Mai V.

asked • 06/25/15

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?

 
 
I only know how to calculate the number of cubes for the specified pattern.
 
1+2×2+3×3+4×4+5×5

Mark M.

I see only Fig. 1 - 3.
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06/25/15

Mai V.

Yes, your eyesight is good, there are only 3 figures: Figure 1, Figure 2, Figure 3.
 
Figure 1 has 1 cube
Figure 2 has 5 cubes
Figure 3 has 14 cubes
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06/25/15

David W.

tutor
Use the formula to guess an N that, when cubed, then doubled, then divides by 6 equals the number.  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
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06/25/15

2 Answers By Expert Tutors

By:

David W. answered • 06/25/15

Tutor
4.7 (69)

Experienced Prof

Mai V.

hmm, what is the thingy thingy thing: ^
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06/25/15

David W.

tutor
The technical term for that symbol is "caret" (see Wikipedia) and it indicates exponentiation.  Thus, N^2 is N2.. 
 
It is one way to write expressions for a computer to evaluate (usually computers do not recognize superscripts or subscripts).  This is what Excel uses.
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06/25/15

David W.

tutor
Use the formula to guess an N that, when cubed, then doubled, then divides by 6 equals the number.  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
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06/25/15

Mai V.

I'm Primary School (or "Elementary School" in US), and this method seems like Secondary School (or "High School" in US). Any other methods easier for me?
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06/26/15

David W.

tutor
Answering questions on WyzAnt's forum is sometimes difficult (even though I often read student's other questions).  Also, tutors don't know the student, don't know what book is used, don't know what topic is being studied, etc.
 
I'm very impressed that you are studying this.  Math is a subject where information is provided in small portions and carefully built into larger and larger knowledge and skills (and thanks for asking about "^" by the way).  That's why missing class (sick or other reason) makes it so hard to catch up.
 
I can't think of a simpler formula or easier method, but I am glad that the "pyramid" pictures are introducing the sum of squares because later math will include it.  So, for now, guessing might mean adding a long list of squares.  With lots of repetitions, most students quickly learn all the squares of numbers from 1 to 20.  Note: this also helps when you need to find square roots in other problems.
 
After working hard using one method, a student is delighted to learn a better method (or shortcut).  My best illustration is the sum of the numbers from 1 to N.  The story is that, in Primary School, young Carl Friedrich Gauss found a formula for this sum instead of tediously adding all the terms.  The formula N*(N+1)/2 is very important in later math and I have used it in many, many cases in computer programs.
 
Keep asking questions!  That's how students learn and that's how teachers improve.
 
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06/26/15

Son N. answered • 06/25/15

Tutor
4.9 (86)

Math and Physics Tutor

Mai V.

In this community, I like your method because it's easy to understand. However, I dislike your method because this is Guess & Check.
 
If they ask for which figure has 9999 cubes, am I going to do one by one? No, definitely I won't, and I believe there'll be a fixed solution solution.
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06/25/15

David W.

tutor
Note:  Using the formula   N*(N+1)*(2*N+1)/6  (not explained here), you could guess a number that when cubed and the result doubled, then divided by 6 approximates 9999 (the equation resembles 2*N3/6).  Ah, that is 14, because (2*14^3) / 6 = 915  (this is the guess part).  And, checking, 14*15*29/6 = 1015.  The 14th term is 1015.
 
p.s., you may want to watch:   https://www.youtube.com/watch?v=3IX-ukfgZdk
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06/25/15

David W.

tutor
Use the formula  N*(N+1)*(2*N+1)/6  and try to guess an N that when cubed, doubled, then divided by 6 (the formula is approximately 2*N ^3/6) produces 9999.   Let's see, 2*14^3/6 = 915, so 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
Report

06/25/15

David W.

tutor
Use the formula to guess an N that, when cubed, then doubled, then divides 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
Report

06/25/15

Son N.

You make a good point here. There is a formula for the exact sum of the squared numbers, but I think the derivation is beyond the scope of your course and my expertise. 
 
If we let Sn be the sum of the square up to the nth term:
 
Sn = 1 + 22 + 32 + ... + n2
 
Then, the sum equates to:
 
Sn = n3/3 + n2/2 + n/6
 
So, for Sn = 204  you should that n = 8 works perfectly. However, finding the exact solution to n is a little difficult as this is a third degree polynomial. Since this equation clearly can't be factor, finding the exact solution for n is beyond my level of expertise. 
 
For cases where n is small the guess and check method is much easier to work out. Also, you shouldn't look down on the guess and check method. It's actually a very powerful method for solving problems in math and physics, but it helps to know how to guess.
 
I hope this helps.
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06/26/15

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