8/ It’s incredible how the players’ successes become highly correlated by tying their choices to the shuffle’s properties — all with a simple strategy and no communication about the deck's state required.

Details:
en.wikipedia.org/wiki/100_pri...

7/ Why This Works:
- If all cycles in the shuffled deck are <=26, all players will find their cards.
- The math (linked later) shows that only about 70% of random shuffles contain a circle that is longer than 26.
- So, the group has approximately a 30% chance to win using this strategy!

6/ Understanding Cycles:

The shuffled deck creates "cycles" where each card points to the next position. Players follow these cycles, starting from their assigned position.

A player will always find their card if it’s part of a cycle of length 26 or less.

5/ A simple example:

Player with card 2 (Ah) starts:
Reveal position 2 → Card 51 (2d)
Reveal position 51 → Card 1 (As)
Reveal position 1 → Card 2 (Ah) - found it!

This forms a cycle: 2 ➔ 51 ➔ 1 ➔ 2

4/ The Strategy:
1️⃣ Player with assigned card X starts by checking the card at position X.
2️⃣ If card Y is revealed and it's not the player's card, continue to check position Y next.
3️⃣ Repeat: Go to the position matching the number of the last revealed card until they find their card or hit 26 checks.

3/ First, each card is assigned a number from 1 to 52. For example:
As = Card 1, Ah = 2,.., 2d = 51, 2c = 52

Similarly, positions in the deck are numbered from 1 (top) to 52 (bottom).

2/ First, the key insight: Players can align their choices to make their successes correlate.

Instead of acting randomly or independently, they use a shared system based on the deck’s structure. This strategy massively improves their odds. Ready to see how? Let’s break it down!

Finally, using ChipEV does NOT follow from ICM being imperfect.

ChipEV is thoroughly flawed for tournament calculations.
Pointing out inaccuracies in ICM doesn't automatically make it worse than ChipEV.

Even if ICM isn't perfect, it's still the best tool available for now.

Third, the equities ICM produces aren't perfect. This is a valid point and the actual source of inaccuracies.

However, the "deal next hand" angle is a red herring. It adds no substance to the discussion and doesn't undermine using ICM in early or mid stages oftournaments.

It's the same here—we use the ICM estimate to say, "If you end the hand with that stack, it'll be worth approximately this much."

It doesn't assume you'll finish the tournament for exactly that amount—that's silly. We focus on EV, knowing actual results for continuing the tournament will vary.

Second, yes, we use the estimated tournament equity as the payoff at the end of the hand.

But substituting a tournament branch with its EV doesn't mean we assume a literal deal is made. Players routinely replace uncertain outcomes with their EV when making decisions.

First, ICM doesn't actually assume the tournament ends immediately.

It models the probability of each player's finishing positions as the tournament plays out, converting chip stacks into their tournament equity based on the prize structure.

@thinkingpoker.bsky.social maintains a poker starter pack.

The original version I saw had me distracted with ideas like modifying positions within chests or other angleshoots to communicate via some side channel. In hindsight, I think it was just poorly worded rather than an issue with the original setting.

The key point to focus on is that the shuffle will remain the same throughout the entire challenge, even though the players can't communicate any information about the deck's state.

At first, I thought strategies might only slightly improve the random 1 in 2^52 odds. But the correct strategy aligns the players' successes so well that it completely defies intuition. The level of coordination possible is absolutely wild. Still one of the most elegant puzzles I’ve ever seen. 2/2

The players can reveal their assigned cards in the initial phase, yes.