GPREP PS- 1

[abhasjha]

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



can some one draw the diagramm and then make me understand ... how you find solution ?
Also any short cuts - (formula to solve this ) ???

[GMATinsight]

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



can some one draw the diagramm and then make me understand ... how you find solution ?
Also any short cuts - (formula to solve this ) ???

Since every row is a square therefore if the bottom layer has 81 squares i.e. 9^2 then the next row will have 8^2 [because every row has one lesser box]

Therefore the cubes in rows will be
81 = 9^2
64 = 8^2
49 = 7^2...
.....and so on
4 = 2^2
1 = 1^2

Sum of all Squares = 9^2 + 8^2 + 7^2 + .....+ 3^2 + 2^2 + 1^1

Sum of Squares of first n positive Consecutive Integers = (1/6)(n)(n+1)(2n+1)

Sum of Squares of first 9 positive Consecutive Integers = (1/6)(9)(9+1)(2x9+1) = 285

Answer: Option E

[GMATGuruNY]

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

The 81 cubes that form the bottom layer look as follows:


The next layer is formed by removing one entire row, along with one box from each of the remaining rows.
This is the equivalent of removing the pink shaded region in the figure above.
The result is the following figure:

In this layer, the number of cubes = 8² = 64.

To form the next layer, we must remove the blue shaded region.
The result in the following figure:

In this layer, the number of cubes = 7² = 49.

By now we should see the pattern.
81, 64 and 49 are consecutive PERFECT SQUARES.
Implication:
The total number of cubes is equal to the sum of the perfect squares between 81 and 1, inclusive:
81+64+49+36+25+16+9+4+1 = 285.

The correct answer is E.

[Scott@TargetTestPrep]

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



can some one draw the diagramm and then make me understand ... how you find solution ?
Also any short cuts - (formula to solve this ) ???

We see that the bottommost layer has 9 x 9 = 81 boxes and the topmost layer is 1 x 1 = 1 box. Therefore, each layer must be a perfect square number of boxes from 1 to 81, inclusive. That is, if we count the layers from top to bottom, they have 1, 4, 9, 16, 25, 36, 49, 64, and 81 boxes. Therefore, the total number of boxes is:

1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81

Instead of adding the actual numbers, let’s add their units digits:

1 + 4 + 9 + 6 + 5 + 6 + 9 + 4 + 1 = 5 + 15 + 5 + 15 + 5 = 45

We see the sum must have a units digit of 5; therefore, choice E is the correct answer.

Answer: E