Growing Stairs
Read the puzzle
Tiles form a staircase. Step 1 is a single tile. Step 2 has columns of 1 and 2 tiles (3 tiles). Step 3 has columns of 1, 2 and 3 tiles (6 tiles). Each step adds a new column one tile taller than the last. 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 new step adds a column one tile taller than the last.
55
1 + 2 + 3 + ... + 10 = 55 tiles.
How to solve it
Look at what each step adds. Step 1 has a column of 1. Step 2 adds a column of 2. Step 3 adds a column of 3. So step n is made of columns 1, 2, 3 and so on up to n.
For step 10 the columns are 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 tiles tall. Add them in pairs from the outside in: 1 + 10 = 11, 2 + 9 = 11, 3 + 8 = 11, 4 + 7 = 11, 5 + 6 = 11. That is 5 pairs of 11, so 5 x 11 = 55 tiles.
As a formula, step n has n x (n + 1) / 2 tiles: 1, 3 and 6 for the steps shown, and 10 x 11 / 2 = 55 for step 10.
The usual wrong answer comes from treating the gaps as fixed. The jumps are 2 and then 3, and someone who keeps adding 3 gets 6 + 7 x 3 = 27.
Takeaway: when each step adds one more than the last, the totals are the triangular numbers.
Learn the method: Number Sequence Puzzles: How to Find What Comes Next
