Growing Comb
Read the puzzle
Tiles form a comb: a bottom bar of 5 tiles with three upright teeth at the left end, middle and right end. Step 1 has teeth 1 tile tall (8 tiles), step 2 has teeth 2 tall (11 tiles), step 3 has teeth 3 tall (14 tiles). Steps 4 to 9 are not shown. How many tiles will step 10 have?
Steps 4 to 9 are not shown. How many tiles will step 10 have?
Each step adds 3 tiles: one on each tooth.
35
A bar of 5, plus 3 teeth of 10 tiles: 5 + 30 = 35.
How to solve it
Split the comb into the part that stays and the part that grows.
The bottom bar is 5 tiles in every step. Above it stand three teeth, and in step n each tooth is n tiles tall. Check: step 1 is 5 + 3 x 1 = 8, step 2 is 5 + 3 x 2 = 11, step 3 is 5 + 3 x 3 = 14. All three match the labels.
So the rule is tiles = 5 + 3n. For step 10, each tooth is 10 tiles tall: 3 x 10 = 30 tooth tiles, plus the bar of 5, gives 5 + 30 = 35 tiles.
You can confirm by adding 3 per step from step 3: 14, 17, 20, 23, 26, 29, 32, 35, which lands on 35 at step 10.
The common wrong answer is 38, from taking step 1's 8 tiles and adding 3 x 10 = 30. That counts step 1's teeth twice. Another slip is 80, from thinking that if step 1 has 8, step 10 has 10 times as many, but the bar does not grow.
Takeaway: split a growing pattern into its fixed part and its growing part, write the rule, and test it on the steps you can see.
Learn the method: Number Sequence Puzzles: How to Find What Comes Next
