Welcome to The Riddler. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability. Two puzzles are presented each week: the Riddler Express for those of you who want something bite-size and the Riddler Classic for those of you in the slow-puzzle movement. Submit a correct answer for either, and you may get a shoutout in next weeks column. Please wait until Monday to publicly share your answers! If you need a hint or have a favorite puzzle collecting dust in your attic, find me on Twitter.
First, an unlucky puzzle that comes a week late:
Depending on the year, there can be one, two or three Friday the 13ths. Last week happened to be the second Friday the 13th of 2020.
What is the greatest number of Friday the 13ths that can occur over the course of four consecutive calendar years?
Extra credit: Whats the greatest number of Friday the 13ths that can occur over a four-year period (i.e., a period that doesnt necessarily begin on January 1)?
Submit your answer
From Patrick Lopatto comes a riddle we can all be thankful for:
To celebrate Thanksgiving, you and 19 of your family members are seated at a circular table (socially distanced, of course). Everyone at the table would like a helping of cranberry sauce, which happens to be in front of you at the moment.
Instead of passing the sauce around in a circle, you pass it randomly to the person seated directly to your left or to your right. They then do the same, passing it randomly either to the person to their left or right. This continues until everyone has, at some point, received the cranberry sauce.
Of the 20 people in the circle, who has the greatest chance of being the last to receive the cranberry sauce?
Submit your answer
Congratulations to Nathan Ainslie of Bloomington, Indiana, winner of last weeks Riddler Express.
Last week, you were a contestant on the TV show Jeopardy! You were competing in the (single) Jeopardy! round and your opponents were simply no match for you. You chose first and never relinquished control, working your way horizontally across the board by first selecting all six $200 clues, then all six $400 clues, and so on, until you finally selected all the $1,000 clues. You responded to each clue correctly before either of your opponents could.
One randomly selected clue was a Daily Double. Rather than award you the prize money associated with that clue, it instead allowed you to double your winnings (up to that point) or wager up to $1,000 should you have less than that. Being the aggressive player you are, you always bet the most you could have. (In reality, the Daily Double was more likely to appear in certain locations on the board than others, but for this problem you assumed it had an equal chance of appearing anywhere on the board.)
How much money did you expect to win during the Jeopardy! Round?
There were a total of 30 clues on the board, and any one of those 30 clues could have been the Daily Double. That meant there were 30 cases to consider: when the Daily Double was the first clue you selected, when it was the second clue, the third clue, and so on. And when it was one of the first six clues selected, the Daily Double was worth $1,000; otherwise, it doubled your money. (Technically, if it was the sixth clue, it was worth $1,000 and doubled your money, since you had won exactly $1,000 by that point.)
Many solvers listed out all 30 cases and averaged the winnings. Alternatively, you could have added up the total amount of prize money across all the cases and then divided by 30. Across these cases, each clue was the Daily Double once, which meant it wasnt the daily double 29 times. Adding up the values of all the clues without worrying about the Daily Double gave you 6(200 + 400 + 600 + 800 + 1000), or $18,000, which meant that 29 boards of clues were worth $522,000.
Now to add up the Daily Doubles. As we already said, for the first six cases the Daily Double was worth $1,000. When the Daily Double was the seventh clue, it was worth $1,200. From there, its value increased by $400 until it was $3,600 as the 13th clue. Then, it increased by $600 until it was $7,200 as the 19th clue. Next, it increased by $800 until it was $12,000 as the 25th clue. Finally, it increased by $1,000 until it was a whopping $17,000 as the 30th and final clue. That was some James Holzhauer wagering right there.
Even without a spreadsheet, you needed some hefty addition to tally up these Daily Doubles. Their total value across the 30 cases turned out to be $192,000 roughly 36 percent of what the non-Daily Double clues had been worth.
Combining the regular clues with the Daily Doubles gave a sum of $714,000 for all 30 cases. That meant the average the amount youd expect to win was $23,800.
For extra credit, instead of working your way horizontally across the board, you selected random clues from anywhere on the board, one at a time. Now how much money did you expect to win during the Jeopardy! Round?
This was a much thornier version of the problem. First, there were five rows in which the Daily Double might have appeared. Then, for each row, you had to consider the order in which you worked your way across the board. So instead of 30 cases, there were in fact 530!/((6!)45!), or about 4.11019 cases to consider.
From here, most solvers used Monte Carlo methods, simulating the game of Jeopardy! thousands or even millions of times to approximate the answer. As a few brave coders (like Josh Silverman, Lowell Vaughn and Alex Vornsand) found the answer was approximately $26,150. (This was quite close to the result youd get $26,146.67 if you assumed every clue had been worth the average amount of $600. The answer was slightly larger than this, since the Daily Double more than doubled your winnings whenever you had less than $1,000.)
