Weekend Challenge for those of you not taking the October test

If you're taking the test tomorrow morning, you should ignore this and REST YOUR BRAIN. If you're not taking it tomorrow, though, then this is just another regular weekend for you, and you should work your brain HARD.

The prize, as it has been every week for weeks and weeks, is access to the coveted PWN the SAT Math Guide Beta. If you wanna win, you have to comment in such a way that I can contact you. Preferably using Facebook, Twitter, or Gmail.

A warning: this one is pretty tough.
In a hand in his weekly poker game, Mike won money from both John and Sean. Mike's percent gain on the hand was equal in magnitude to John's percent loss. John bet twice as much as did Sean. Mike and John together held \$200 in chips before the hand began. If Sean bet \$20 in chips on the hand, what was Mike's chip total after the hand was over?
Good luck!

UPDATE: Nobody got this yet, so I'm gonna leave it up for now unanswered. If you solve it, book access is yours.

SECOND UPDATE: OK, Eowyn got it. Nice. Solution below the cut.

Good luck tomorrow, everyone.

Note: all of this was going to be a comment on this post at The Fat Envelope Blog but after I typed it all up I got server error after server error trying to submit it. Because I was so sufficiently fired up, I decided to post it here instead. My apologies if you didn't come here for politics. We'll be back to regularly scheduled math shortly.

The backstory, if you don't want to click over, is that a columnist I'd not heard of before today, Linda Chavez, wrote a truly scummy piece in which she used the unending debate about the SAT as a bludgeon to further her political agenda.

I'm not a proponent of abolishing the test (far from it!) but Ms. Chavez's piece is...awful. Just awful. And biased as the day is long. She approaches the entire debate from a political standpoint, not an educational one. If the elephants on the page weren't enough to clue you in, she reveals her agenda quite clearly here (my emphasis):

The movement away from requiring the SAT has picked up steam in the last few years, ostensibly driven by the desire to increase racial and ethnic diversity at colleges. If it's true, this would be troubling enough, since the desire to achieve a predetermined ethnic or racial mix should play no role in determining who gets into college. But, in any event, the real motive behind the SAT-optional movement is more complicated and self-serving.

This is the final Weekend Challenge of my 20s. Good grief.

I turn 30 years old tomorrow. 30 f'ing years old. CLING TO EVERY SECOND OF YOUR YOUTH. SQUEEZE EVERY OUNCE OF JOY OUT OF IT LIKE JUICE FROM YOUR FAVORITE FRUIT. SOMEDAY YOU WILL BE OLD LIKE ME.

The prize for the challenge: Free access to the PWN the SAT Math Guide Beta.
Week 1: Served customers numbered a through b
Week 2: Served customers numbered c through d
Week 3: Served customers numbered e through f
Week 4: Served customers numbered g through h
Week 5: Served customers numbered i through j

The proprietor of a deli is trying to project how many customers he usually has on Mondays, so that he can order enough roast beef, but not order too much because nobody likes rotten roast beef. At the deli, customers take numbers before they are served, so he plans to collect data over the next 5 Mondays in the format of the list above, and is looking for an expression for average number of customers that he can plug numbers into as he collects them. What is the expression for the average number of customers (in terms of a through j) that are served in the deli on those 5 Mondays?
Good luck, children. I'll post the solution Monday.

UPDATE: Congrats, Serrilius. I hope you enjoy the book.

Solution below the cut.

This Weekend Challenge is child's play.

Instead of a weekend challenge question last week, I posted a question writing contest that I thought would be a big hit, but which has thus far generated much enthusiastic sound and fury, and only one actual question. Ah well. I maintain my position that if you are able to synthesize good SAT questions, you'll be in very good shape to do well on test day. But if you don't want to try it, I won't shed any additional* tears.

Anyway, a return to form: This weekend challenge is designed to frustrate, and then eventually satisfy. The prize for the first correct response will be Beta Access to my Math Guide.

A precocious toddler is making a pattern with his alphabet blocks, each of which displays a letter (A-Z) in one of 5 colors: red, blue, green, orange, yellow. He has a LOT of blocks to work with, so he sets about putting them in alphabetical order, and also color order. The pattern goes as follows: Red A, Blue B, Green C, Orange D, Yellow E, Red F, etc. When he gets to the end of the alphabet, he starts at the beginning again. Assuming he doesn't screw up the pattern, the nth block will be his first Green N. What is n?

UPDATE: Nice work, Robin. I hope you enjoy the Beta!

Solution below the cut.

Counting: Oh, the possibilities

 Source.
One technique-able counting problem type that you might come across on the SAT is what I'll call a “possibilities” problem*. It might involve cards (but not playing cards – the SAT doesn't like those), or pictures being lined up on a wall in different orders. Your job will be to determine the number of possible outcomes given a particular scenario. Like so:

1. Mike is arranging seven of his various awards and commendations on a shelf in his office. If he insists that his hard-fought Class of 1999 Math Award be placed in the center, in how many different orders could he arrange the seven items?

(A) 60
(B) 72
(C) 120
(D) 720
(E) 1440

The counting questions that are really just listing questions

An unfortunate truth about the SAT is that while many questions can be answered with snappy tricks (many of which can be found on these pages), not all of them can. Most "counting" questions (and probability questions, for that matter) fall into this category.

Yes, I'm serious. Most.

Basically, if you don't see within 15 seconds or so that you're dealing with a matching problem, or a possibilities problem where you can just set up hangman blanks and count, then you should bail on looking for shortcuts and just start listing things. That's right. List them.

