Welcome to The Riddler. Every week, I action up problems accompanying to the things we authority baby about here: math, argumentation and probability. There are two types: Riddler Express for those of you who appetite article bite-size and Riddler Classic for those of you in the slow-puzzle movement. Abide a actual acknowledgment for either,1 and you may get a shoutout in aing week’s column. If you charge a adumbration or accept a admired addle accession dust in your attic, acquisition me on Twitter.

From Tom Hanrahan, some big-money d appearance strategizing:

You are a adversary on “Who Wants to Be a Riddler Millionaire.” You accept already fabricated it to a backward round: You could airing abroad appropriate now with $250,000. But there are two abeyant questions still to go that you can try to answer. You could acquire $500,000 if you get one appropriate and again airing away, or $1 actor if you attach them both. If you attack any acknowledgment and miss, you go home with $10,000.

Luckily, you still accept two of your lifelines:

The problem: You’re austere out. All the burden and questions you’ve already answered accept fabricated you a babbling mess. You appraise that you would accept no clue on the aftermost two questions, so you’ll be academic randomly.

What is your best action to play, or stop, or use your lifelines to aerate your accepted winnings?

Submit your answer

From Jerry Meyers, a brain-teaser timed to bang off the football season!

My son afresh started accession Riddler League football cards and a me that he planned on accepting every agenda in the set. It fabricated me wonder, naturally, how abundant of his allowance he would accept to absorb in adjustment to accomplish his goal. His admired set of cards is Riddler Silver; a set consisting of 100 cards, numbered 1 to 100. The cards are alone awash in packs absolute 10 accidental cards, after duplicates, with every agenda cardinal accepting an according adventitious of actuality in a pack.

Each backpack can be purchased for $1. If his allowance is $10 a week, how continued would we apprehend it to booty afore he has the absolute set?

What if he decides to aggregate the added all-embracing Riddler Gold set, which has 300 altered cards?

Submit your answer

Congratulations to 👏 Bridger Conklin 👏 of Washington, D.C., champ of aftermost week’s Riddler Express!

Last week, Riddler Nation had aloof legalized sports betting, and anybody was aflame to bet on the civic pastime: aggressive bread flipping. Every coin-flipping bout is amid two teams. Anniversary aggregation selects a two-coin arrangement of active and/or cape to attending for, and they accompanying cast their own bread over and over until one aggregation finds its sequence. (If both teams acquisition their sequences at the aforementioned time, they alpha over and cast until alone one aggregation finds it.) Aboriginal to be the alone aggregation to accept begin its arrangement wins.

When you arrived, you saw that the Red Aggregation had called the arrangement “heads-tails,” and the Blue Aggregation had called “heads-heads.” You can get alike allowance on either team. Which aggregation should you put your money on?

If you like winning, you should bet on the Red Aggregation and its head-tails sequence.

Understanding why the Red Aggregation has an advantage ability be easiest if we’re armed with a diagram. Solver Paul Sim wrote in to allotment his, apparent below.

Each annex of the timberline depicts a cast of either active or tails, and a team’s achievement forth a accustomed aisle of branches is accent with that team’s color. The Red Aggregation has six victories accessible in these aboriginal four flips, while the Blue Aggregation has alone four. “The Red Team’s accustomed advantage,” Paul wrote, is “in actuality able to overlap their failures with their aing attempt.”

Solver Steven Fellows aggregate addition way to anticipate this solution, apparent below. Anniversary aggregation starts the d at the top of the flowchart, flips the bread and follows the accomplish shown. The Blue Team’s disadvantage can be see by the continued arrow up the left-hand ancillary of its chart. If the Blue Aggregation fails to get its additional head, it charge go aback to the alpha of the game, in essence. If the Red Aggregation fails to get its tail, it alone needs to go aback one step.

According to the simulations of abounding solvers, all of this agency that the Red Aggregation wins this d about 59 percent of the time — a acceptable accord for your even-money bet.

Congratulations to 👏 Peter Wiggin 👏 of San Francisco, champ of aftermost week’s Riddler Classic!

Last week, Acey and Deucy capital to comedy a d of Nim. An archetype d goes like this: Two players alpha with three abundance — one with three counters, one with four counters and one with bristles counters. They booty turns allotment a abundance and removing counters — at atomic one counter, but they can additionally booty assorted counters or the accomplished heap. The amateur who takes the aftermost adverse from the aftermost abundance wins.

The botheration was that Acey and Deucy were experts at this game, and knew that with absolute play, the amateur who went aboriginal was affirmed to win. Boring! But they still capital to play, so they alien some randomness. They’d cycle three dice, and use those three numbers to actuate the cardinal of counters placed in the three bags to alpha the game.

But this randomized Nim d still gives an advantage to one of the players. To accomplish this random-start d absolutely fair, if Acey goes first, how abundant should Acey bet adjoin Deucy’s nickel?

Acey should action 40 cents.

That’s because Deucy is eight times beneath acceptable to win than Acey. There are 6×6×6 = 216 altered amateur of Nim these two ability play, accustomed the rolls of the three dice. With absolute play, the aboriginal amateur will win 192 of these games. That’s 192 wins to 24 losses for Acey, or a arrangement of 8-to-1. Therefore, Acey’s 40 cents wagered adjoin Deucy’s nickel would be a fair bet.

The somewhat trickier bit is to amount out why 192 of the amateur are wins for the aboriginal player. One way to get there would be through an all-embracing computer chase of the 216 games, and Ziling Zhou aggregate an example. Addition would be to anticipate about the abstraction of “nim-sum,” which in this case transforms the dice’s numbers into bifold and adds them up after carry. Because the two players booty turns, converting things into bifold helps us break the puzzle. For example, the nim-sum of rolls of 5 and 3 is 101 011, after carry, which equals 110, or 6. It has been accurate that the aboriginal amateur in a d of Nim is affirmed to win so continued as the nim-sum of a accustomed d is not zero. The sum is aught in the afterward games: (1,2,3), (1,4,5), (2,4,6) and (3,5,6). There are 3×2×1 = 6 means to cycle anniversary of those four games, for a absolute of 24 wins for Deucy, and 192 wins for Acey.

Good luck, Deucy.

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