# PP292 — The declared card as a wild card

Jeffery Lyn Huckstead · Cerebral Graphix · 29 September 2026 · site release v3.5 · working note · Part of *Project 51: Reading the Record Across Rules*, DOI [10.5281/zenodo.23048482](https://doi.org/10.5281/zenodo.23048482)

Status words: **PROVED** (argument given here), **COMPUTED** (exact enumeration or seeded check, with a script and receipt), **OPEN**, **DESIGN** (a choice, not a result). This is a card model. There is no zero data and nothing here concerns the Riemann hypothesis. "Riemann" is the name of a published rule-based dealer. RH STATUS: OPEN.

## 0. The question

Poker 51 declares one card, Ⅰ = 2♣, and keeps it out of the deal. The published working paper proves that removing it leaves five categories at their exact 52-card probabilities:

- one pair,
- two pair,
- three of a kind,
- full house,
- four of a kind.

These are the categories defined only by repeated ranks.

The question here is what happens when Ⅰ comes back into play with a rule. The four rules studied put Ⅰ back in the deck (52 cards). Every natural card keeps its meaning. A hand holding Ⅰ is worth the best hand it can make:

| Variant | Ⅰ may stand for |
|---|---|
| **Wild Ⅰ** | any card; with four of a rank, five of a kind |
| **The bug** | an ace, or any card that completes a straight, flush or straight flush; with four aces, five aces |
| **Ⅰ any suit** | a two of any suit |
| **Ⅰ any rank** | a club of any rank |

Rules and evaluator: `area-51/poker-51/poker51-variants-v1.js`. Five of a kind ranks above a straight flush.

## 1. Exact counts (COMPUTED)

Every hand is counted:

- **Variants:** C(52,5) = 2,598,960 hands, of which 249,900 hold Ⅰ.
- **Standard game:** C(51,5) = 2,349,060 hands.

Script: `wild-census-v1.mjs`; receipt: `WILD_CENSUS_v1.json`. An independent recount of the 249,900 Ⅰ-hands in `tests/v3_5/verify-poker51-variants-v1.mjs` reproduces every number.

| Hand | Standard (51) | Wild Ⅰ | The bug | Ⅰ any suit | Ⅰ any rank |
|---|---:|---:|---:|---:|---:|
| Five of a kind | — | 12 | 1 | — | — |
| Straight flush | 38 | 195 | 195 | 46 | 72 |
| Four of a kind | 564 | 2,868 | 763 | 624 | 1,188 |
| Full house | 3,384 | 5,976 | 3,978 | 3,744 | **6,678** |
| Flush | 4,615 | 7,098 | 7,098 | 6,587 | **5,076** |
| Straight | 9,690 | 19,581 | 19,581 | 10,194 | 19,704 |
| Three of a kind | 49,632 | **125,664** | 57,640 | 54,912 | 89,628 |
| Two pair | 111,672 | **111,672** | 125,730 | 123,552 | 146,718 |
| One pair | 992,640 | 1,149,069 | 1,100,868 | 1,098,240 | 1,153,071 |
| High card | 1,176,825 | 1,176,825 | 1,283,106 | 1,301,061 | 1,176,825 |

How often Ⅰ improves a hand (COMPUTED), counted against the same five cards with Ⅰ read as a plain 2♣, out of the 249,900 Ⅰ-hands:

| Variant | Hands improved |
|---|---:|
| Wild Ⅰ | 205,244 |
| Ⅰ any rank | 196,757 |
| The bug | 69,277 |
| Ⅰ any suit | 1,485 |

## 2. Wild Ⅰ admits no consistent ranking (COMPUTED)

With the standard ranking, three of a kind (125,664 hands) is more common than two pair (111,672), yet it ranks higher.

Now rank two pair above three of a kind instead. The 76,032 hands holding exactly one natural pair plus Ⅰ are then read as two pair. That gives 187,704 two pairs against 49,632 threes of a kind, and the order is inverted again.

This is the wild-card paradox of S. Gadbois ("Poker with wild cards — a paradox?", *Mathematics Magazine* 69 (1996), 283–285); see also "Reconciling wild cards and poker", *CHANCE* (2011), doi:10.1007/s00144-011-0035-4. Here it is counted exactly on the declared-card deck.

The note goes one step further than the two-step flip, with an exhaustive search (`ranking-search-v1.mjs`, receipt `RANKING_SEARCH_v1.json`).

**The test.** For each ordering of the categories that occur (10! = 3,628,800 orderings when five of a kind is possible, 9! = 362,880 otherwise), every Ⅰ-hand is read as its highest-ranked reachable category under that ordering. The ordering is consistent if every category is strictly rarer than each category below it.

| Variant | Orderings tried | Consistent orderings |
|---|---:|---|
| Wild Ⅰ | 3,628,800 | **none** |
| The bug | 3,628,800 | exactly one: the standard ranking, with five aces on top |
| Ⅰ any suit | 362,880 | exactly one: the standard ranking |
| Ⅰ any rank | 362,880 | exactly one: the standard ranking with **flush above full house**, as short-deck poker ranks them |

**Scope.** This is exact for this deck and these rules. Players are assumed to read Ⅰ for the highest-ranked category they can reach; kickers do not affect category counts.

