Cube tower

Geometry Level pending

I glue ten cubes together, one on top of the other, to make a tower in which the top cube has side length 1, the second cube has side length 2, the third cube has side length 3 and so on until the bottom cube has side length 10. What is the surface area of the resulting 3D shape? (Assume that the when two cubes are glued together they meet at a face and the smaller cube is directly on top of the larger cube.)


The answer is 1740.

Melissa Quail by Melissa Quail , 21, United Kingdom

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

3 solutions

It's easy! 1st find area of individual cubes 6 (1^2 +'2^2 + 3^2+...+10^2) Now deduct the area which will be mutually trapped : square faces from 1 to 9th cube. (1^2 + 2^2 + 3^2+...+9^2) *Tip: sum of square of 1st n natural numbers is S = [n × (n+1) × (2n+1)]/6 Cheers!

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Yeah this is exactly how I worked the answer out except you need to take off the trapped areas twice because they are trapped from two cubes so they have been counted twice. Brock Brown's code was very good though and would be very useful if we were dealing with large numbers of cubes. By the way, my phone just updated Brilliant and now I can't work out how to see my profile or post anything else. Could you tell me how to do this on the new update? Thanks!

Melissa Quail - 6 years, 5 months ago

Log in to reply

actually i solve problems through browser, so i don't know about it. and to the point subtracting the area twice; i meant that mutually trapped areas of 2 faces of cube since both would be covered while placing one above the other.

Agastya Chandrakant - 6 years, 5 months ago

Log in to reply

OK thanks anyway. I have found what I was looking for now so I can carry on posting problems :-)

Melissa Quail - 6 years, 5 months ago
Saya Suka
May 21, 2019

Answer.
= (4 surrounding areas)x(10x11x21/6) + (2 top n bottom faces)x10^2.
= 1540 + 200.
= 1740.
Got it wrong the first time around, on the assumption that the bottom face is excluded, since the tower would be placed upright on something so it won't be seen.



0 Helpful 0 Interesting 0 Brilliant 0 Confused
Brock Brown
Dec 29, 2014

This code ought to solve the surface area of this kind of structure for any amount of cubes: 

1
2
3
4
5
6
7
 
blocks = 10
area = blocks**2
i = 1
while i <= blocks:
    area += (i**2-(i-1)**2) + i**2*4
    i += 1
print "Answer:", area

Or you could use the following summation, which represents what the program does:

f(n) = n 2 n^{2} + s u m i = 1 n sum_{i=1}^{n} i 2 i^{2} - ( i − 1 ) 2 (i - 1)^{2} + 4 i 2 i^{2}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...