It made sense that picking clues randomly would net you more winnings on average, because you now had a greater chance of selecting higher-value clues (like the $800 and $1,000 rows) before hitting the Daily Double.
For this puzzle, well let Alex Trebek have the last word. We miss you.
Congratulations to Alex Zorn of Brooklyn, New York, winner of last weeks Riddler Classic.
Last week, you modeled blown football leads, something the Atlanta Falcons know a thing or two (or three) about. The Georgia Birds and the Michigan Felines were playing a game in which a fair coin was flipped 101 times. In the end, if heads came up at least 51 times, the Birds won; but if tails came up at least 51 times, the Felines won.
What was the probability that the Birds had at least a 99 percent chance of winning at some point during the game meaning their probability of victory was 99 percent or greater given the flips that remained and then proceeded to lose?
This was a challenging riddle, to be sure. For starters, you first had to make sense of what it meant to have at least a 99 percent chance of winning, while avoiding the tempting answer of 1 percent. Before any flips were made, the Birds had a 50 percent chance of winning. But suppose, through incredible luck, that the first 50 tosses all came up heads. From there, the Birds could still technically lose if the final 51 tosses all came up tails an event whose probability was 1/251.
That was just one (albeit very unlikely) way for the Birds to have a 99 percent chance of winning and then blow the game. But there were many other, more likely scenarios, each of which involved an excess of heads flipped toward the beginning. One way to add up the probabilities of these scenarios was to analyze (preferably via code) what was happening at each combination of wins (W) and losses (L) for the Birds.
Heres a graph that shows the pairs (W, L), shown in red, from which the Birds had at least a 99 percent chance of winning. You could determine these directly using combinatorics, or working backwards recursively, noting that the probability of winning from (W, L) was the average of the probabilities from (W+1, L) and (W, L+1).
Next, you wanted to find the probability that the Birds passed through at least one of the red locations in the graph above. That is, for each (W, L), what was the probability that at some point the Birds had a 99 percent chance of winning?
This graph was very similar to the first. But there were some places in green and aqua where the Birds had ventured into the 99 percent region and then back out. These paths through the graph were what made it possible for the Birds to blow their lead.
From here, you had to focus on which of these paths resulted in an overall loss for the Birds, and work backwards. The final graph below shows the Birds chances of reaching a 99 percent chance of victory at some point and then blowing the lead, for each (W, L).
These chances were greatest when the Birds had won 50 games and lost 51, when there was a roughly 2 percent chance that they had blown a 99 percent lead somewhere along the way. Working backwards, their chances of blowing a lead when they had zero wins and zero losses were about 10 times smaller, or 0.21 percent. This was the solution to the riddle: the Birds chances of at some point having a 99 percent chance at victory and then proceeding to lose.
For extra credit, instead of 101 total flips, there were many, many more (i.e., you had to consider the limit as the number of flips went to infinity). Again, the Birds won if heads came up at least half the time. Now what was the probability that the Birds had a win probability of at least 99 percent at some point and then proceeded to lose?
To solve this, some coders continued increasing the number of flips beyond 101 and looked for asymptotic behavior. This weeks winner, Alex Zorn, approached the extra credit analytically by first defining three key probabilities:
With these three variables defined, Alex was able to set up a few equations. For example, w = 0.99h and c = 0.01h, since thats what was meant by a 99 percent win probability. Moreover, the probability that the Birds were the ultimate winners was 0.5, which had to equal w. That meant h = 50/99, which in turn meant that c the probability of blowing that 99 percent lead, was 1/198.
Finally, solver Allen Gu noticed some rather peculiar oscillating behaviors as the number of coin flips increased and the threshold for choking was closer to 50 percent:
Personally, Id be very curious to see this plotted on a logarithmic graph. My hunch is that the oscillations are due to the discrete nature of the problem, when the border between the yellow and blue regimes in that first graph shifts.
In any case, I have been informed that the Birds are doing a little better as of late. Those Los Angeles Lightning Bolts, on the other hand not so much.
Well, arent you lucky? Theres a whole book full of the best puzzles from this column and some never-before-seen head-scratchers. Its called The Riddler, and its in stores now!
