### Author Topic: GLAT - Google Labs Aptitude Test  (Read 60684 times)

#### fzy

• Hero Member
• Posts: 520
##### GLAT - Google Labs Aptitude Test
« on: 十一月 18, 2004, 10:53:06 am »
Has anybody here given serious thoughts about these questions in the Google Test? Some are jokes. But how about the rest?

Here are the 21 questions:

1. Solve this cryptic equation, realizing of course that values for M and E could be interchanged. No leading zeros are allowed.

2. Write a haiku describing possible methods for predicting search traffic seasonality.

3.
1
1 1
2 1
1 2 1 1
1 1 1 2 2 1

What's the next line?

4. You are in a maze of twisty little passages, all alike.  There is a dusty laptop here with a weak wireless connection.  There are dull, lifeless gnomes strolling around.  What dost thou do?

A) Wander aimlessly, bumping into obstacles until you are eaten by a grue.
B) Use the laptop as a digging device to tunnel to the next level.
C) Play MPoRPG until the battery dies along with your hopes.
D) Use the computer to map the nodes of the maze and discover an exit path.
E) Email your resume to Google, tell the lead gnome you quit and find yourself in whole different world.

5. What's broken with Unix?
How would you fix it?

6. On your first day at Google, you discover that your cubicle mate wrote the textbook you used as a primary resource in your first year of graduate school. Do you:

A) Fawn obsequiously and ask if you can have an autograph.
B) Sit perfectly still and use only soft keystrokes to avoid disturbing her concentration
C) Leave her daily offerings of granola and English toffee from the food bins.
D) Quote your favorite formula from the textbook and explain how it's now your mantra.
E) Show her how example 17b could have been solved with 34 fewer lines of code.

7. Which of the following expresses Google's over-arching philosophy?

A) "I'm feeling lucky"
B) "Don't be evil"
C) "Oh, I already fixed that"
D) "You should never be more than 50 feet from food"
E) All of the above

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

9. This space left intentionally blank.  Please fill it with something that improves upon emptiness.

10. On an infinite, two-dimensional, rectangular lattice of 1-ohm resistors, what is the resistance between two nodes that are a knight's move away?

11. It's 2PM on a sunny Sunday afternoon in the Bay Area.  You're minutes from the Pacific Ocean, redwood forest hiking trails and world class cultural attractions.  What do you do?

12. In your opinion, what is the most beautiful math equation ever derived?

13. Which of the following is NOT an actual interest group formed by Google employees?

B. Buffy fans
C. Cricketeers
D. Nobel winners
E. Wine club

14. What will be the next great improvement in search technology?

15. What is the optimal size of a project team, above which additional members do not contribute productivity equivalent to the percentage increase in the staff size?

16. Given a triangle ABC, how would you use only a compass and straight edge to find a point P such that triangles ABP, ACP and BCP have equal perimeters?  (Assume that ABC is constructed so that a solution does exist.)

17. Consider a function which, for a given whole number n, returns the number of ones required when writing out all numbers between 0 and n.  For example, f(13)=6.  Notice that f(1)=1.  What is the next largest n such that f(n)=n?

18.  What is the coolest hack you've ever written?

19. 'Tis known in refined company, that choosing K things out of N can be done in ways as many as choosing N minus K from N: I pick K, you the remaining.
Find though a cooler bijection, where you show a knack uncanny, of making your choices contain all K of mine.  Oh, for pedantry: let K be no more than half N.

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

21. In 29 words or fewer, describe what you would strive to accomplish if you worked at Google Labs.

#### 万精油

• Hero Member
• Posts: 1831
##### GLAT - Google Labs Aptitude Test
« Reply #1 on: 十一月 18, 2004, 11:23:51 am »
Quote
12. In your opinion, what is the most beautiful math equation ever derived?

The common agreement for this one is the equation:

This simple equation contains the most important five numbers in mathematics:

0, 1, e, pi, i.

#### idiot94

• Sr. Member
• Posts: 484
##### Re: GLAT - Google Labs Aptitude Test
« Reply #2 on: 十一月 18, 2004, 01:11:31 pm »
Some are jokes. But how about the rest?

Really?

Here are the 21 questions:

1. Solve this cryptic equation, realizing of course that values for M and E could be interchanged. No leading zeros are allowed.

(www. = idiot)

2. Write a haiku describing possible methods for predicting search traffic seasonality.
If (thisspring > spring_traffic_level_history) spring_traffic_level_history++
If (thissummer> summer_traffic_level_history) summer ....

