would require more than five fingers.

**Frames in bowling. **Two balls ain’t enough. Give everyone three attempts to knock down all ten pins.

**Cheerleader weigh-ins. **Really, folks? The 15th century called, and they want their misogyny back. One anonymous NFL cheerleader wrote that she was banned from performing because she weighed more than 122 pounds. While we’re at it, ban weigh-ins for jockeys, too. The Kentucky Derby — which apparently has one of the more liberal weight allowances — caps the weight at 126 pounds; that includes 7 pounds for the jockey’s gear, so the jockey can’t tip the scale at more than 119 pounds.

**Taunting.** Allow it everywhere. In college football, the rule is just stupid. Admittedly, one player shouldn’t be allowed to stand over another player while making insulting comments about their mother; but “taunting” according to the NCAA Rule Book includes spinning or spiking the ball, choreographed acts, and the player altering stride when approaching the end zone. C’mon! Further, I’d like to see taunting *encouraged* a bit more at some events, such as math competitions. Wouldn’t it be great if one participant walked up to another and said, “You can’t even spell Q.E.D.!”

### Mr. Consistency, Khris Davis

If you flipped four coins, the probability of getting exactly one head would be 0.25.

But the probability of doing that four times in a row is much lower, somewhere closer to 0.0039, or about 1 in 250.

Now, imagine flipping 100 coins four times, and getting the same number of heads each time. The odds of that happening are only slightly better than impossible. In fact, if every person *in the entire world* were to flip 100 coins four times, it would still be highly unlikely that this would ever happen.

That’s how rare it is, and it gives you some idea of what Major League Baseball player Khris Davis just pulled off. The Oakland Athletics outfielder just finished his fourth consecutive season with a batting average of .247. That’s right — the same average four seasons in a row.

Davis had some advantage over our coins, though. For starters, he wasn’t required to have the same number of at-bats every year. Moreover, batting averages are rounded to three decimal places, so his average wasn’t *exactly* the same during those four years; it was just really, really close:

**2015**: .24745 (97 hits in 392 at-bats)**2016**: .24685 (137 in 555)**2017**: .24735 (140 in 566)**2018**: .24653 (142 in 576)

How could something like this happen? According to Davis, “I guess it was meant to be.“

Perhaps it *was* predestination, but I prefer to put my faith in numbers.

Empirically, we can look at the data. From 1876 to present, there have been 19,103 players in the major leagues. The average length of an MLB career is about 5.6 years, which means that an average player would have about three chances to record the same batting average four seasons in a row. It’s then reasonable to say that there have been approximately 3 × 19,103 = 57,309 opportunities for this to happen, yet Khris Davis is the only one to accomplish this feat. So experimentally, the probability is about 1 in 60,000.

Theoretically, we can look at the number of ways a player could finish a season with a .247 batting average. In 2007, the Phillies’ Jimmy Rollins recorded an astounding 716 at-bats. That’s the most ever by a Major League Baseball player. So using a sample space from 1 to 716 at-bats, I determined the number of ways to achieve a .247 batting average:

- 18 hits, 73 at-bats
- 19 hits, 77 at-bats
- 20 hits, 81 at-bats
- 21 hits, 85 at-bats
- 22 hits, 89 at-bats
- 36 hits, 146 at-bats
- …
- 161 hits, 652 at-bats
- 161 hits, 653 at-bats
- …
- 177 hits, 716 at-bats

And, of course, there are the examples above from Davis’s last four seasons.

It’s interesting that it’s not possible to obtain a batting average of .247 if the number of at-bats is anywhere from 90 to 145; yet it’s possible to hit .247 with 161 hits for either 652 or 653 at-bats. I guess it’s like Ernie said: “That’s how the numbers go.“

All told, **there are 245 different ways to hit .247** if the number of at-bats is 716 or fewer.

That may sound like a lot, but consider the alternative: there are 256,441 ways to **not** hit .247 with 716 or fewer at-bats.

So, yeah. No matter how you look at it, what Davis did is pretty ridiculous. Almost as ridiculous as what happened to Saul…

Saul is working in his store when he hears a voice from above. “Saul, sell your business,” the voice says. He ignores it. His business is doing well, and he’s happy. “Saul, sell your business,” the voice repeats. The voice goes on like this for days, then weeks. “Saul, sell your business.” Finally, Saul can’t take it any more. He finds a buyer and sells his business for a nice profit.

