The figures below show solid figures formed with 1-cm cubes. (a) How many 1-cm cubes are needed to build Figure 5? (b) Which figure is made up of 204 cubes?

Question

Figure: http://imgur.com/L7O22VF

 

 

I only know how to calculate the number of cubes for the specified pattern.

 

1+2×2+3×3+4×4+5×5

Answer 1

First, mathematicians usually write it backwards:

         Figure N has     N^2 + (N-1)^2 + … + 1    cubes

So, figure 5 has   5^2 + 4^2 + 3^2 + 2^2 + 1^2   =  25 + 16 + 9 + 4 + 1 = 55 cubes

Now, the sum of the squares of the numbers from 1 to n is

    N^2  + (N-1)^2 + (N-2)^2 + … + 1^2   = N*(N+1)*(2*N+1)/6   ( proof not given here)

So, to find the N for which there are 204 cubes:

       204 = N*(N+1)*(2*N+1)/6  

(again, not solving a cubic equation),

    When N=8     8*9*17 / 6 = 204  

 

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Use the formula to guess an N that, when cubed, then doubled, then divided by 6 equals 9999.  Let's guess 14, because 2*14^3/6 ≈ 915.   Let's check:   14*15*29/6 = 1015.    The 14th term is 1015.
 
p.s., you might want to watch:      https://www.youtube.com/watch?v=3IX-ukfgZdk

Answer 2

Ok, so if I assume that the pattern goes on, figure 4 has 16 cubes on the bottom level and figure 5 has 25 cubes on the bottom level. 

 

Then, the formula your have for the number of cubes in figure 5 is correct, and that will give you the answer to part a of this problem. Now you want to find which figure contains 204 cubes, so just continue doing what you did for figure 5.

 

for example:

Figure 6 will have 1 + 2² + 3² + 4² + 5² + 6² cubes. This equals 91

 

Now that we've established a pattern you don't need to do every single calculation. We know that the next figure (figure 7) will just have an additional 7² cubes, so we simply need to add 49 to our previous sum, 91.

 

Then:

Figure 7: 140

Figure 8: 140 + 8² = 140 + 64 = 204

 

And we have our answer for part b.

 

Figure 8 will contain 204 cubes