You got the idea, ponk!

3.
1
1 1
2 1
1 2 1 1
1 1 1 2 2 1

What's the next line?

2433545645342

4. You are in a maze of twisty little passages, all alike.  There is a dusty laptop here with a weak wireless connection.  There are dull, lifeless gnomes strolling around.  What dost thou do?

A) Wander aimlessly, bumping into obstacles until you are eaten by a grue.
B) Use the laptop as a digging device to tunnel to the next level.
C) Play MPoRPG until the battery dies along with your hopes.
D) Use the computer to map the nodes of the maze and discover an exit path.
E) Email your resume to Google, tell the lead gnome you quit and find yourself in whole different world.

I love this one, I guess you would like the answer of E. But replace that google with something cooler for ME!

5. What's broken with Unix?
How would you fix it?
My mahince was broken with Unix.. I will buy a new one from Ebay

6. On your first day at Google, you discover that your cubicle mate wrote the textbook you used as a primary resource in your first year of graduate school. Do you:

A) Fawn obsequiously and ask if you can have an autograph.
B) Sit perfectly still and use only soft keystrokes to avoid disturbing her concentration
C) Leave her daily offerings of granola and English toffee from the food bins.
D) Quote your favorite formula from the textbook and explain how it's now your mantra.
E) Show her how example 17b could have been solved with 34 fewer lines of code.

Khmm.. excuse me, is she cute?

7. Which of the following expresses Google's over-arching philosophy?

A) "I'm feeling lucky"
B) "Don't be evil"
C) "Oh, I already fixed that"
D) "You should never be more than 50 feet from food"
E) All of the above

E

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

What is an icosahaerdf.,d.agdagfs?

9. This space left intentionally blank.  Please fill it with something that improves upon emptiness.

zzzzz ZZZZZZZ

10. On an infinite, two-dimensional, rectangular lattice of 1-ohm resistors, what is the resistance between two nodes that are a knight's move away?

Hah .... isn't this a physics competition problem for kids? You gotta be kidding me, dude!

11. It's 2PM on a sunny Sunday afternoon in the Bay Area.  You're minutes from the Pacific Ocean, redwood forest hiking trails and world class cultural attractions.  What do you do?

I will stay home and play Diablo II on US west Realm and beat the bastard Baal.

12. In your opinion, what is the most beautiful math equation ever derived?

1+2=3? or 3x7=21?

13. Which of the following is NOT an actual interest group formed by Google employees?

B. Buffy fans
C. Cricketeers
D. Nobel winners
E. Wine club

A, geeks do not play basketball

14. What will be the next great improvement in search technology?

Personality match.
()

15. What is the optimal size of a project team, above which additional members do not contribute productivity equivalent to the percentage increase in the staff size?

3. Two cutties and me.

16. Given a triangle ABC, how would you use only a compass and straight edge to find a point P such that triangles ABP, ACP and BCP have equal perimeters?  (Assume that ABC is constructed so that a solution does exist.)

oh man, you got me here!

17. Consider a function which, for a given whole number n, returns the number of ones required when writing out all numbers between 0 and n.  For example, f(13)=6.  Notice that f(1)=1.  What is the next largest n such that f(n)=n?

hmm... I gotta go home soon.

18.  What is the coolest hack you've ever written?

I opened my machine box and clean the fan of CPU last night.... it is much COOLER now ...

19. 'Tis known in refined company, that choosing K things out of N can be done in ways as many as choosing N minus K from N: I pick K, you the remaining.
Find though a cooler bijection, where you show a knack uncanny, of making your choices contain all K of mine.  Oh, for pedantry: let K be no more than half N.

#\$*#(*(^%

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

LoL ... I like B

21. In 29 words or fewer, describe what you would strive to accomplish if you worked at Google Labs.

29 words? You kidding me, no? 1 word: money!
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

#### 万精油

• Hero Member
• Posts: 1831
##### GLAT - Google Labs Aptitude Test
« Reply #3 on: 十一月 18, 2004, 01:48:27 pm »
Quote
17. Consider a function which, for a given whole number n, returns the number of ones required when writing out all numbers between 0 and n. For example, f(13)=6. Notice that f(1)=1. What is the next largest n such that f(n)=n?

