Wes Carroll's Puzzler

A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers choose their spaces at random from among the available spaces. Santa Claus then arrives in his oversized and very full sleigh, which requires two adjacent spaces. What is the probability that there’s a place for him? // Spiciness: **** out of ****

What’s the largest 2-digit prime factor of “200 choose 100”? // Spiciness: *** out of ****

Last week, two of my friends and I went to a restaurant and had a lovely meal. We decided to evenly split the check, so we asked the waiter to just combine the totals. However, when the waiter came with the check, he revealed that there had been a mistake and instead of recording the complete total, the computer only returned a list of the totals of each pair of people. // Spiciness: * out of ****

A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. For example, after traveling one mile the odometer changed from 000039 to 000050. If the odometer now reads 001729, how many miles has the car actually traveled? // Spiciness: *** out of ****

How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits? // Spiciness: ** out of ****

A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token into a discard pile. The game ends when some player runs out of tokens. Players A, B, and C start with 15, 14, and 13 tokens, respectively. // How many rounds will there be in the game? // Spiciness: ** out of ****

Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. // What is the length of the track in meters? // (Also, a challenge for all listeners.) // Spiciness: *** out of ****

My perpetually tricky friend told me that while she was walking through town she saw four particularly vibrant houses. There was an auburn one, a brick one, a cherry one, and one the shade of dogwood rose. She wanted me to figure out the order of the houses. She said that the the auburn came before the the brick one while the cherry one came before the dogwood rose, but the cherry and the dogwood rose were not adjacent. I told her that she hadn’t given me enough information, so she just laughed and told me that she could tell me the color of the first one or the color of the last one, but it wouldn’t help if she did either one. // What color was the second house? // Spiciness: *** out of **** // Note: Not having paper makes this one especially difficult. If you give yourself paper, I think you can rate this puzzle as two chili peppers of spiciness instead of three.

A friend of mine told me that she can walk a mile south, a mile east, a mile north and end up back home. I first thought she lived at the north pole, but she laughed and told me that, since there was no land there, she would be unable to make the walk. She asked me to try again, so I thought for a few minutes before finally saying that I knew how to get within a few minutes of her house, but couldn’t give her an exact location. // Where does she live? // Spiciness: * out of ****

Working alone, I put two coats of paint on a wall, one before lunch and one after. Yesterday, I began at the usual time. Two hours before lunch I was joined by my good friend Aidan, who paints at the rate of 600 sqft per workday, and who left just as the first coat was finished. I promptly began the second coat, and had lunch at the usual time. One hour before quitting time, I had painted a second coat everywhere except where Aidan had painted that morning. If we each have the same workday, and if each of us works at a constant rate (albeit not necessarily the same rate as the other), what was the area of the wall? // Spiciness: *** out of ****

Earlier this week I was rob...er...exploring tombs and I accidently triggered a trap that locked me in a room. With me are a pair of plates, a few thousand tiny statues of gnats and a puzzle that should lead to my escape. I need to place specific numbers of gnats onto each of the two plates. The number of gnats on the left plate needs to be a 3-digit palindrome, while the number on the right needs to be a 4-digit palindrome, with a difference between them of 22. I remember that a palindromic number is one where if you read it forwards and backwards, it looks the same. For example, 43534 and 5885 are both palindromes. // Please send in solutions; I want to get out of here. // Spiciness: ** out of ****

Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly? // // (Spiciness: * out of ****)

For years you were a lonely prisoner here. But earlier today, you were brought to a courtyard to join the others, where you are all addressed by the Warden. There have been budget cuts, he explains, and the one hundred of you need to leave this facility. Whether you will be sent to another high-security facility, or set free, depends on whether you pass the following test of cleverness and teamwork. // There is a secret room not far from here, and like your individual cells, it is soundproof, lightproof, and in all other ways impervious to communication. The only object in this room is a single light switch, not connected to anything. It is currently in the off position. // In an hour, you will each be sent back to your cells. One of you will be selected at random to visit the room. While there, that prisoner may choose to flip the switch or not. No other actions will be permitted. Then another prisoner will be chosen at random. And again and again and again, over and over, always at random. // At any point, any of you may declare that all of you have visited the room. If the declaration is true, you will all go free. If not, then you will never again see the light of day. // You have one hour to formulate your strategy. // How will you arrange for everyone to go free? // Note: you have no idea how often prisoners will be sent to the room. Any solution whereby you try to “run out the clock” will be considered incorrect. A correct solution is one for which a declaration proves that all prisoners have visited the room at least once each. // Oh, one last thing: if it’s still not enough of a challenge for you, try solving the variant in which the switch starts in a random position. // (Spiciness: **** out of ****)

I have four lengths of rope. I hold them so that you can see all eight ends, but you can’t tell which end connects to which other end. You pick a pair of ends, and I tie them together. We repeat -- you pick, I tie -- until we run out of ends. // What’s the expected value of the number of loops you’ll have at the end? Or, in plain English, if we play this game a zillion times, what’s the average number of loops I’ll get per game? Note: the correct answer is not a whole number.

In the eight-term sequence "a, b, c, d, e, f, g, h", c represents 5, and the sum of any three consecutive terms is 30. What’s a+h? (Spiciness: ** out of ****)

We’re going to play a simple coin-flip game. We take turns flipping a fair coin. The first one to get “heads” wins. You go first. // What’s your chance of winning? // Spiciness: *** out of ****

A friend of mine has pictures of his three daughters on his mantle. He took the pictures when each of the girls was a particularly adorable age — the same age for all three of them, as it happens. Unfortunately, this made it impossible for me to determine which was the oldest. So I had to ask him. Since my friend is a puzzle junkie, however, he declined to answer directly, telling me only that the product of their current ages was 72. “However,” he added, “since that isn’t enough information to determine their ages, I’ll also tell you that the sum of their ages happens also to be the number of our street address.” (Of course, I understood that each daughter’s age was to be considered a whole number for purposes of this puzzle.) I darted outside to check the number on his mailbox. I was daunted to discover that I still didn’t have enough information to determine their ages, and I returned to tell him so. “That is an astute observation,” he said, smiling. “So you’ll be glad to know that my oldest daughter prefers strawberry ice cream.” Finally! I knew their ages. Do you? // Spiciness: *** out of ****

Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages? // Spiciness: * out of **** // (Today’s puzzler comes from the 2009 AMC 10 exam. Learn more at dtmath.com/amc.)

You have just tested positive for a condition known to affect 1% of the population. However, your doctor assures you that the test for this condition is only 90% accurate. You’re not sure whether that’s supposed to make you feel better or not. So, you tell me: assuming no other information, what’s the chance that you have the condition? // Spiciness: *** out of ****

Ted has three numbered statement for us to consider, and he wants to know whether the third one is true. Here they are: // 1. There are three numbered statements. 2. Two of the three statements are false. 3. You know the answer to the question. // So: is Statement 3 true? // Spiciness: ** out of ****