“Saul, take your money, and go to Las Vegas,” the voice says.

“But why?” asks Saul. “I have enough to retire!”

“Saul, take your money to Las Vegas,” the voice repeats. It is incessant. Finally, Saul relents and heads to Vegas.

“Saul, go to the blackjack table and bet all your money on one hand.”

He hesitates for a moment, but he knows the voice won’t stop. So, he places his bet. He’s dealt 18, while the dealer has a 6 showing. “Saul, take a card.”

“What? The dealer has…”

“Saul, take a card!” the voice booms.

Saul hits. He gets an ace, 19. He sighs in relief.

“Saul, take another card.”

“You’ve got to be kidding me!” he pleads.

“Saul, take another card.”

He asks for another card. Another ace, 20.

“Saul, take another card,” the voice demands.

But I have 20!” Saul shouts.

“TAKE ANOTHER CARD, SAUL!”

“Hit me,” Saul says meekly. He gets another ace, 21.

And the voice says, “Un-fucking-believable!”

### Which is Closest?

Not too long ago, I published a blog post about end-to-end comparisons, those silly feats of computational gymnastics that try to reduce an overwhelming statistic to something more tangible. Something like this:

If each piece of candy corn sold in a year by Brach’s — the top manufacturer of the waxy confection — were laid end to end, they would circle the Earth 4.25 times.

In writing that post, I inadvertently formulated a statistic that rather surprised me:

If all the players on an NFL team were laid end to end, they’d stretch from the back of one end zone to the opposite goal line.

That the players would almost line the entire field struck me as an amazing coincidence. And it got me to thinking — might this be true for other sports?

Not one to let sleeping dogs — or professional athletes — lie, I decided to investigate. Based on that research, here’s a simple, one-question quiz for you.

**Which of the following comparisons is the most accurate?**

- If all of the players on an
**NHL (hockey)**roster were laid end to end, they would reach from**one end of the rink to the other**. - If all of the players on an
**NBA (basketball)**roster were laid end to end, they would reach from**one end of the court to the other**. - If all of the players on an
**NFL (football)**roster were laid end to end, they would reach from**one end line to the other**. - If all of the players on an
**MLB (baseball)**roster were laid end to end, they would reach from**home plate to second base**. - If all of the players on an
**MLS (soccer)**roster were laid end to end, they would reach from**one end to the other**.

As you begin to think about that question, some notes:

- Every professional baseball stadium has different measurements. Fenway Park (Boston) is a mere 310′ from home plate to the right field wall, whereas Comerica Park (Chicago) extends 420′ from home plate to straightaway center. Consequently, the distance from second to home is used in the fourth answer choice, because it’s the same for every field.
- To my surprise, MLS stadiums are not uniform in length and width. Who knew? The length of the field must be at least 100 meters, at most 110 meters, and anywhere in between is fine. Assume an average length of 105 meters for the fifth answer choice.

Before you read much further, let me say how much fun I’ve had discussing this question around the dinner table and at the local pub. In spite of hard facts, there is resolute disagreement about player height, roster size, and field dimensions. And the shocking (or should I say predictable?) results raise an eyebrow every time. I only mention that to persuade you to think about the question, alone or with some friends, before continuing.

Okay, you’ve cogitated? Then let’s roll.

In researching the answer to the question, I was struck by how close the total length of all players on the roster is to the length of the field, court, or rink. Coincidence? Of course, a larger field requires more players, so perhaps this is the evolution of roster size that one would expect.

To answer the question, you need to know the height of an average player, the number of players on a roster, and the dimensions of professional venues. All of that data can be found in a matter of minutes with an online search, but I’ll save you the trouble.