This was a problem in my Guofeng puzzle column many years ago, and here's an answer given by Yihua (who's also on this site with a different name

---------------------------------------------------

上 期 的 问 题 很 能 检 验 一 个 人 做 学 问 的 能 力 。 它 需 要 有 推 理 ， 有 耐 心 ， 还 要 仔 细 。 总 共 有 三 个 人 答 对 。 其 中 一 华 的 答 案 最 全 面 。 所 以 我 就 选 用 他 的 答 案 贴 在 这 里 ， 我 只 是 做 一 些 小 改 动 ， 省 了 我 不 少 事 。 事 实 上 ， 如 果 真 让 我 来 写 答 案 ， 不 见 得 能 写 得 这 么 清 楚 。

下 面 是 一 华 的 答 案 ：

设 我 们 数 的 数 为 Ｎ ， 数 到 Ｎ 我 们 共 写 了 Ｓ 个 １ 。

首 先 找 出 一 个 规 律 ：

当 Ｎ 等 于 　 １ ０ ＾ ｋ 　 时 ， Ｓ 等 于 　 ｋ ★ １ ０ ＾ （ ｋ － １ ） ＋ １ 。 这 一 点 很 容 易 证 明 ， 如 果 把 ０ 也 考 虑 进 去 ， 小 于 １ ０ ＾ ｋ 的 数 共 有 　 ｋ ★ １ ０ ＾ ｋ 　 个 位 置 ， 这 些 位 置 由 ０ 到 ９ 这 十 个 数 字 来 分 ， 每 个 数 字 分 到 十 分 之 一 ， 也 就 是 　 ｋ ★ １ ０ ＾ （ ｋ － １ ） 。

满 足 　 Ｎ ＝ Ｓ 　 的 第 一 个 数 是 １ ， 第 二 个 是 １ ９ ９ ９ ８ １ ， １ ９ ９ ９ ８ ２ － １ ９ ９ ９ ９ ０ 也 都 是 。

易 证 ， 当 Ｎ ＞ ＝ １ ０ ＾ １ ０ 时 ， Ｓ 将 始 终 大 于 Ｎ ， 即 不 可 能 满 足 Ｎ ＝ Ｓ 。 这 说 明 满 足 　 Ｎ ＝ Ｓ 　 的 最 大 数 Ｎ 是 存 在 的 。

显 然 有 理 由 推 测 Ｎ 在 ９ ９ ９ ９ ９ ９ ９ ９ ９ 与 ９ ９ ９ ９ ９ ９ ９ ９ ９ ９ 之 间 。

而 从 ９ ９ ９ ９ ９ ９ ９ ９ ９ ９ 往 下 减 ９ ， Ｓ 则 减 １ ；
减 ９ ９ ， Ｓ 减 ２ ０ 。
Ｎ 减 到 １ ９ ９ ９ ９ ９ ９ ９ ９ ９ 时 ，
Ｓ ＝ １ ０ ０ ０ ０ ０ ０ ０ ０ ０ ０ － ８ ★ ９ ０ ０ ０ ０ ０ ０ ０ ０
＝ ２ ８ ０ ０ ０ ０ ０ ０ ０ ０ ；

而 ２ ０ ０ ０ ０ ０ ０ ０ ０ ０ － ９ ９ ９ ９ ９ ９ ９ ９ ９ ９ 区 间 里 Ｓ 也 始 终 大 于 Ｎ ， 这 样 我 们 把 上 限 下 移 到 １ ９ ９ ９ ９ ９ ９ ９ ９ ９ 。

当 Ｎ ＝ ９ ９ ９ ９ ９ ９ ９ ９ ９ 时 ， Ｓ ＝ ９ ０ ０ ０ ０ ０ ０ ０ ０ ， 即 Ｓ ＜ Ｎ 。 因 此 在 此 区 间 有 可 能 找 到 Ｎ 。 当 然 也 可 能 出 现 这 种 情 况 ： 当 Ｎ ＝ Ａ 时 ， Ｎ ＞ Ｓ ； 而 当 Ｎ ＝ Ａ ＋ １ 时 ， Ｎ ＜ Ｓ 。 如 果 这 样 的 话 ， 我 们 就 只 得 到 下 面 的 区 间 找 Ｎ 了 。 所 幸 的 是 在 此 区 间 找 到 了 。

