## Betting on football with future knowledge

This week’s Fiddler is a football-themed puzzle with a twist: you can see the future! Sort of.

You know ahead of time that your football team will win 8 of their 12 remaining games, but you don’t know which ones. You can place bets on every game, placing bets either for or against your team. You can bet any amount up to how much you currently have. You want to implement a betting strategy that guarantees you’ll have as much money as possible after the 12 games are complete. If you did so, then after the 12 games how much money would you be guaranteed to have if you started with $100? My solution: [Show Solution] A more elegant alternative solution, due in large part to a clever observation by Vince Vatter. [Show Solution] ## Loteria This week’s Riddler Classic is about Lotería, also known as Mexican bingo! A thousand people are playing Lotería, also known as Mexican bingo. The game consists of a deck of 54 cards, each with a unique picture. Each player has a board with 16 of the 54 pictures, arranged in a 4-by-4 grid. The boards are randomly generated, such that each board has 16 distinct pictures that are equally likely to be any of the 54. During the game, one card from the deck is drawn at a time, and anyone whose board includes that card’s picture marks it on their board. A player wins by marking four pictures that form one of four patterns, as exemplified below: any entire row, any entire column, the four corners of the grid and any 2-by-2 square. After the fourth card has been drawn, there are no winners. What is the probability that there will be exactly one winner when the fifth card is drawn? My solution: [Show Solution] ## Desert escape This week’s Riddler classic is about geometry and probability, and desert escape! Here is the (paraphrased) problem: There are$n$travelers who are trapped on a thin and narrow oasis. They each independently pick a random location in the oasis from which to start and a random direction in which to travel. What is the probability that none of their paths will intersect, in terms of$n$? My solution: [Show Solution] ## Cutting a ruler into pieces This week’s Riddler Classic is a paradoxical question about cutting a ruler into smaller pieces. Recently, there was an issue with the production of foot-long rulers. It seems that each ruler was accidentally sliced at three random points along the ruler, resulting in four pieces. Looking on the bright side, that means there are now four times as many rulers — they just happen to have different lengths. On average, how long are the pieces that contain the 6-inch mark? With four cuts, each piece will be on average 3 inches long, but that can’t be the answer, can it? Here is my solution: [Show Solution] ## Flip to freedom This week’s Riddler Classic is a problem about coin flipping. The text of the original problem is quite long, so I will paraphrase it here: There are$n$prisoners, each with access to a random number generator (generates uniform random numbers in$[0,1]$) and a fair coin. Each prisoner is given the opportunity to flip their coin once if they so choose. If all of the flipped coins come up Heads, all prisoners are released. But if any of the flipped coins come up Tails, or if no coins are flipped at all, the prisoners are not released. If the prisoners cannot communicate in any way and do not know who is flipping their coin or not, how can they maximize their chances of being released? Here is my solution: [Show Solution] ## Mismatched socks This week’s Riddler Classic is a problem familiar to many… I have$n$pairs of socks in a drawer. Each pair is distinct from another and consists of two matching socks. Alas, I’m negligent when it comes to folding my laundry, and so the socks are not folded into pairs. This morning, fumbling around in the dark, I pull the socks out of the drawer, randomly and one at a time, until I have a matching pair of socks among the ones I’ve removed from the drawer. On average, how many socks will I pull out of the drawer in order to get my first matching pair? Here is my solution: [Show Solution] ## Gift card puzzle Here is a puzzle from the Riddler about gift cards: You’ve won two gift cards, each loaded with 50 free drinks from your favorite coffee shop. The cards look identical, and because you’re not one for record-keeping, you randomly pick one of the cards to pay with each time you get a drink. One day, the clerk tells you that he can’t accept the card you presented to him because it doesn’t have any drink credits left on it. What is the probability that the other card still has free drinks on it? How many free drinks can you expect are still available? Here is my solution: [Show Solution] ## Elf music This holiday-themed Riddler problem is about probability: In Santa’s workshop, elves make toys during a shift each day. On the overhead radio, Christmas music plays, with a program randomly selecting songs from a large playlist. During any given shift, the elves hear 100 songs. A cranky elf named Cranky has taken to throwing snowballs at everyone if he hears the same song twice. This has happened during about half of the shifts. One day, a mathematically inclined elf named Mathy tires of Cranky’s sodden outbursts. So Mathy decides to use what he knows to figure out how large Santa’s playlist actually is. Help Mathy out: How large is Santa’s playlist? Here is my solution: [Show Solution] ## Sniff out the spies This interesting problem appeared on the Riddler blog. Here it goes: There are N agents and K of them are spies. Your job is to identify all the spies. You can send a given number of agents to a “retreat” on a remote island. If all K spies are present at the retreat, they will meet to strategize. If even one spy is missing, this spy meeting will not take place. The only information you get from a retreat is whether or not the spy meeting happened. You can send as many agents as you like to the retreat, and the retreat can happen as many times as needed. You know the values of N and K. What’s the minimum number of retreats needed to guarantee you can identify all K spies? If each retreat costs \$1,000 per person, what is the total cost to identify all K spies?

To begin with, let’s assume you know that N = 1,024 and K = 17.

Here is my solution for $K=1$:
[Show Solution]

And here is a partial solution for $K \gt 1$:
[Show Solution]

## Paths to work

This Riddler puzzle is a classic problem: how many lattice paths connect two points on a grid? Here is a paraphrased version of the problem.

The streets of Riddler City are laid out in a perfect grid. You walk to work each morning and back home each evening. Restless and inquisitive mathematician that you are, you prefer to walk a different path along the streets each time. How many different paths are there? (Assume you don’t take paths that are longer than required). Your home is M blocks west and N blocks south of the office.

Here is my solution:
[Show Solution]