# How many orientations are there for pawns (for a single player) on a chess board?

How many orientations are there for pawns on a chess board?

1. Pawns can only move forward or diagonally forward.
2. No two pawns may exist on the same square.
3. Pawns start on the second row, and cannot sit on the 8th row (due to Pawn Promotion). This leaves a 6x8 area where they can exist.
4. Also, since all pawns are identical, two orientations are identical if any two pawns are flipped.
5. Finally, any pawn may be completely missing from the board, due to having been captured by the opponent.

How many possible states can this 6x8 board of pawns exist in?

Side note: This is a partial solve for the total states for a complete chess board.

• Since pawns only move diagonally when capturing, are we including or excluding pawn orientations that would require more than the possible number of captures? – Dennis Meng Jun 29 '14 at 5:04

## The number of legal $n$-pawn depth $d$ positions

We can assume (a) pawns always move one square at a time, (b) if a capture occurs, it occurs before any pawn moves, (c) no en passant captures.

Given $n$ pawns having made $d$ moves in total, the number of legal arrangements is computed below, giving $$444957311$$ positions in total.

Note that this includes positions in which the opponent might not have enough material to capture, and positions where the opponent has enough material to capture, but it might not be possible to maneuver the material to the right squares at the right time (this might make things much more complicated computationally).

$$\tiny\begin{array}{r|rrrrrrrrr} & n=0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline d=0 & 1 & 8 & 28 & 56 & 70 & 56 & 28 & 8 & 1 \\ 1 & & 8 & 64 & 222 & 430 & 500 & 348 & 134 & 22 \\ 2 & & 8 & 92 & 448 & 1212 & 1960 & 1882 & 988 & 218 \\ 3 & & 8 & 128 & 792 & 2662 & 5412 & 6686 & 4618 & 1362 \\ 4 & & 8 & 156 & 1184 & 4864 & 12090 & 18488 & 16194 & 6266 \\ 5 & & 8 & 192 & 1696 & 8048 & 23370 & 42706 & 46102 & 22996 \\ 6 & & & 156 & 2040 & 11920 & 40690 & 87074 & 112610 & 70798 \\ 7 & & & 128 & 2208 & 15744 & 63424 & 158740 & 243020 & 189048 \\ 8 & & & 92 & 2208 & 19238 & 90756 & 263276 & 471700 & 447053 \\ 9 & & & 64 & 2040 & 21632 & 119602 & 401172 & 835220 & 951372 \\ 10 & & & 28 & 1696 & 22704 & 147160 & 567262 & 1361974 & 1842988 \\ 11 & & & & 1184 & 21632 & 168120 & 748388 & 2063920 & 3282988 \\ 12 & & & & 792 & 19238 & 179488 & 923354 & 2918850 & 5417010 \\ 13 & & & & 448 & 15744 & 179488 & 1071114 & 3873526 & 8330680 \\ 14 & & & & 224 & 11936 & 168176 & 1169486 & 4837538 & 11999230 \\ 15 & & & & 56 & 8064 & 147288 & 1205608 & 5706282 & 16254208 \\ 16 & & & & & 4886 & 119840 & 1170566 & 6365120 & 20775717 \\ 17 & & & & & 2688 & 91168 & 1073856 & 6727246 & 25117210 \\ 18 & & & & & 1232 & 63952 & 928172 & 6743406 & 28789402 \\ 19 & & & & & 448 & 41328 & 755776 & 6414618 & 31330584 \\ 20 & & & & & 70 & 23912 & 576632 & 5789860 & 32421808 \\ 21 & & & & & & 12544 & 411712 & 4953312 & 31918138 \\ 22 & & & & & & 5712 & 273784 & 4016028 & 29920510 \\ 23 & & & & & & 2128 & 168000 & 3076146 & 26701774 \\ 24 & & & & & & 560 & 94276 & 2221960 & 22685065 \\ 25 & & & & & & 56 & 47488 & 1505392 & 18324954 \\ 26 & & & & & & & 21392 & 954576 & 14061406 \\ 27 & & & & & & & 8064 & 561232 & 10227850 \\ 28 & & & & & & & 2408 & 303752 & 7031438 \\ 29 & & & & & & & 448 & 149072 & 4552764 \\ 30 & & & & & & & 28 & 65408 & 2762590 \\ 31 & & & & & & & & 24976 & 1563128 \\ 32 & & & & & & & & 7728 & 816117 \\ 33 & & & & & & & & 1792 & 389824 \\ 34 & & & & & & & & 224 & 167104 \\ 35 & & & & & & & & 8 & 63232 \\ 36 & & & & & & & & & 20084 \\ 37 & & & & & & & & & 4992 \\ 38 & & & & & & & & & 848 \\ 39 & & & & & & & & & 64 \\ 40 & & & & & & & & & 1 \\ \end{array}$$

## Impossible $3$-pawn and $4$-pawn positions.

The $3$-pawn position $$\begin{array}{|ccc} \cdot & \cdot & \cdot \\ p & \cdot & \cdot \\ p & p & \cdot \\ \hline \end{array}$$ (and the mirror image) cannot be achieved legally. Impossible $4$-pawn positions are (a) the $2 \times (6 \times 8-3)=90$ extensions of the above impossible $3$-pawn position, (b) the $6$ shifts of $$\begin{array}{ccc} \cdot & \cdot & \cdot \\ \cdot & p & \cdot \\ p & p & p \\ \hline \end{array}$$ and (c) $11 \times 2$ corner cases $$\begin{array}{|ccc} \cdot & \cdot & \cdot \\ p & p & \cdot \\ p & \cdot & p \\ \hline \end{array} \quad \begin{array}{|ccc} \cdot & \cdot & \cdot \\ p & p & \cdot \\ \cdot & p & p \\ \hline \end{array} \quad \begin{array}{|ccc} p & \cdot & \cdot \\ \cdot & \cdot & \cdot \\ p & p & p \\ \hline \end{array} \quad \begin{array}{|ccc} p & \cdot & \cdot \\ p & \cdot & \cdot \\ p & \cdot & p \\ \hline \end{array} \quad \begin{array}{|ccc} p & \cdot & \cdot \\ p & \cdot & \cdot \\ \cdot & p & p \\ \hline \end{array} \quad \begin{array}{|ccc} p & \cdot & \cdot \\ \cdot & p & \cdot \\ p & p & \cdot \\ \hline \end{array}$$

$$\begin{array}{|ccc} p & \cdot & \cdot \\ \cdot & p & \cdot \\ p & \cdot & p \\ \hline \end{array} \quad \begin{array}{|ccc} p & \cdot & \cdot \\ \cdot & p & \cdot \\ \cdot & p & p \\ \hline \end{array} \quad \begin{array}{|ccc} p & \cdot & \cdot \\ p & p & \cdot \\ p & \cdot & \cdot \\ \hline \end{array} \quad \begin{array}{|ccc} p & \cdot & \cdot \\ p & p & \cdot \\ \cdot & p & \cdot \\ \hline \end{array} \quad \begin{array}{|ccc} p & \cdot & \cdot \\ p & p & \cdot \\ \cdot & \cdot & p \\ \hline \end{array}$$ (including their mirror images).