Kua number 5, and the number you use instead
Published 2026-08-20
If your hand calculation came out at 5, nothing is wrong with your arithmetic. There is no trigram numbered 5 and therefore no set of directions attached to it, so Eight Mansions (八宅, Ba Zhai) needs a replacement number. Standard practice substitutes 2 for men and 8 for women.
That substitution is not an afterthought on this site. kuaNumber in src/lib/kua.ts applies it as its
last step, so the number 5 is never returned and never displayed anywhere. Ask the
calculator about a birth year that lands on it and you get Kua 2 or Kua 8 back
with no remark, which is worth knowing about our own tool: it gives you the substituted answer without
telling you a substitution happened.
Why there is no fifth number
The eight-direction map has eight seats, one per trigram, and the numbers running around it are 1, 2, 3,
4, 6, 7, 8 and 9. In the nine-square grid those numbers are drawn on, the middle square holds 5 — and the
middle square is not a direction. There is nothing to face. TRIGRAMS in the code has keys for the eight
numbers and no key for 5; asking the direction tables for it returns nothing rather than a table of
guesses.
Which birth years land on it
The arithmetic hits 5 once every nine years for each sex. Counted over 1900–2100, men land on it in 22 of those years and women in 23, and no single year lands on it for both.
| Sex | Years that produce 5 before substitution |
|---|---|
| Men | 1932, 1941, 1950, 1959, 1968, 1977, 1986, 1995, 2004, 2013, 2022 |
| Women | 1936, 1945, 1954, 1963, 1972, 1981, 1990, 1999, 2008, 2017, 2026 |
One trap in reading that table. The year in it is the year counted from the beginning of spring, not your
passport year. A man born in January 2005 is counted as 2004, so he is in the substituted set even though
his birth certificate says 2005. A man born in January 1986 is counted as 1985 and is not, even though
1986 is in the list. Both of those come straight out of effectiveYear.
The directions you actually work from
Once substituted, you use the replacement number’s table with nothing carried over from the 5. Generated
from AUSPICIOUS and INAUSPICIOUS:
| Sheng Chi | Tian Yi | Yan Nian | Fu Wei | The four to avoid | |
|---|---|---|---|---|---|
| Men, Kua 2 (Kun 坤) | Northeast | West | Northwest | Southwest | North, East, Southeast, South |
| Women, Kua 8 (Gen 艮) | Southwest | Northwest | West | Northeast | North, East, Southeast, South |
Both replacements are West group numbers, so anyone whose arithmetic produces 5 ends up West group, whichever formula got them there. That has a practical edge. West is the only cardinal direction among the four, so in a bedroom square to the compass a headboard can reach exactly one of them — the other three are diagonals and a bed flat against a wall cannot point at a diagonal. Which direction should the head of your bed face covers that squeeze.
Men and women get the same four directions and the same four to avoid; only the ranking differs, and even there Tian Yi and Yan Nian simply trade places.
A side effect worth knowing
Because the substitution funnels one number into another, the population is lopsided. Counting every year from 1900 to 2100, the men’s formula returns 2 in 44 of them and each of its other numbers in about 22. The women’s formula returns 8 in 45. So those two numbers are roughly twice as common as any other, and that is an artifact of the substitution rather than anything in the tradition.
It also means Kua 2 men are two populations merged: those whose arithmetic genuinely gave 2, such as 1971 and 1989 births, and those substituted from 5, such as 1986 and 2004 births. Eight Mansions treats them identically from that point on, and no source we know of assigns them different directions.
Where practice is split
The substitution above is the standard one and the one this site implements. Some teachers handle it differently, and we are not in a position to compute what their rule gives — the honest thing is to tell you what changes if your teacher or your book gives you a different number.
It changes everything, because Kua numbers are not on a scale. There is no near miss. If your source substitutes something that puts you in the East group, the four directions you are told to favor are the four this site tells you to avoid, and the two answers cannot be split down the middle. So if you are working from a printed chart that disagrees with our output, do not average them. Pick the source you are following and stay with it, and know that our number came from the 2-for-men and 8-for-women rule.
What the substituted number does not do
Substituting a number gives you a table to work from. It does not make the table a prediction. Eight Mansions names Northeast as Sheng Chi for Kua 2 and associates it with prosperity; what it does not do is tell you what happens to you, and neither do we. See method and assumptions for the rules the site implements and the range they hold over.