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Luoshu Square (洛书)

Describes a palace of the Luoshu magic square (洛书, Lo Shu): its colour name (一白 … 九紫), later-heaven trigram, direction, element, season and family member, together with the full grid, the magic constant 15, the Hetu (河图) partner number and the opposite palace.

When to use

You need the meaning of a Luoshu / Flying Star number 1–9, the palace grid, or which trigram and direction a number stands for.

Do not use when: You need a personal Kua number (use ming-gua) or the annual/monthly flying stars of a period (use nine-star-feng-shui).

Formula

grid = [4 9 2; 3 5 7; 8 1 6], magic constant = 3·(1+…+9)/9 = 15; opposite = 10 − number (5 → 5); Hetu partner = number ± 5

Later-heaven mapping 坎1 N, 坤2 SW, 震3 E, 巽4 SE, 5 centre, 乾6 NW, 兑7 W, 艮8 NE, 离9 S; colours from the 紫白 (purple-white) flying-star tradition. Traditional cultural reference calculation; not a prediction and not medical, legal or financial advice.

Inputs

Parameter	Type	Unit	Required	Description
`number`	integer		default 5	Luoshu number 1–9 (5 = centre). Range: ≥ 1, ≤ 9

Outputs

Output	Type	Description
`number`	integer	The palace number.
`color_name`	string	Nine-star colour name: 一白二黑三碧四绿五黄六白七赤八白九紫.
`trigram`	string	Later-heaven trigram of the palace (none for the centre).
`direction`	string	Compass direction of the palace.
`element`	string	Five-element attribution (1 Water, 2/5/8 Earth, 3/4 Wood, 6/7 Metal, 9 Fire).
`season`	string	Season of the palace's trigram.
`family_member`	string	Family role of the palace's trigram.
`grid`	string_list	The square as three rows in the traditional orientation with south (9) at the top: 4 9 2 / 3 5 7 / 8 1 6.
`magic_constant`	integer	Sum of every row, column and diagonal.
`hetu_pair`	integer	Partner number in the River Map pairs 1–6 水, 2–7 火, 3–8 木, 4–9 金, 5–10 土.
`hetu_element`	string	Element of the Hetu pair.
`opposite_palace`	integer	Palace across the centre; the two numbers sum to 10 (the centre is its own opposite).
`all_palaces`	list	One row per number: colour name, trigram, direction, element.

Example

Palace 1: {"number":1} → {"color_name":"1 一白 (Yi Bai, White)","trigram":"坎 Kan ☵ (Water)","direction":"North (N, 北)","element":"Water (水)","magic_constant":15,"hetu_pair":6,"opposite_palace":9}

Palace 5 (centre): {"number":5} → {"color_name":"5 五黄 (Wu Huang, Yellow)","direction":"Centre (中)","element":"Earth (土)","hetu_pair":10,"opposite_palace":5,"grid":["4 9 2","3 5 7","8 1 6"]}

GET https://tttkmbb.com/api/v1/calculate/luoshu-square?number=1

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/luoshu-square (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/luoshu-square · Markdown: https://tttkmbb.com/yijing/luoshu-square.md · JSON definition: https://tttkmbb.com/yijing/luoshu-square.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="luoshu-square"
OpenAPI operationId: describe_luoshu_square
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Lo Shu Square (reference)
Wikipedia – Bagua (reference)

FAQ

Why is 9 at the top of the grid?

Chinese diagrams put south at the top; 9 (离, south) is therefore the top-centre cell and 1 (坎, north) the bottom-centre cell. Rotating the grid does not change the sums.

What is the Hetu partner?

The River Map (河图) pairs 1–6, 2–7, 3–8, 4–9 and 5–10 as the generating and completing numbers of Water, Fire, Wood, Metal and Earth; 10 is not a Luoshu palace.

Related calculators

Bagua Trigram Lookup (八卦) — Full attributes of the palace's trigram.
Ming Gua Calculator (命卦 / Eight Mansions) — Personal Kua number defined by these palaces.
Five Elements Relation (五行生克) — Relations between the elements of two palaces.