League |
Average Height (in.) |
Players on a Roster |
Combined Height, Laid End to End (ft.) |
Dimensions |

NHL | 73 | 23 | 140 | 200 feet (from end to end) |

NFL | 74 | 53 | 327 | 120 yards (360 feet, from end to end) |

NBA |
79 |
14 |
92 |
94 feet (from end to end) |

MLB | 73 | 23 | 140 | 127 feet (from home to second) |

MLS | 71 | 28 | 166 | 105 meters (345 feet, from end to end) |

As it turns out, the MLS comparison is the least accurate. The combined heights of soccer players is only 48% of the length of their field. The NHL comparison is a little better, with players’ heights extending 70% of the length of the field. But the NFL and MLB are both very close, with the players’ heights equalling 91% of the field length and 110% of the distance from home to second, respectively. Astoundingly, if the players on an NBA team were laid end to end, they’d come just 22 inches short of covering the entire court, accounting for a miraculous 98% of the length!

So there you have it. **D**, final answer.

One last thought about this. I play ultimate frisbee, a sport with a field that measures 120 yards (360 feet). For tournaments, our rosters are capped at 29 players, and I suspect my amateur teammates are, on average, shorter than most professional athletes. If we assume a height of 5’10” for a typical frisbee player, then the combined height is 172 feet. That puts us in the realm of soccer, with our combined length covering just 48% of the field.

If, like me, you play a sport that isn’t one of the Big 5 in the U.S., I’d love to hear about your sport’s field and roster size, and how it ranks with the comparisons above.

### Nationals Win Probability, and Other Meaningless Statistics

The first pitch of last night’s Nationals-Phillies game was 8:08 p.m. That’s pretty late for me on a school night, and when a 38-minute rain delay interrupted the 4th inning, well, that made a late night even later.

The Phillies scored 4 runs in the top of the 5th to take a 6‑2 lead. When the Nationals failed to score in the bottom of the 5th, I asked my friends, “What are the chances that the Nationals come back?” With only grunts in response and 10:43 glowing from the scoreboard, we decided to leave.

On the drive home, we listened as the Nationals scored 3 runs to bring it to 6‑5. That’s where the score stood in the middle of the 8th inning when I arrived home, and with the Nats only down by 1, I thought it might be worth tuning in.

The Nats then scored 3 runs in the bottom of the 8th to take an 8-6 lead. And that’s when an awesome stat flashed on the television screen:

Nats Win Probability

- Down 6-2 in the 6th: 6%
- Up 8-6 in the 8th: 93%

Seeing that statistic reminded me of a Dilbert cartoon from a quarter-century ago:

I often share Dogbert’s reaction to statistics that I read in the newspaper or hear on TV or — *egad!* — are sent to me via email.

I had this kind of reaction to the stat about the Nationals win probability.

For a weather forecast, a 20% chance of rain means it will rain on 20% of the days with exactly the same atmospheric conditions. Does the Nats 6% win probability mean that *any team* has a 6% chance of winning when they trail 6-2 in the 6th inning?

Or does it more specifically mean that the Nationals trailing 6-2 in the 6th inning to the Phillies would only win 1 out of 17 times?

Or is it far more specific still, meaning that this particular lineup of Nationals players playing against this particular lineup of Phillies players, late on a Sunday night at Nationals Stadium, during the last week of June, with 29,314 fans in attendance, with a 38-minute rain delay in the 4th inning during which I consumed a soft pretzel and a beer… are **those** the right “atmospheric conditions” such that the Nats have a 6% chance of winning?

As it turns out, the win probability actually includes lots of factors: whether a team is home or away, inning, number of outs, which bases are occupied, and the score difference. It does not, however, take into account the cost or caloric content of my mid-game snack.

A few other stupid statistics I’ve heard:

- Fifty percent of all people are below average.
- Everyone who has ever died has breathed oxygen.
- Of all car accidents in Canada, 0.3% involve a moose.
- Any time Detroit scores more than 100 points and holds the other team below 100 points, they almost always win.

**Have you heard a dumb stat recently?** Let us know in the comments.

### Guess the Graph on Square Root Day

Today is **Opening Day** in Major League Baseball, and 13 games will be played today.

It’s also **Square Root Day**, because the date 4/4/16 transforms to 4 × 4 = 16.

With those two things in mind, here’s a trivia question that seems appropriate. **Identify the data set used to create the graph below.** I’ll give you some hints:

- The data set contains 4,906 elements.
- It’s based on a real-world phenomenon from 2015.
- The special points marked by A, B, and C won’t help you identify the data set, but they will be discussed below.

Got a guess?