当 Ｎ ＝ １ １ １ １ １ １ １ １ １ ０ 时 ，

Ｓ ＝ （ ９ ０ ０ ０ ０ ０ ０ ０ ０ 　 ＋ 　 　 　 　 　 　 　 　 　 １ ） ＋
（ １ ０ ０ ０ ０ ０ ０ ０ ０ 　 ＋ ８ ０ ０ ０ ０ ０ ０ ０ ＋ １ ） ＋
（ ２ ★ １ ０ ０ ０ ０ ０ ０ ０ ＋ ７ ０ ０ ０ ０ ０ ０ 　 ＋ １ ） ＋
（ ３ ★ １ ０ ０ ０ ０ ０ ０ 　 ＋ ６ ０ ０ ０ ０ ０ 　 　 ＋ １ ） ＋
（ ４ ★ １ ０ ０ ０ ０ ０ 　 　 ＋ ５ ０ ０ ０ ０ 　 　 　 ＋ １ ） ＋
（ ５ ★ １ ０ ０ ０ ０ 　 　 　 ＋ ４ ０ ０ ０ 　 　 　 　 ＋ １ ） ＋
（ ６ ★ １ ０ ０ ０ 　 　 　 　 ＋ ３ ０ ０ 　 　 　 　 　 ＋ １ ） ＋
（ ７ ★ １ ０ ０ 　 　 　 　 　 ＋ ２ ０ 　 　 　 　 　 　 ＋ １ ） ＋
（ ８ ★ １ ０ 　 　 　 　 　 　 ＋ １ 　 　 　 　 　 　 　 ＋ １ ）
＝ 　 １ １ １ １ １ １ １ １ １ ０

即 Ｎ ＝ Ｓ 。 当 然 找 的 过 程 没 那 么 容 易 ， 要 从 高 位 一 位 位 往 下 推 ， 有 一 定 的 计 算 量 。 当 推 到 前 四 位 都 是 １ 时 ， 我 猜 测 全 是 １ ， 一 算 ， Ｓ 大 了 ９ ， 把 个 位 改 成 ０ ， Ｎ 少 １ ， Ｓ 少 １ ０ ， 正 好 。 这 就 减 少 了 一 些 计 算 量 。

结 论 ： Ｎ ＝ １ １ １ １ １ １ １ １ １ ０ 。

上 面 这 些 推 论 虽 然 写 得 很 仔 细 ， 但 要 想 看 懂 ， 还 得 要 静 下 心 来 自 己 慢 慢 推 。 否 则 一 看 这 些 ９ 头 就 大 了 。 当 你 一 步 一 步 推 出 Ｎ ＝ Ｓ 时 ， 我 想 你 所 感 受 到 的 欣 慰 将 是 很 大 的 。

#### fzy

• Hero Member
• Posts: 520
##### GLAT - Google Labs Aptitude Test
« Reply #4 on: 十一月 24, 2004, 10:38:09 am »
Among the 21 questions, 8 are math or math related: Nos 1, 3, 8, 10, 12, 16, 17, and 20. 1 and 17 are fairly easy. (The professor provided more than enough for 17 here.) 3 and 20 are joking problems and you really cannot take them seriously. 12 is, as is said, "In your opinion". The rest, 8, 10, and 16, are very difficult. (I could not do any one of them.   ) Any body wants to give a try?

#### mundane

• Newbie
• Posts: 4
##### Re: GLAT - Google Labs Aptitude Test
« Reply #5 on: 十一月 24, 2004, 12:58:18 pm »
3.
1
1 1
2 1
1 2 1 1
1 1 1 2 2 1

What's the next line?

312211，hehe
Nobody is perfect.
I am nobody.

#### Anonymous

• Guest
##### Re: GLAT - Google Labs Aptitude Test
« Reply #6 on: 十一月 24, 2004, 07:49:13 pm »
Quote from: mundane
3.
1
1 1
2 1
1 2 1 1
1 1 1 2 2 1

What's the next line?

312211，hehe

Any number is OK here...

#### 万精油

• Hero Member
• Posts: 1831
##### GLAT - Google Labs Aptitude Test
« Reply #7 on: 十一月 24, 2004, 08:00:21 pm »
Quote
Any number is OK here

Generally speaking, I don't like this kind of "what's next?" questions, since we can always give an arbitary number and find a reason for it. That's why I didn't pay attention to it. However, after seeing the answer, 312211, I realized what the sequence was, and it brought quite a laugh. As idiot94 always say, LOL. For this particular example, I find it very amusing, actually, this is one of the fews that I like.   The next number would be

13112221

I believe if you see what the sequence is, you will find it funny too.

