Question

For a display, identical cubic boxes are stacked in square layers. Each layer consists of cubic boxes arranged in rows that form a square, and each layer has 1 fewer row and 1 fewer box in each remaining row than the layer directly below it. If the bottom of the layer has 81 boxes and the top of the layer has only 1 box, how many boxes are in display?

A. 236
B. 260
C. 269
D. 276
E. 285

Answer 1

E. 285

Explanation:

The display forms a pyramid with each layer having a square based with one fewer row and one fewer column than the previous layer.

Since the first layer has 81 boxes, it is a 9x9 square. The next layer is an 8x8 square, followed by a 7x7 and so on.

Therefore, the total number of boxes is:

`N = (9*9)+(8*8)+(7*7)+(6*6)+(5*5)+(4*4) +(3*3)+(2*2) +1\\N=81 + 64+49+36+25+9+4+1\\N=285`

There are 285 boxes in display.