Growing Frame
Read the puzzle
Tiles form a hollow square frame one tile thick. Step 1 is a 3 by 3 frame (8 tiles), step 2 is a 4 by 4 frame (12 tiles), step 3 is a 5 by 5 frame (16 tiles). Steps 4 to 19 are not shown. How many tiles will step 20 have?
Steps 4 to 19 are not shown. How many tiles will step 20 have?
Look at the side length of each frame. How does it grow from step to step?
84
Step 20 is a 22 × 22 frame: 4 sides of 21 tiles, 4 × 21 = 84.
How to solve it
First find how the side length grows. Step 1 is 3 tiles across, step 2 is 4, step 3 is 5. The side is always 2 more than the step number, so step 20 is a 22 by 22 frame.
Now count the tiles round a frame of side 22 without counting any corner twice. One neat way: walk round the frame in four equal runs, each run starting at a corner and stopping just before the next corner. Each run is 22 - 1 = 21 tiles, so the frame has 4 x 21 = 84 tiles.
Check the method on a visible step. Step 3 has side 5: 4 x (5 - 1) = 4 x 4 = 16, which matches its label. The totals also rise by 4 each step (8, 12, 16), and 8 + 19 x 4 = 84 for step 20.
The usual wrong answer is 88, from multiplying 4 sides by 22 tiles. That counts each of the 4 corner tiles twice, once for each side it sits on. Another slip is 80, from using a side of 20 instead of 22.
Takeaway: a hollow square of side s has 4 x (s - 1) tiles, and so the tile count for step n here is 4n + 4.
Learn the method: Number Sequence Puzzles: How to Find What Comes Next