Stop complaining. Listing them isn't "the long way"! Sitting there with your leg shaking and your hands on your head trying to see a shortcut where there is no shortcut while time ticks away is "the long way"!
The long way is the short way
(Grid-in)
1. How many positive integers less than 100 are not divisible by 7?

Instead of a weekend challenge, a contest.

I just didn't get it together to write a weekend challenge question this week, but I didn't want to pass on the opportunity to give away access to the Beta. So instead of writing a question, I've decided to run a bit of a contest (inspired by work I've been doing with Debbie Stier at the Perfect Score Project).

I want YOU to write an SAT question. It should be pretty hard, but not so hard that there's no way it could appear on the SAT. Post your best stuff in the comments, and I'll choose my favorite question (or questions) and award those authors access to the Math Guide. I'll let this contest run until Wednesday, 9/14 Saturday, 9/17 whenever. If you write a good question and post it in the comments here, I'll see it. If I think it's good, I'll email you access to the Beta. Easy, right?

Some guidelines and stuff:

• The question must be your own original work (duh).
• Formatting matters. It needs to feel real for me to like it. And it can't have typos.
• The idea here is to simulate the SAT, not to write the hardest question you can write. Don't use any concept that isn't tested on the SAT.
• Make sure the incorrect answer choices you choose aren't random -- make sure they anticipate possible missteps a hapless student could make.
• If your answer choices have numbers, make sure those numbers go in order from least to greatest to greatest to least...just like on the SAT.
• If your question requires a diagram, it's up to you to figure out how that's going to work. Suggestion: Use a free image uploading service and post a link in the comments. To make it easy on me and whoever else is trying to solve the question, maybe include the text of the question in the image?
• SAT questions often test multiple concepts at once (ex: a circle question that also involves special right triangles). Just sayin'.
• There could be 1 winner, or 10. I'm not setting limits. If I think your question is awesome, you're a winner.
• If your question is REALLY good, I might even ask you if I can use it in my book.
Alright, that's it (I think). Start posting your questions in the comments. I can't wait to see what y'all come up with.

The SAT is not like other tests.

If you've been wondering why things have been a bit quieter around here for the past few weeks, there are 3 reasons:
1. I've been scrambling to finish this book so that I can ship it off to print.
2. I've been trying to keep up with all the great questions I've been getting at qa.pwnthesat.com.
3. I started grad school this month and I've been trying to adjust after close to 10 years away from academia.
The third reason is pertinent right now, because for a class on finance (which I think will be quite good) I was assigned some preparatory work over the summer, which included some online quizzes. I want to quote a question from one of them here, and then explain why it's a bad question. And then hopefully parlay my personal anecdote into insightful test prep advice for you, using my extreme perspicacity. Wouldn't that be splendid?
If you have a present value and an interest rate, you can calculate a future value.

True / False
I don't expect y'all to be experts in finance (nor should you expect me to be until a few months from now) but the formula being referred to here is a simple one: FV = PV(1 + i)n, where FV = future value, PV = present value, i = interest rate, and n = number of periods the investment will earn interest.

So if you have PV and i, you should be able to calculate FV for any n. In other words, if you know the  present value of an investment is \$1000, and you know the interest rate is 5%, you should be able to tell how much that investment will be worth in 1 year, 2 years, 10 years, 100 years, etc. You should be able to find "a future value." I picked "True."

The answer they were looking for is "False" because you can't solve the equation without also knowing n.

So yeah, I got a question wrong. Alert the media. But also take note of the imprecision in the way the question was written. The question asks whether one could find "a future value" given a present value and an interest rate. I maintain that one can: the value 2 years out is a future value, as is the value 3 years out and the value 8 years out. I can find all of them. So I can find "a future value." Many, in fact.

If the question writer really wanted an answer of "False," she should have asked whether one could find "the future value." You cannot find "the future value" without knowing how many periods into the future the interest will compound.

This drove me absolutely nuts for days, but the bottom line is that not everybody writes tests the same way, and I had to adjust to the way these dumb online tests are written just like you have to adjust to the way your US History or Physics teacher writes tests, and just like you must adjust to the way the SAT is written.

In each question, in all three subjects, every single word matters, both in the questions, and in the answer choices. Every single word has been wrangled over and vetted by multiple test writing professionals, and is in there for a reason. The wording of questions on the SAT is extremely deliberate and precise. You will never, never have to wonder whether a test writer wrote "a" when he meant to write "the." The sentence means exactly what it means. I take comfort in that, and I wish every other test I'll ever have to take would follow the same guidelines.

Just like that simple finance question drove me nuts, the absolute precision of the SAT might drive you nuts for a while. But if you want to get to a place where you can dominate this test, you'll have to adjust. Each and every word matters

Weekend Challenge - Labor Day edition

This Labor Day weekend I'm going to the wedding of one of my oldest friends from my hometown. Combine the fact that the number of my friends who aren't married is dwindling dangerously close to zero and the fact that I wasn't able to attend this year's fantasy football draft so I had to autodraft and the fact that this weekend marks the end of the summer and OMG WHAT IS HAPPENING TO MY LIFE.

This weekend's prize: you guessed it -- free access to the Math Guide Beta. Make sure you don't comment anonymously if you want to win.

In a certain youth soccer league, each team plays each other team exactly one time per season. If, during a certain season, there are 231 games played, how many teams were in the league that season?

UPDATE: Nice work, "AP FRQ Solutions" (whoever you are). I've shared the Beta document with you. Use it in good health.

Solution below the cut.