## 3. Ⅰ any suit keeps five categories at their exact 52-card counts (PROVED)

**Proposition.** Under the Ⅰ-any-suit rule, the number of hands in each of one pair, two pair, three of a kind, full house and four of a kind equals its count in the ordinary 52-card deck:

| Category | Count |
|---|---:|
| One pair | 1,098,240 |
| Two pair | 123,552 |
| Three of a kind | 54,912 |
| Full house | 3,744 |
| Four of a kind | 624 |

*Proof.* Read Ⅰ as the natural 2♣. This gives a bijection between the variant's hands and the ordinary 52-card hands. Choosing Ⅰ's suit never changes the hand's multiset of ranks, since Ⅰ is a two either way. A hand with a repeated rank is never a flush, because a flush has five distinct ranks. So the suit choice cannot move a hand into or out of any of the five repeated-rank categories. Each such hand is in the same category under both readings, and the counts agree. ∎

The suit choice matters only when the four natural cards share a suit. It turns exactly **1,479 high-card hands into flushes** and **6 straights into straight flushes**; everything else is unchanged (COMPUTED).

This pairs with the paper's removal theorem:

| Operation on the declared card | The five repeated-rank categories |
|---|---|
| Remove it (51 cards) | keep their exact 52-card **probabilities** |
| Make it suit-wild (52 cards) | keep their exact 52-card **counts** |

The mechanism is the same in both. The five categories depend only on which ranks repeat; everything that moves is in the order-dependent categories: straights, flushes, straight flushes and high cards.

## 4. Riemann in the variants

**Draw rule (DESIGN).** Riemann always keeps Ⅰ. Among its natural cards it keeps what the published House rule keeps in the hand that Ⅰ makes at its best. The standard game's rule is unchanged.

**Reads (COMPUTED).** The v0.3 policy is unchanged; only its inputs change. Its belief table is rebuilt for each variant by `build-riemann-belief-variants-v1.mjs`: seed 51, the same sample sizes as the standard table, the variant's deck, evaluator and draw rule.

| Variant | Sampled deals | Table sha256 |
|---|---:|---|
| Wild Ⅰ | 10,852,510 | `74bc82e2…` |
| The bug | 11,484,425 | `cfbce677…` |

The verifier checks both hashes, and with `--regenerate` rebuilds both tables byte for byte.

**Why rebuilding matters (COMPUTED, seeded).** Over 20,000 after-draw spots per variant, the same policy was run with the standard table in place of the rebuilt one:

| Variant | Decisions that change | Average read error |
|---|---:|---|
| Wild Ⅰ | 4.5% | overstates its hand by 2.4 points |
| The bug | 4.2% | almost no overall bias |

Against a random hand, Riemann's reference row, some standard hands sit lower under Wild Ⅰ:

| Hand | Standard game | Wild Ⅰ |
|---|---:|---:|
| Pair of jacks | 0.795 | 0.732 |
| Two pair, kings up | 0.958 | 0.926 |
| Three sevens | 0.979 | 0.956 |

## 5. What the site does with this (DESIGN)

- **Playable.** Poker 51 v0.4 adds a game switch: Standard · Wild Ⅰ · The bug.
  - **Wild Ⅰ** is the conventional wild game. It keeps the standard ranking, as table games do, and the Rule Card states that no consistent ranking exists.
  - **The bug** is the preset this research recommends: the only one of the four rules where Ⅰ changes play substantially (69,277 improved hands) and the standard ranking stays the unique consistent one.
- **Research only.** Ⅰ any suit is the most interesting result for the papers, but it changes too few hands (1,485) to be worth a mode. Ⅰ any rank is kept here for its flush/full-house swap.
- **Not claimed.** That either variant, or Riemann's play in it, is fair, optimal or an equilibrium. Strategy returns are unmeasured (OPEN).

## 6. Status

| Claim | Status | Where |
|---|---|---|
| The counts in §1 | COMPUTED | census + independent recount |
| Wild Ⅰ: no ordering of the ten categories is consistent | COMPUTED | all 3,628,800 orderings |
| The bug, Ⅰ any suit: the standard ranking is the unique consistent ordering | COMPUTED | all orderings |
| Ⅰ any rank: unique consistent ordering puts flush above full house | COMPUTED | all orderings |
| Ⅰ any suit keeps the five repeated-rank categories at their exact 52-card counts | PROVED | §3 |
| Riemann's rebuilt tables change about 1 in 22 after-draw decisions | COMPUTED (seeded) | verifier |
| Returns, optimality or equilibrium in either playable variant | OPEN / not claimed | — |

## 7. Reproduce

From the site root, with Node 20+ and no packages:

```sh
node research/project-51/poker-51/wild/wild-census-v1.mjs          # ~3 minutes
node research/project-51/poker-51/wild/ranking-search-v1.mjs       # ~1 minute
node tests/v3_5/verify-poker51-variants-v1.mjs [--regenerate]
```

RH STATUS: OPEN.