## Posts tagged ‘exponential’

### 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.*

### Exponentially Smarter, Literally

To show my sons what Siri can do, I asked her (it?) the following question:

What is 6 + 4?

Siri told me, “The answer is 10.” But she also provided a bunch of other information pulled from Wolfram Alpha, including the following data:

This data appears to be taken from dissertation research by B. A. Fierman which was furthered by psychologist Mark H. Ashcraft. What it shows is that we get exponentially smarter — or at least faster at calculating — as we get older.

According to Excel, this data can be modeled exponentially by *y* = 8.36 · *e*^{–0.129x}, though this model has obvious limitations. For example, it implies that a one-year-old would be able compute this sum in 7.35 seconds, yet I know no one-year-old who understands addition. Further, it claims that it would take me 0.03 seconds to compute the sum, but I would argue first that I don’t compute the sum, I merely recall it; and second, my reaction time when asked for the sum would be greater than 0.03 seconds.

Playing around with the generic function *y* = *ab*^{x} + *c* using the world’s best graphing calculator from Desmos, I found a model that may approximate the data a little better:

*y* = 57 · 0.65^{x} + 0.9

With this model, it would take a one-year-old 37.95 seconds to compute sum. That’s still not reasonable for any one-year-old that I know, but at least the model says it would take me 0.9 seconds to recall the fact, a far more reasonable estimate than the 0.03 seconds given by the Excel model above.

Interestingly, How To Geek claims that Siri uses Wolfram Alpha for 25% of its searches. Yet if you ask Siri, “What is the meaning of life?” it will respond,

I can’t answer that right now, but give me some very long time to write a play in which nothing happens.

or

Try and be nice to people. Avoid eating fat. Read a good book every now and then, get some walking in, and try to live together in peace and harmony with people of all creeds and nations.

On the other hand, if you ask Wolfram Alpha, “What is the meaning of life?” it will respond,

42.

Proper.

All this talk of exponentials reminds me of a joke.

Q: How do you know that your dentist studied algebra?

A: She tells you that candy will lead to exponential decay.

Perhaps the most famous joke about exponentials is not one of which I’m terribly fond. I share it here only to honor my mission of providing math jokes to the world, not because I think any of you will enjoy it.

Several functions are sitting in a bar, bragging about how fast they go to zero at infinity. Suddenly, one hollers, “Look out! Derivation is coming!” All of the functions immediately cower under the table, but the exponential function sits calmly on the chair.

The derivation comes in, sees the exponential function, and says, “Don’t you fear me?”

“No, I’m

e,” says the exponential confidently.^{x}“That’s all well and good,” replies the derivation, “but who says I differentiate with respect to

x?”

### Why Did Vi Hart Go to Khan Academy?

I love Vi Hart. And with over 300,000 subscribers and 25 million views on her YouTube channel, I’m clearly not alone.

But perhaps you don’t know who she is. Maybe you’ve been living under a rock. Maybe you’re still using dial-up. Or maybe you’ve just been posing as a mathy folk, only visiting this blog because you think the author is hot. (Of course, you’d be correct in your assessment, but you shouldn’t let hot authors guide your tour through the blogosphere.)

If you don’t know who Vi Hart is, you can check out her Binary Trees video below (from her now famous Doodling in Math Class series).

Pretty awesome, huh?

In the video, she makes the following statement:

…if the [math] curriculum wasn’t so appalling and the teaching methods weren’t so atrocious, you wouldn’t have to entertain yourself with these stories and games.

She also implies that many math classes are

…fuzzy, unfocused, and altogether not very good.

Some educators don’t like these videos. Some don’t like that a brash, young woman is criticizing what they do and how they do it. Some find her statements offensive.

Not me.

I think she’s spot on.

Too many math classrooms still look like the math classrooms of yesteryear, devoid of excitement and technology and filled with endless hours of meaningless practice.

But here’s where I have trouble. On the About Vi page of her site, she says:

I am now a full-time mathemusician at Khan Academy! It’s pretty exciting.

If she is truly opposed to appalling curriculum, why would she work for a company that creates the video version of a 1950’s textbook?

Maybe I’m being too harsh. But I don’t think so. Though she now creates recreational math videos for Khan Academy that are awesome, the vast majority of videos on the site are nothing more than math lectures of topics that probably should have been removed from the curriculum years ago. When I asked a colleague his thoughts, he had this to say:

Vi’s videos show such polish and cleverness, while Khan’s were so obviously made by someone who just took an exercise from a textbook and sat down at a computer and improvised. About the only thing [Khan Academy] has going for it is that it’s free. I suppose it can have some good use in getting kids an opportunity to learn and practice skills they need, but having them practice skills for no particularly good reason… it’s just reinforcing everything that’s wrong with math education.

In her Binary Trees video, Vi Hart makes fun of the boring presentation of exponential functions that typically occurs in math classes. Yet the Khan Academy video Exponential Growth Functions uses the same examples and “atrocious teaching methods” that would be found in many of the math classes that are “fuzzy, unfocused, and altogether not very good.”

So, what’s up, Vi? How can you rail against bad teaching but then go to work for a place that delivers bad teaching in spades? Your work is amazing, and you had such an opportunity. I hope your intent is to make change from within rather than assimilate.

**How do you feel about Vi Hart’s move to Khan Academy? And what do you think about Khan Academy in general?**