The Rotating 3×3 Access Board
During a security exercise, a training room displays a 3×3 board of nine tiles. Each tile carries one of three symbols: a triangle (▲), a circle (●) or a square (■). The board is built so that every row and every column contains each of the three symbols exactly once. The current board, read from top-left to bottom-right, is: Row 1: ▲ ● ■ Row 2: ● ■ ▲ Row 3: ■ ▲ ? After each round the whole board is rotated 90° clockwise as a single unit; tiles keep their symbols. The facilitator asks the trainees to deduce the missing tile before the first rotation, then to state what appears in the top-left and centre tiles after that rotation. A trainee also claims that any rotation of the completed board will still satisfy the one-of-each rule for every row and column.
1. Which symbol replaces the question mark? Give the reasoning, not just the symbol.
Read the worked answer
The missing symbol is ●.
Use the row rule first. Row 3 already contains ■ and ▲. Since every row must contain each of the three symbols exactly once, the third cell must be ●.
Check against the column rule. Column 3 contains ■ in row 1, ▲ in row 2 and the missing cell in row 3. The missing symbol must again be ●. Both constraints agree, so the completed board is:
▲ ● ■ ● ■ ▲ ■ ▲ ●
The agreement is the important part: if the row answer and column answer had differed, at least one of the stated constraints would have been misread.
2. After one 90° clockwise rotation of the completed board, which symbol is in the top-left tile and which is in the centre tile? Explain how you tracked the positions.
Read the worked answer
After a 90° clockwise rotation:
- The new top-left tile is the old bottom-left tile. In the completed board, old bottom-left = ■. So the new top-left is ■.
- The centre tile stays in the centre under a 90° rotation. The old centre is ■, so the new centre is ■.
A useful tracking rule for a 90° clockwise rotation is: new row 1 is the old column 1 read from bottom to top; new column 3 is the old row 1 read from left to right. Applying that rule gives the rotated board:
■ ● ▲ ▲ ■ ● ● ▲ ■
The top-left and centre entries match the position-tracking answers. The check guards against the common error of rotating the symbols themselves rather than moving the tiles as a block.
3. A trainee says: 'Because every row and column already contains one of each symbol, any rotation of the completed board will also satisfy the rule.' Is the trainee correct? Explain.
Read the worked answer
Yes, the trainee is correct.
A rotation does not create or destroy symbols; it permutes the cells. Under a 90° rotation, each old row becomes a new column, and each old column becomes a new row. Since every old row contained exactly one ▲, one ● and one ■, every new column contains exactly one of each. Since every old column contained exactly one of each, every new row does. The same reasoning applies to 180° and 270° rotations.
This is a property of the Latin-square structure: the one-of-each constraint is rotation-invariant. The trainee's conclusion would not follow for an arbitrary change such as swapping two symbols in one cell, because that could duplicate a symbol in a row or column. For rotation, however, the conclusion is sound.