Email Zach Wissner-Gross at firstname.lastname@example.org
- The R of R.B.G. - The New York Times - December 1st, 2020
- Five new Steam games you probably missed (November 30, 2020) - PC Gamer - December 1st, 2020
- Starry Garden: Collect Constellations and Adorable Animals - TechAcute - November 29th, 2020
- Every Xbox Series X Launch Game Ranked from Worst To Best (According To Metacritic) - GameRant - November 29th, 2020
- Games with Gold: Gat out of Hell, The Raven Set Their Sights on Xbox Series X Owners - CBR - Comic Book Resources - November 27th, 2020
- 9 Online Puzzles That Will Give Your Brain a Workout - Study Breaks - November 27th, 2020
- SwitchArcade Round-Up: 'Girabox', 'More Dark', and Today's Other New Releases and Sales - Touch Arcade - November 27th, 2020
- Itch.io's Black Friday sale has Among Us and loads of indie goodies - Rock Paper Shotgun - November 27th, 2020
- The Legend of Bum-bo emerges from the sewer after a very difficult year - PCGamesN - November 27th, 2020
- Global Games and Puzzles Market 2020 to Witness Splendid Growth by 2027: Trefl SA, Springbok Puzzles, Eurographics, Inc., Ravensburger AG, Heye Puzzle... - November 27th, 2020
- Ten Puzzle Games You Must Have Played - Gameindustry.com - GameIndustry.com - November 26th, 2020
- Cook up a storm with this feast of games - Nintendo - November 26th, 2020
- Babe in the Woods - The New York Times - November 26th, 2020
- 'Spirit Of The North Enhanced Edition' Review: An Ethereal Escape - Electric Bento - November 26th, 2020
- Bored Of Playing Multiplayer Games? 7 Alternatives For You To Try - Our Culture - Our Culture Mag - November 26th, 2020
- 5 Memorable Viral Video Games (& 5 That Were Overhyped) - FandomWire - November 26th, 2020
- Home for the Holidays - toys and games | Holiday | unionleader.com - The Union Leader - November 26th, 2020
- First-Person Games vs. Third-Person Games: What Are the Differences? - MakeUseOf - November 26th, 2020
- Local Toy Store Thrive Amid Pandemic's Return To Board Games, Chess And Card Collecting - Rockland County Business Journal - November 26th, 2020
- Conferences join forces on virus testing procedures for refs - The Coastland Times - The Coastland Times - November 25th, 2020
- Xbox Live Games with Gold reveals final batch of games for 2020 - SlashGear - November 25th, 2020
- Buffalo Games adds pieces to its puzzle through two deals - Buffalo News - November 25th, 2020
- Satisfying Puzzles And Tunes Make 'Carto' Perfect On The Switch - UPROXX - November 25th, 2020
- Best new mobile games on iOS and Android - November 2020 round-up - Metro.co.uk - November 25th, 2020
- Everyone Has Gone Missing In Point And Click Tiny Room Stories: Town Mystery - TheGamer - November 25th, 2020
- Call of Duty: Black Ops Cold War review: a game as bloated and overwhelming as its name - The Verge - November 25th, 2020
- United Way of Berks County guide has ways you can volunteer or give gifts this holiday season - The Mercury - November 25th, 2020
- The Best PSP Games of All Time - Best gaming pro - November 25th, 2020
- The Latest Games To Play In Indonesia & Win - Inside Pulse - November 25th, 2020
- We reviewed (nearly) every game out now for the PlayStation 5 and Xbox Series X - The A.V. Club - November 25th, 2020
- Tristan Thompson is the missing piece to the Boston Celtics championship puzzle - NBA IN - November 25th, 2020
- More Dark Is A Puzzle-Platformer 'Intended For Adults', And It Launches On Switch This Week - Nintendo Life - November 25th, 2020
- STAFF OPINION: Hop in, loser. We're warping our sense of reality - The Beacon - November 23rd, 2020
- Rez and Tetris Effect Producer Tetsuya Mizuguchi Working on a "New Adventure with Synesthesia" with Enhance Games - Niche Gamer - November 23rd, 2020
- Mourinho: When the puzzle around him is perfect, Kane is fantastic - AS English - November 23rd, 2020
- Quick Trip to a Store and Back - The New York Times - November 23rd, 2020
- 10 Holiday Gifts to Give Any Game of Thrones Fans Missing the Show - Winter is Coming - November 23rd, 2020
- 'Call of Duty Mobile' & 'Among Us' Will Fight For The Title of Best Mobile Game 2020 - Digital Information World - November 23rd, 2020
- The Greatest Google Stadia Video games for 2020 - The Shepherd of the Hills Gazette - November 23rd, 2020
- Impact of Covid 19 On Augmented Reality Gaming Market 2020 Industry Challenges Business Overview And Forecast Research Study 2026 - The Market Feed - November 23rd, 2020
