Count Every Cube
Read the puzzle
A picture shows a stepped stack of identical cubes on a 4 by 4 floor grid. Every cube rests on the floor or on another cube, so there are no hidden gaps. Read as column heights, the back row is 4, 3, 3 and 1 cubes tall, the next row 3, 3, 2 and 1, the third row 2, 2, 1 and 0, and the front row 1, 1, 0 and 0. How many cubes are in the stack?
How many cubes are in the stack?
Count column by column, not cube by cube.
27
Column heights, row by row: 4+3+3+1 = 11, 3+3+2+1 = 9, 2+2+1 = 5, 1+1 = 2. Total 27, hidden cubes included.
How to solve it
Count column by column instead of cube by cube. Every visible top face sits on a column that reaches down to the floor, and the puzzle promises there are no gaps, so a column's height tells you exactly how many cubes it holds.
Read the heights row by row, starting at the back where the single top cube stands: 4 + 3 + 3 + 1 = 11. The next row: 3 + 3 + 2 + 1 = 9. The third row: 2 + 2 + 1 = 5. The front row: 1 + 1 = 2. The three empty floor squares at the front corner add nothing.
Add the rows: 11 + 9 + 5 + 2 = 27 cubes.
The usual wrong answer counts only the cubes you can see, which comes out far smaller, because the cubes holding up the higher steps are hidden behind and beneath the front ones. Quick check: the stack has 13 filled columns, and 27 cubes means an average height of just over 2, which fits a stack that is 4 high at the back and 1 high at the front.
Learn the method: Cube Counting Puzzles: How to Count Every Cube in a Stack