- The horizontal axis represents “Distance (Feet)” and the vertical axis represents “Frequency.”

Still not sure? Final hints:

- Point A on the graph represents Ruben Tejada’s 231-foot inside-the-park home run on September 2, 2015.
- Point B on the graph represents the shortest distance to the wall in any Major League Baseball park — a mere 302 feet to the right field fence at Boston’s Fenway Park.
- Point C represents the longest distance to a Major League wall — a preposterous 436 feet to the deepest part of center field at Minute Maid Park in Houston.

Okay, you’ve probably guessed by now that the data underlying the graph is **the distance of all home runs hit in Major League Baseball during the 2015 season**. That’s right, there were 4,906 home runs last year, of which 11 were the inside-the-park variety. The distances ranged from 231 to 484 feet, with the average stretching the tape to 398 feet, and the most common distance being 412 feet (86 HRs traveled that far). The data set includes 105 outliers (based on the 1.5 × IQR convention), which explains why a box plot of the data looks so freaky:

The variety of shapes and sizes of MLB parks helps to explain the data. Like all math folks, I love a good graphic, and this one from Louis J. Spirito at the Thirty81Project.com is both awesome and enlightening:

Here are some more baseball-related trivia you can use to impress your friends at a cocktail party or math department mixer.

**Who holds the record for most inside-the-park home runs in MLB history?**

*Jesse Burkett, 55 (which is 20 more than he hit outside-the-park)*

**Which stadium has the tallest wall?**

*The left field fence at Fenway Park (a.k.a., the “Green Monster”) is 37 feet tall.***Which stadium has the shortest wall?**

*This honor also belongs to Fenway Park, whose right field wall is only 3 feet tall.***Although only 1 in 446 home runs was an inside-the-park home run in 2015, throughout all of MLB history, inside-the-park home runs have represented 1 in ____ home runs.**

158**Name all the ways to get on first base without getting a hit.**

*This is a topic of much debate, and conversations about it have taken me and my friends at the local pub well into the wee hours of the morning. I have variously heard that there are 8, 9, 11, and 23 different ways to get on base without getting a hit. I think there are 8; below is my list.*

*(1) Walk*

*(2) Hit by Pitch*

*(3) Error*

*(4) Fielder’s Choice*

*(5) Interference*

*(6) Obstruction*

*(7) Uncaught Third Strike*

*(8) Pinch Runner*

**What is the fewest games a team can win and still make the playoffs?**

*39. The five teams in a division play 19 games against each of the other four teams in their division. Assume that each of those teams lose all of the 86 games against teams***not**in their division. Then they could finish with 39, 38, 38, 38, and 37 wins, respectively, and the team with 39 wins would make the playoffs by winning the division.**Bases loaded in the bottom of the ninth of a scoreless game, and the batter hits a triple. What’s the final score of the game?**

*1-0. By rule, the game ends when the first player touches home plate.***In a 9-inning game, the visiting team scores 1 run per inning, and the home team scores 2 runs per inning. What is the final score?**

*16-9. The home team would not bat in the bottom of the ninth, since they were leading.*

### 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.*

### Jon Lester, Eugenio Vélez, and Hitlessness

I was reminded of my second-favorite joke, which is only mildly mathy, while watching the Cubs-Nationals game last night…

What do you do with an elephant who has three balls?

Walk him, and pitch to the rhino.

(If you’re wondering what my favorite joke is, read all about it in Make Your Own (Math) Joke.)

When it comes to pitching and hitting, Jon Lester is clearly better at one than the other. His ERA is an impressive 3.30, but his batting average is .000. That’s right, he’s never gotten a hit in 9 Major League seasons. With two more outs last night — the first, a deep fly ball that was caught by Denard Span, which you can watch on Yahoo sports; the latter, a strikeout — he “improved” to an incredible 0-for-59, a Major League record for futility with the longest hitless streak to start a career.

Lester’s hitless streak is the longest ever by a pitcher. But pitchers aren’t paid to hit. The dubious distinction of the longest hitless streak for a position (non-pitching) player in the Major Leagues is held by Eugenio Vélez, who didn’t get a hit in 46 consecutive at-bats during the 2010-2011 seasons.