- Everything Else Leaked in the Capcom Hack | Game Rant - GameRant - November 23rd, 2020
- 5 Most Memorable Viral Video Games Of All Time (& 5 That Were Really Stupid, Now That We Think About It) - Screen Rant - November 21st, 2020
- TouchArcade Game of the Week: 'Bridge Constructor: The Walking Dead' - Touch Arcade - November 21st, 2020
- Tetris movie officially in the works, will star Rocketman's Taron Egerton - Destructoid - November 21st, 2020
- 3 Of The Hottest Video Games Dropping In December 2020 - Gstyle magazine - November 21st, 2020
- SwitchArcade Round-Up: 'Hyrule Warriors: Age of Calamity', 'Micetopia', and Today's Other New Releases and Sales - Touch Arcade - November 21st, 2020
- Behind the mask: A chat with the developers of Maskmaker - Blogdottv - November 21st, 2020
- The 30 best iPad games you need to play - The DHT News - November 21st, 2020
- Review: Microsofts new Xbox consoles are powerful and speedy, but are they necessary? - The Dallas Morning News - November 21st, 2020
- How a Solo Indie Developer Built the Best Rated Game on Oculus Quest - Road to VR - November 19th, 2020
- DREAMO, A LOVE LETTER TO THE NARRATIVE PUZZLE GENRE - Gamasutra - November 19th, 2020
- Blair Witch: Experience the Horror of the Movie By...Solving Puzzles? - CBR - Comic Book Resources - November 19th, 2020
- Out Now: 'Bridge Constructor: The Walking Dead', 'The First Tree', 'Moonlighter', 'One Night at Flumpty's', 'Tomb Toad', 'DeLight: The Journey Home',... - November 19th, 2020
- Elite Dangerous and The World Next Door are free on the Epic Games Store this week - Neowin - November 19th, 2020
- The Pathless review: The joy of the unknown | Entertainment - UT Daily Beacon - November 19th, 2020
- Every PS5 launch game ranked worst to best - Looper - November 19th, 2020
- Call of the Sea sets sail in December - Gamesradar - November 19th, 2020
- 'When the Past was Around' confirmed for launch on Switch / PlayStation and Xbox mid-December 2020 - Gamasutra - November 19th, 2020
- 'Myst' (yes, that 'Myst') is coming back as an Oculus VR game in December - Mashable - November 19th, 2020
- It's Hitman's 20th anniversary, so here are 20 facts about the silly stealth sim - Rock Paper Shotgun - November 19th, 2020
- Xbox head Phil Spencer on video games as a way of life, even for the powerful - CNET - November 18th, 2020
- Agate, Telkom team up for Bali-themed puzzle adventure game - The Jakarta Post - Jakarta Post - November 18th, 2020
- Puyo Puyo Tetris 2 Hands-On Preview (PS4) - Still A Very Good Puzzler At Its Core - PlayStation Universe - November 18th, 2020
- Call of the Sea is an adventure inspired by Lovecraft, but it's no horror game - PC Gamer - November 18th, 2020
- The best family board games - PC Gamer - November 18th, 2020
- Here Are The Games Coming Out In December 2020 - IGN India - November 18th, 2020
- 10 Puzzle Games That Are Impossible To Beat Without A Guide - GameRant - November 15th, 2020
- Video Games to Relax - The New York Times - November 15th, 2020
- Announcing Light The Fires, an award-winning hypercasual puzzle game - Pocket Gamer - November 15th, 2020
- Telling the Difference Between PS5 and PS4 Games Is Next-Gen's Toughest Puzzle - Push Square - November 15th, 2020
- Why the PS5 and Xbox Series X need to break the video game mold - CNET - November 15th, 2020
- Video Games for Relaxation - The New York Times - Aviation Analysis Wing - November 15th, 2020
- WiiWare's Greatest Hits. A celebration of the indie games of | by Shawn Laib | SUPERJUMP | Nov, 2020 - Medium - November 15th, 2020
- Bugsnax review you are what you eat | Games - The Guardian - November 15th, 2020
- Games And Puzzles Market Analysis, COVID-19 Impact,Outlook, Opportunities, Size, Share Forecast and Supply Demand 2021-2027|Trusted Business Insights... - November 15th, 2020
- Arkansas women to be tested early in SEC play - Arkansas Online - November 15th, 2020
- The Xbox Series X and the PS5 need to make video games weird and interesting - CNET - November 11th, 2020
- Game IQ the Gold Standard for game analytics and classification - MobileMarketing Magazine - November 11th, 2020
- The Digital Fix Gaming's Generation Awards: Best Puzzle, Simulation, PSVR, and RPG - The Digital Fix - November 11th, 2020
- Tales of the Neon Sea review - "Stunning but literally puzzling" - Pocket Gamer - November 11th, 2020