For a display, identical cubic boxes are stacked in square layers. Each layer consists of cubic boxes arranged in rows

[BTGmoderatorDC]

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



OA E

Source: GMAT Prep

[Jay@ManhattanReview]

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

OA E

Source: GMAT Prep

Since it is given that the no. of cubes in the bottom row is 81, and they are arranged in a square, there are √81 = 9 rows.

Let's find out the no. of cubes in each layer.

• No. of cubes in the bottom/9th row = 81 = 9^2;
• No. of cubes in the 8th row = (9 – 1)*(9 – 1) = 8^2 = 64;
• No. of cubes in the 7th row = 7^2 = 49;
• No. of cubes in the 6th row = 6^2 = 36;
• No. of cubes in the 5th row = 5^2 = 25;
• No. of cubes in the 4th row = 4^2 = 16;
• No. of cubes in the 3rd row = 3^2 = 9;
• No. of cubes in the 2nd row = 2^2 = 4;
• No. of cubes in the 1st row = 1^2 = 1

Sum of all cubes = 81 + 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 285

Alternatively, we apply the formula for the sum of squares of the first n positive integers.

SUM = n(n+1)(2n + 1)/6; where n = no. of positive integers, here n = 9

Thus, SUM = 9*(9+1)(2*9 + 1)/6 = 285

The correct answer: E

Hope this helps!

-Jay
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[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



OA E

Source: GMAT Prep

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 the 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