Growing Corner
Read the puzzle
Tiles form an L shape with an extra tile in the inside corner. Step 1 is a 2 by 2 block (4 tiles). Step 2 has a column of 3 and a row of 3 sharing the corner tile, plus the inner tile (6 tiles). Step 3 has arms of 4 (8 tiles). Steps 4 to 14 are not shown. How many tiles will step 15 have?
Steps 4 to 14 are not shown. How many tiles will step 15 have?
Each step adds 2 tiles: one on each arm.
32
Two arms of 16 share the corner (31), plus the inner tile: 32.
How to solve it
Find how long each arm is at each step. In step 1 each arm is 2 tiles, in step 2 each is 3, and in step 3 each is 4. So in step n each arm is n + 1 tiles, and in step 15 each arm is 16 tiles.
The two arms share the corner tile, so together they hold 16 + 16 - 1 = 31 tiles. Then add the one inner tile that sits in the angle of the L and never moves: 31 + 1 = 32 tiles.
Check on the visible steps. Step 3: arms of 4 give 4 + 4 - 1 = 7, plus 1 inner tile is 8, which matches. You can also add 2 tiles per step: step 1 has 4, so step 15 has 4 + 14 x 2 = 32. The rule is 2n + 2.
The usual wrong answer is 33, from adding both arms of 16 and the inner tile without removing the shared corner. Another slip is 30, from forgetting the inner tile, or 34, from using 2 x 15 + 4.
Takeaway: when two lines of tiles meet, the meeting tile belongs to both, so count it once.
Learn the method: Number Sequence Puzzles: How to Find What Comes Next
