### Author Topic: GLAT - Google Labs Aptitude Test  (Read 54213 times)

#### JeffG

• Guest
##### GLAT - Google Labs Aptitude Test
« Reply #15 on: 十二月 02, 2004, 12:42:16 pm »
Quote
Construct three circles centered at A, B, C, with radius b+c-a, a+c-b, and a+b-c.

Should it be: Construct three circles centered at A, B, C, with radius (b+c-a)/2, (a+c-b)/2, and (a+b-c)/2 ?

#### idiot94

• Sr. Member
• Posts: 484
##### Re: GLAT - Google Labs Aptitude Test
« Reply #16 on: 十二月 02, 2004, 12:45:08 pm »
Quote from: fzy
I promised I94 (or I95, how old are you, anyway?) that I will post answers. So I start now.

I am not that old lol (or that young... you know what I am saying )

I am in mid 30.

Quote from: fzy

This is what I got from the net:

Construct three circles centered at A, B, C, with radius b+c-a, a+c-b, and a+b-c. Then construct the outer tangent circle of the three. Its center is the isoperimetric point, and satisfies the requirement. For details, see http://mathworld.wolfram.com/IsoperimetricPoint.html and http://www.ajur.uni.edu/v3n1/Gisch%20and%20Ribando.pdf.

Other answers will follow.

Thanks a lot. Jesus, no wonder I cannot figure it out... Even with this solution, it will take me a while to understand why it is the right point. Further, to draw the outer tangent cirlce is a very complicated trick as well, which I failed to figure out on my own either. (that was almost 15 years ago, but later I found the answer on a book ... the ancient greek answer. )
In general, the men of lower intelligence won out. Afraid of of their own shortcomings ... they boldly moved into action. Their enemies, ...  thought there was no need to take by action what they could win by their brains. Thucydides, History

#### fzy

• Hero Member
• Posts: 520
##### GLAT - Google Labs Aptitude Test
« Reply #17 on: 十二月 02, 2004, 12:58:43 pm »
Quote from: JeffG

Should it be: Construct three circles centered at A, B, C, with radius (b+c-a)/2, (a+c-b)/2, and (a+b-c)/2 ?

Yes. Or "with diameters b+c-a, a+c-b, and a+b-c". Thanks for the correction.

#### fzy

• Hero Member
• Posts: 520
##### Re: GLAT - Google Labs Aptitude Test
« Reply #18 on: 十二月 07, 2004, 01:57:01 pm »
Quote

20. What number comes next in the sequence: 10, 9, 60, 90, 70, 66, ?

A) 96
B) 1000000000000000000000000000000000
0000000000000000000000000000000000
000000000000000000000000000000000
C) Either of the above
D) None of the above

This problem is harder for us than native English speakers. We can see the sequence better if we write it as "ten, nine, sixty, ninety, seventy, sixty six". We see that it is defined by a(n) = {largest number whose name has n + 2 letters}. So the answer is ninety six. The problem is that the question itself is wrong: The fourth number is obviously "google" rather than ninety. (Can't believe these guys forgot their own name!   )

#### idiot94

• Sr. Member
• Posts: 484
##### GLAT - Google Labs Aptitude Test
« Reply #19 on: 十二月 07, 2004, 03:29:45 pm »
lol ... google .... I like that one
In general, the men of lower intelligence won out. Afraid of of their own shortcomings ... they boldly moved into action. Their enemies, ...  thought there was no need to take by action what they could win by their brains. Thucydides, History

#### Dr Kevin Wang

• Full Member
• Posts: 129
##### Re: GLAT - Google Labs Aptitude Test
« Reply #20 on: 十二月 13, 2004, 06:35:38 pm »
Quote from: fzy
Quote

20. What number comes next in the sequence: 10, 9, 60, 90, 70, 66, ?

A) 96
B) 1000000000000000000000000000000000
0000000000000000000000000000000000
000000000000000000000000000000000
C) Either of the above
D) None of the above

This problem is harder for us than native English speakers. We can see the sequence better if we write it as "ten, nine, sixty, ninety, seventy, sixty six". We see that it is defined by a(n) = {largest number whose name has n + 2 letters}. So the answer is ninety six. The problem is that the question itself is wrong: The fourth number is obviously "google" rather than ninety. (Can't believe these guys forgot their own name!   )

The number is "googol", which is answer (B).  The observation stands that the fourth number should be googol.

#### fzy

• Hero Member
• Posts: 520
##### Re: GLAT - Google Labs Aptitude Test
« Reply #21 on: 十二月 14, 2004, 10:16:15 am »
Quote from: SiYue

The number is "googol", which is answer (B).  The observation stands that the fourth number should be googol.

I know the number is spelled as googol. They changed it to google (and googleplex, for their headerquarter compound) to avoid potential copy right problems. But we are entitled to have some fun at them, after what they did to us, right?

#### fzy

• Hero Member
• Posts: 520
##### Re: GLAT - Google Labs Aptitude Test
« Reply #22 on: 一月 04, 2005, 12:17:58 pm »
Quote

8. How many different ways can you color an icosahedron with one of three colors on each face?

It turns out that every regular polyhedron has a coloring polynomial that can be used to calculate the number of different coloring. They are:

tetrahedron: (11/12)*n^2 + (1/12)*n^4
cube: (1/3)*n^2 + (1/2)*n^3 + (1/8 )*n^4 + (1/24)*n^6
... ...
icosahedron: (2/5)*n^4 + (1/3)*n^8 + (1/4)*n^10 + (1/60)*n^20

So for icosahedrons (20 faced regular polyhedron) and three colors, the answer is 58130055.

These polynomials can be proved using the Polya Enumeration Theorem, which I heard of while in college, but never bothered to find out what it is about.

For a little more detail, see http://mathworld.wolfram.com/PolyhedronColoring.html