# PLO paired-board hand selection

Original GTO Gecko editorial examples, created and checked September 6, 2026.
Article: https://gtogecko.com/blog/plo-paired-board-hands

## Question and scope

What is the strongest legal five-card high hand in each of eight completed
four-card Omaha examples? These are constructed teaching cases, not observed
hands, a sampled population, an equity estimate or betting advice.

Use a standard 52-card deck and standard high rankings. Select exactly two
of four hole cards and three of five board cards. There are C(4,2) × C(5,3)
= 6 × 10 = 60 legal selections per example. Check all of them, rank each,
and take the maximum. A displayed witness is one selection attaining that
maximum; other selections can tie it. Rank-score arrays compare
lexicographically, beginning with category 0 (high card) through 8 (straight
flush), followed by category-relevant ranks, with ace = 14 except in the
five-high straight. This is an exact finite check: no random seed or sampling.

The first six examples are separate deals. Examples 7 and 8 also form a
compatible two-player showdown: all thirteen physical cards are distinct,
and kings full of aces beats three kings with ace-queen kickers.

Card notation in the data uses T for ten and J/Q/K/A for jack/queen/king/ace;
the second character is s (spades), h (hearts), d (diamonds) or c (clubs).
For example, Ah is the ace of hearts.

## Files

- [cases.json](cases.json): literal cards, expected scores, plain-language results
  and explicit five-card witnesses.
- [results.csv](results.csv): human-readable results, checked against the JSON.
- [verify.py](verify.py): Python standard-library checker and boundary controls.
- [selection.svg](selection.svg): original source of the article hero diagram.
- [SHA256SUMS](SHA256SUMS): hashes of the other five files, including this README.

## Reproduce

Download these six files into one directory, preserving filenames. With Python
3.10 or newer, run the following commands in that directory:

```text
python verify.py
python -O verify.py
```

The script also works when invoked by its full path from another directory.
It does not use network access, private files or installed packages. Each
run checks all eight maxima and their legal witnesses, all 24 consistent suit
permutations per case, reversal of input order, fourteen high-hand classifier
boundaries, the article's pocket-sevens counterexample, the final showdown's
card compatibility, four rejected malformed inputs and published CSV parity.
Explicit exceptions remain active under optimized Python. Failure exits with
an exception; success prints a PASS summary. Files are read, not overwritten.

For download integrity, compare SHA-256 hashes with SHA256SUMS. Hashes refer
to the delivered UTF-8 files with LF line endings; a text editor may alter them.
With Python, this command checks all manifest entries from the same directory:

```text
python -c "from pathlib import Path; import hashlib; rows=[x.split('  ',1) for x in Path('SHA256SUMS').read_text().splitlines()]; bad=[n for h,n in rows if hashlib.sha256(Path(n).read_bytes()).hexdigest()!=h]; print('PASS' if not bad else 'FAIL: '+str(bad)); raise SystemExit(bool(bad))"
```

## Evidence and checks

Omaha construction: https://www.pokerstars.com/poker/games/omaha/
High rankings: https://www.pokerstars.com/poker/games/rules/hand-rankings/
Independent operator rules:
https://christchurchcasino.co.nz/wp-content/uploads/2018/07/Poker-How-to-Play-Web.pdf
All checked September 6, 2026. PokerStars provided substantive content through
the research browser; direct HTTP from the research host redirected to a
regional exit page. The casino PDF was retrieved and read directly.

An independent five-card implementation also checked the eight maxima and
witnesses during editorial review. Neither implementation is a certified
general-purpose poker engine. Tests target these specific published examples.
The source SVG and numerical examples are original GTO Gecko work; no third-party
rule text or artwork is reproduced in the graphic.

## Limits

These cases establish no frequency of encountering a hand, no winning equity
against an unspecified opponent, and no correct betting action. They exclude
Hi/Lo, Short Deck, wild cards and other hand-construction rules. Position,
stacks, pot size, fees, ranges and later action are not modeled. The visual
illustrates example 1 only; full exact results are in the article and CSV.