The Tiled Courtyard Diagonal Dispute
A landscaper is laying square tiles on a rectangular courtyard that measures 18 feet by 24 feet. The plan calls for a decorative diagonal stripe of contrasting tiles running from one corner to the opposite corner. Each contrasting tile is a square measuring 1 foot on a side, and tiles are laid on a grid aligned with the courtyard edges so that every tile covers exactly one 1 ft by 1 ft cell. The contractor claims the diagonal stripe will cross exactly 24 tiles because the courtyard is 24 feet long. The designer disagrees. Before ordering materials, the crew needs the exact count of 1 ft tiles that the corner-to-corner diagonal passes through, and they also want to know whether the stripe will hit any tile corner exactly (which would change how a tile is scored).
1. How many 1 ft by 1 ft tiles does the corner-to-corner diagonal actually pass through? Justify the count with reasoning, not a diagram.
Read the worked answer
The diagonal passes through a tile whenever it crosses a vertical grid line or a horizontal grid line, plus the starting tile.
- The courtyard is 18 ft wide and 24 ft tall, so there are 17 interior vertical grid lines and 23 interior horizontal grid lines.
- If the diagonal never passed through a lattice point (a tile corner), it would cross 17 + 23 = 40 interior lines, entering a new tile each time, plus the starting tile: 40 + 1 = 41 tiles.
- But when the diagonal passes exactly through a lattice point, it crosses a vertical and horizontal line simultaneously, so it enters only one new tile instead of two. Each such lattice-point crossing reduces the count by 1.
The diagonal from (0,0) to (18,24) passes through lattice points whenever 18t and 24t are both integers. gcd(18,24) = 6, so the interior lattice points occur at t = k/6 for k = 1..5: that is 5 interior lattice points.
Count = 18 + 24 − gcd(18,24) = 42 − 6 = 36 tiles.
Check: 41 − 5 = 36. ✓ The contractor's claim of 24 is wrong; the stripe crosses 36 tiles.
2. Does the stripe hit any tile corner exactly? If so, how many interior tile corners does it strike, and why does this matter for scoring tiles?
Read the worked answer
Yes. The diagonal hits a tile corner exactly at interior lattice points. From the previous work, the diagonal passes through lattice points at t = k/6, k = 1, 2, 3, 4, 5, giving 5 interior corners: (3,4), (6,8), (9,12), (12,16), (15,20).
This matters because at those points the diagonal enters a new tile through a corner rather than through an edge midpoint region. A tile reached only at a corner is touched at a single point; depending on the scoring convention, the crew must decide whether such a tile counts as "crossed." Using the standard line-crossing convention (a tile counts if the segment passes through its interior), the count of 36 already accounts for this, because each lattice-point crossing merges two line-crossings into one. The practical takeaway: order 36 contrasting tiles plus a small waste allowance, and pre-score the 5 tiles at those corner points since the stripe clips them at a corner.