Growing Diamond
Read the puzzle
Tiles form a diamond with rows centred on each other. Step 3 has rows of 1, 3, 5, 3, 1 (13 tiles). Step 4 has rows 1, 3, 5, 7, 5, 3, 1 (25 tiles). Step 5 has rows 1, 3, 5, 7, 9, 7, 5, 3, 1 (41 tiles). Steps 6 and 7 are not shown. How many tiles will step 8 have?
Steps 6 and 7 are not shown. How many tiles will step 8 have?
Each diamond is two square grids of tiles, one a little smaller than the other.
113
Step 8 is an 8 × 8 grid woven with a 7 × 7 grid: 64 + 49 = 113 tiles.
How to solve it
Split each diamond into two tilted square grids woven together. Step 3 is a 3 by 3 grid of 9 tiles plus a 2 by 2 grid of 4 in the gaps: 9 + 4 = 13. Step 4 is 4 x 4 + 3 x 3 = 16 + 9 = 25, and step 5 is 5 x 5 + 4 x 4 = 25 + 16 = 41. All three match.
So step n has n x n + (n - 1) x (n - 1) tiles. For step 8: 8 x 8 + 7 x 7 = 64 + 49 = 113 tiles.
Check by rows instead. Step 8 has rows 1, 3, 5, 7, 9, 11, 13, 15, 13, 11, 9, 7, 5, 3, 1. The top half 1 + 3 + ... + 15 is 8 x 8 = 64, and the bottom half 1 + 3 + ... + 13 is 7 x 7 = 49. Total 113.
The usual wrong answer is 89, from adding 16 every step (41 + 3 x 16). The jumps grow by 4 each time: +20 to 61, +24 to 85, +28 to 113.
Learn the method: Number Sequence Puzzles: How to Find What Comes Next
