Ava and Bianca were bored on a rainy afternoon. Ava, who loved math and language arts equally and had a very impressive vocabulary, challenged Bianca to give her any math-related word to define. Ava said, “Bianca, I’ll bet you a quarter that I can give you a definition for any word that a mathematician might use .”

Bianca relished the challenge. She tried to think of a winning word. She knew she couldn’t trip Ava up with geometry-related words. She knew Ava would know “penumbra” (the partially shaded outer region of the shadow cast by an opaque object) and “quincunx” (an arrangement of five objects with four at the corners of a square or rectangle and the fifth at its center, as the five dots on a die.) But, Bianca thought, perhaps a word related to the math of the game of chess would be a winning play.

Bianca had the perfect word, “zugzwang!” She said triumphantly.

Ava’s face fell. She had lost the bet. “I don’t know that word. Are you sure it is related to math?” Holding her hand out to collect the quarter, Bianca explained, “Yes, it is math-related. In most two-player games, someone has to go first. In some games, like tennis and Connect Four, there is an advantage to going first. But, in other games or game situations, going first means a guaranteed loss. Chess players call the obligation to make a move that will result in a loss “zugzwang.”

Ava was happy to learn a new word but objected to giving up the quarter, “that’s not a math word, that’s a chess word!”

Bianca was surprised at Ava’s objection, “Math and chess are deeply related. There is a whole class of puzzles and problems that are called ‘mathematical chess problems.’ For example, haven’t you heard of the Eight Queens Puzzle? It is simple to explain but difficult to solve. Place eight chess queens on an 8×8 chessboard so that no two queens threaten each other. Can you do it? There are actually 92 solutions!”

Ava liked this—a whole new world of puzzles to explore! But, Ava said, “The Eight Queens Puzzle sounds like a cool problem, but it might be too hard to do as my very first chess math puzzle. Do you have an easier one we can start with?”

“Sure,” said Bianca, “I know a great one. It’s hard too, but I know you can do it! Here is the challenge: On a 4×4 solitaire chess board, there are eight pieces that need to be placed on specific squares. As shown in the image above, three pieces have already been correctly placed on the board. Use the following clues and tell me where the remaining pieces belong:

**Pieces you must use:**

-2 rooks of either color

-2 knights of either color

-1 black king

**Rules you must follow:**

- The queens are adjacent to the king of the same color.
- The kings are on the same diagonal.
- There are no more than two pieces per row or column.
- The rooks do not share a row, column, or diagonal.
- The knights are adjacent.
- The rooks are on adjacent rows.
- Both knights share diagonals with rooks.

## Try to solve it, then click to see the answer!

**Solution :**

Black King – A3

Rooks – B3 and D4

Knights – B2 and C2

The black king must be adjacent to the black queen, so it is either on A3, B3, or B4. Since the kings must share a diagonal, the black king must be on A3.

Since no row or column has more than two pieces, we know that A1, A2, and B1 are empty.

Since the rooks must be on adjacent rows, there must be a rook on row 3. There cannot be a rook on C3 because every other available square shares a row, column, or diagonal with C3, which breaks rule (4). So there is a rook on either B3 or D3.

Assume there is a rook on D3. Then by rule (3), D2, D4, B3, and C3 are empty. This puts the knights, which must be adjacent, on B2 and C2 and the remaining rook on B4 (the only remaining square which does not share a row, column, or diagonal with D3). However, this breaks rule (7) because the knight on B2 does not share a diagonal with a rook.

Therefore, there cannot be a rook on D3, so there is a rook on B3. By rule (3), C3 and D3 are empty. The other rook may be on either D2 or D4. If it is on D2, then any placement of the knights would result in 3 pieces in a single row, which is not allowed. So the other rook must be on D4. The knights are then in the only remaining available squares, B2 and C2.