The Damaged Corner of the 6-by-6 Panel
A tiling contractor is restoring a 6-by-6 panel of square tiles on a station wall. Every tile is either a black diamond or a white circle, and the original design follows a single repeating rule in both directions. Photographs survive for all but the bottom-right 2-by-2 block, which was destroyed when a bench was bolted to the wall. The surviving rows, read left to right, are: Row 1 - white, black, white, white, black, white. Row 2 - black, white, white, black, white, white. Row 3 - white, white, black, white, white, black. Row 4 - white, black, white, white, black, white. Row 5 - black, white, white, black, then two unknown. Row 6 - white, white, black, white, then two unknown. The contractor holds 40 black diamonds and 40 white circles in stock and will reorder only if the panel needs more. A helper suggests copying the 2-by-2 block directly above the gap; a second suggests copying the block to its left. The bench cannot be moved before the panel is finished.
1. What are the four missing tiles, and what rule generates the whole panel? Show the rule working on at least two complete rows.
Read the worked answer
The missing block is, top row white, white; bottom row white, black.
The rule is a diagonal stripe: a tile is a black diamond when its row number plus its column number is a multiple of 3, and a white circle otherwise.
- Row 1: columns 2 and 5 give sums 3 and 6 - black, matching the photograph.
- Row 3: columns 3 and 6 give sums 6 and 9 - black, matching.
- Row 6: columns 3 and 6 give sums 9 and 12 - black, matching.
Now the gap. Cell (5,5) sums to 10 - white. (5,6) sums to 11 - white. (6,5) sums to 11 - white. (6,6) sums to 12 - black.
Equivalently, the black diamonds lie on three parallel diagonal lines running down-right, and the black at (6,6) is the continuation of the line through (3,3). The three-row and three-column period is visible in the photographs: row 4 repeats row 1 exactly.
2. Both helpers propose the same block. Show what each would copy, confirm they agree, and explain why the agreement is not evidence of correctness.
Read the worked answer
Both proposals are white-black over black-white, and both are wrong.
- Copy from above means taking rows 3 and 4 in columns 5 and 6: white and black on top, black and white below.
- Copy from left means taking rows 5 and 6 in columns 3 and 4: also white and black on top, black and white below.
The two suggestions coincide because the panel repeats every three rows and every three columns. Shifting by two lands on a different phase of the diagonal stripe, and shifting by two in either direction happens to produce the same wrong phase. That coincidence is precisely what makes the trap persuasive: two independent people reaching the same answer feels like confirmation, but they reached it by the same faulty method.
The correct block, white-white over white-black, matches neither neighbour. The safe check is to measure the shift against the pattern's period: only a shift of three rows or three columns would reproduce a block exactly, and neither helper shifted by three.
3. How many black and white tiles does the finished panel need, and is the current stock sufficient? Show the counting.
Read the worked answer
The panel needs 36 tiles: 12 black diamonds and 24 white circles. Stock of 40 of each is sufficient, with 8 black and 16 white left over, so no reorder is needed.
The count follows from the rule rather than from the photograph. Row by row, black appears where the row number plus the column number is a multiple of 3, which happens exactly twice in every six-tile row. Six rows at two blacks each gives 12 blacks; 36 minus 12 gives 24 whites.
Check it against the known rows: row 1 has blacks at columns 2 and 5, row 2 at columns 1 and 4, row 3 at columns 3 and 6, and rows 4 to 6 repeat that cycle - six rows, two blacks apiece.
The four missing tiles account for one black and three whites, already included above. Guessing that the panel is roughly half black would have produced an order for about 18 black diamonds, six more than the design can use.