A starting player averages 3.3 at‑bats per game, so Velez’s record is equivalent to 14 games without a hit. Assuming that a player is actually *trying* to hit the ball, a 14-game hitless streak is an impressive accomplishment; and probabilistically, it’s damn near impossible. Not withstanding the likelihood that very few sane general managers would let such a player continue to bat, it also defies the odds that the sun wouldn’t shine at least once on this slumping hitter’s behind.

A while back, I created an **Excel file (XLS) to analyze hit streaks**. But you could also use it to analyze hitless streaks by changing a couple formulas.

Using Eugenio Velez’s career batting average of .241 (which is deflated, because it *includes* his record-breaking streak), a hitless streak of 14 games didn’t occur even once in 500,000 games using the Excel sheet model. With 162 games per season, that’s more than 3,000 seasons. Only the very best pro baseball players have a career that spans 20 seasons; those players who hit only .241 have careers that are far shorter, so 46 consecutive at-bats without a hit is impressive, indeed.

Have fun playing with the spreadsheet. Now for a trivia question…

Who is the only Major League player to have 7 hits in one 9-inning game?

Rennie Stennett, Pittsburgh Pirates, September 16, 1975. The Pirates won the game 22‑0 against the Cubs. (Johnny Burnett had 9 hits in an 18-inning game in 1932; three other players have had 7 hits in a game, but all of them required extra innings.)

And a joke…

Why was the calculus teacher bad at baseball?

He was better at fitting curves than hitting them.

And a quote…

Slowest pitch in baseball to reach the catcher? 30 mph, thrown at a 45° angle. Any slower at any other angle hits ground.

— Neil deGrasse Tyson

### Math Jokes for Pitchers and Catchers

Pitchers and catchers in Major League Baseball report to spring training this week. Here’s a math joke for baseball players. (And while I’m not certain who said this originally, it’s a solid bet that this quote comes from Dave Barry, though I haven’t been able to confirm that.)

Everyone knows that if a mathematician had to choose between solving a difficult story problem and catching a fly ball, he would surely solve the problem without thinking twice about whether the Infield Fly Rule was in effect.

And since we’re talking baseball…

What do you do with an elephant with three balls?

Walk him, and pitch to the rhino.

Here are some baseball and math trivia questions for you **[updated 2/10/2013]**, some based on problems from Erich Friedman’s Baseball Puzzles page:

- In the bottom of the ninth inning in a game with no score, the bases are loaded. The batter hits a ball that rolls into the right field corner. Before the ball is thrown home, all four base runners cross the plate. What is the final score of the game?
- If the visiting team scores 2 runs per inning, and the home team scores 3 runs per inning, what is the final score of a nine-inning game?
- A player has four at-bats in a game, and he got a hit in his last of these at-bats. His batting averages for the season (rounded to three decimal places, as usual) at three different times during the game do not have any digits in common. What was his batting average at the end of the game?
- What is the minimum number of games a Major League Baseball team must win to make the playoffs?

Answers

- The final score is
**1-0**. Once the first player touches the plate, the home team wins, and the game is officially over. - The final score is
**24-18**. With a lead in the middle of the ninth inning, the home team does not bat in the bottom of the ninth. - At the end of the game, his average was
**.409**. He was 5 for 18 (.278) at the beginning of the game, 6 for 19 (.316) after one at-bat, and 9 for 22 (.409) after three more at-bats. - A team only has to win
**28 games**. There are four teams in the American League West division, and the requirement in Major League Baseball is that every team must play either 18 or 19 games against the other teams in their division. That means that each of these four teams could play as few as 54 divisional games. If each of them loses the 108 non-divisional games that they play, then the four teams could conceivably finish with 28, 27, 27, and 26 wins, respectively. Under this scenario, the division winner is the team with just 28 victories.

There are many who complain that, because of the Electoral College, it’s possible for a candidate to be elected President with less than 30% of the popular vote. Yet this situation is worse — a team could make the playoffs by winning just 17% of its games. (Of course, the Presidential election has slightly more consequence. Maybe.)

### Pizza in the Park

Yesterday, the Washington Nationals — now my surrogate team, since being a Pirates fan is just too damn painful — finished the 2012 season with the National League East division title.

The day before yesterday, I was at the Nationals-Phillies game. A friend took my sons and me to the game, so in return, I offered to buy dinner at Nationals Park. When I asked his sons what we should get, they answered just as you’d expect any 5- and 9-year-olds to answer: “Pizza!”

I was happy to oblige. I assumed I was getting off cheap.

I was wrong.

At the concession stand, I gasped when I saw that pizza slices were $6 each. Admittedly, they were big slices — an 18″ pizza was divided into 6 slices — but that’s still a lot of money for 42 square inches of pizza. If you think of it as just 14 cents per square inch, it doesn’t feel quite so bad. Until you realize that you could get an entire pie outside the stadium for the cost of 2 slices inside the stadium.

Whatever. The tickets were free, so I ordered a pie. But as I did, I asked the clerk, “Why are pies $36 if slices are $6 each? Shouldn’t there be a discount for buying an entire pizza?” He shrugged his shoulders and gave me the same look I usually get from checkout people when I ask similar questions. His eyes said, “Sorry, dude, I just work here.”

The man behind the counter who was cooking the pizzas must have heard me. When my pie came out of the oven, he used his cutter to divide the pie into 8 slices instead of just 6. I guess he thought I’d feel better if the slices only cost $4.50 each. Never mind that each slice was only 3/4 the size of a regular slice.

Since I felt like I was in the middle of a bad math joke, I figured I ought to deliver the punch line.

“What’d you do that for?” I asked. “I’m not very hungry! There’s no way I’ll be able to eat 8 slices!”

The upside? The Nationals won, 4-2, and they earned the top seed in the playoffs. Go, Nats!

### When I Die…

Did you hear about Tommy Lasorda’s wish? The retired manager of the Los Angeles Dodgers is recovering from a heart attack he suffered in June, and he recently told the *L. A. Times*:

I’ve already told my wife that when I do go, I want our home schedule attached to my tombstone. I want people who are in the cemetery visiting their loved ones to say, “Let’s go to Lasorda’s grave and see if the Dodgers are playing home or away.”

That could get expensive, since it would have to be updated annually. I suggest one of these instead:

Old baseball players never die; they just go batty.

Old baseball players never die; they just do one more lap around the bases.

Old baseball players never die; they just get traded to the Blue Jays. (Sorry, Toronto!)

This got me thinking about what I’d want on my tombstone when I leave.

Don’t meet my end with gasps and shrieks;

I left you with a book (and blog) for geeks.

Given the likelihood of my eternal destination, I take comfort in the following advice:

Go to Heaven for the climate,

to Hell for the company.

Contemplating what should appear on my tombstone puts me in good company. Many mathematicians have pondered the same question.

At age 18, **Carl Friedrich Gauss** showed that it was possible to draw a regular 17‑gon with compass and straightedge. Proud of his accomplishment, he later requested that a 17‑gon be inscribed on his tombstone. Although his wish was not granted, a memorial to him in his hometown of Braunschweig, Germany, now bears a small 17‑gon just below his right foot.

Imagine a sphere inscribed in a cylinder whose height is equal to its diameter. **Archimedes** discovered that the ratio of the surface area of the cylinder to the surface area of the sphere is 3:2 and also that the ratio of the volume of the cylinder to the volume of the sphere is 3:2. Despite his many accomplishments in mathematics, this is the one for which he wished to be remembered. He asked that a cylinder with inscribed sphere be displayed on his tombstone, with the ratio 3:2 inside.

**Jacob Bernoulli** was so enamored with the logarithmic spiral that he wanted one inscribed on his tombstone. Unfortunately, the engraver mistakenly carved an Archimedean spiral (shown below).

An apocryphal story is that the following poem appears on the tombstone of **Diophantus**. In fact, this problem appeared in a fifth-century Greek anthology of puzzles.

Here lies Diophantus, the wonder behold.

Through art algebraic, the stone tells how old:

“God gave him his boyhood one-sixth of his life,

One-twelfth more as youth while whiskers grew rife;

And then yet one-seventh ere marriage begun;

In five years there came a bouncing new son.

Alas, the dear child of master and sage

After attaining half the measure of his father’s life, chill fate took him.

After consoling his fate by the science of numbers for four years, he ended his life.”

Good luck solving the puzzle before you meet your end!