I think it is something we can give to small children.

#### idiot95

• Guest
##### GLAT - Google Labs Aptitude Test
« Reply #8 on: 十一月 25, 2004, 03:16:15 pm »
the next next would be:
1113213211

#### Anonymous

• Guest
##### GLAT - Google Labs Aptitude Test
« Reply #9 on: 十一月 25, 2004, 07:27:43 pm »
Quote from: 万精油
Quote
Any number is OK here

Generally speaking, I don't like this kind of "what's next?" questions, since we can always give an arbitary number and find a reason for it. That's why I didn't pay attention to it. However, after seeing the answer, 312211, I realized what the sequence was, and it brought quite a laugh. As idiot94 always say, LOL. For this particular example, I find it very amusing, actually, this is one of the fews that I like.   The next number would be

13112221

I believe if you see what the sequence is, you will find it funny too.

I think it is something we can give to small children.

I knew what the sequence is meant to be.  Yes, it is only a problem for children.

There are quite a lot of people who believe that the answer for a what's next problem is unique...

#### idiot95

• Guest
##### GLAT - Google Labs Aptitude Test
« Reply #10 on: 十一月 26, 2004, 10:45:02 am »
hehe.. LOL again, can we go further on this "silly" "what is next" problem?

The question is, according to this rule, is there any finite cirlce? i.e.,
a1 -> a2 -> a3 ... ->a1 again?

22
Then, this is a trivial circle, every next sequence will be 22

But the question is, is there any other (in particular, non-trivial) circles?

(I guess this is Prof. W's favorite type of questions .. :p )

(OH, btw, happy thanksgiving everyone ..

#### froid

• Newbie
• Posts: 16
##### GLAT - Google Labs Aptitude Test
« Reply #11 on: 十一月 26, 2004, 03:51:54 pm »
Quote from: idiot95
hehe.. LOL again, can we go further on this "silly" "what is next" problem?

The question is, according to this rule, is there any finite cirlce? i.e.,
a1 -> a2 -> a3 ... ->a1 again?

22
Then, this is a trivial circle, every next sequence will be 22

But the question is, is there any other (in particular, non-trivial) circles?

(I guess this is Prof. W's favorite type of questions .. :p )

(OH, btw, happy thanksgiving everyone ..

There is no other circle. And for any starting string x, note s(x) its transformation, if x is not 22, then
length(s^(n+1)(x))/length(s^n(x)) -> 1.303577.... when n->infinity

That's infamous Conway's cosmological theorm and the constant called Conway's constant. There is a very beautiful theory behind this "silly" sequence.

#### 万精油

• Hero Member
• Posts: 1831
##### GLAT - Google Labs Aptitude Test
« Reply #12 on: 十一月 26, 2004, 04:48:59 pm »

The weird and wonderful chemistry of audioactive decay -- Alex Kontorovich, March 23, 2004

In 1987, Prof. John Conway wrote on the progression of the so-called Look-And-Say sequence: 1, 11, 21, 1211, 111221, 312211,... (Can you guess the next entry?) He found that the sequence decomposed into certain recurring strings. Categorizing these 92 strings and labeling them by the atoms of the periodic table (from Hydrogen to Uranium), Conway was able to prove that the asymptotic length of the sequence grows exponentially, where the growth factor (now known as Conway's Constant) is found by computing the largest eigenvalue of a 92x92 transition matrix. Even more remarkable is the Cosmological Theorem, which states that regardless of the starting string, every Look-And-Say sequence will eventually decay into a compound of these 92 atoms, in a bounded number of steps. Conway writes that, although two independent proofs of the Cosmological Theorem were verified, they were lost in writing! It wasn't until a decade later that Doron Zeilberger's paper (coauthored with his computer, Shalosh B. Ekhad) gave a tangible proof of the theorem. We will discuss this weird and wonderful chemistry, and some philosophical consequences. The only prerequisite is basic linear algebra.

#### idiot95

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##### GLAT - Google Labs Aptitude Test
« Reply #13 on: 十一月 26, 2004, 04:57:17 pm »
holy kitten ...!

Maybe the Universe isn't so far away from my pizza? :p

#### fzy

• Hero Member
• Posts: 520
##### Re: GLAT - Google Labs Aptitude Test
« Reply #14 on: 十二月 02, 2004, 12:12:33 pm »
I promised I94 (or I95, how old are you, anyway?) that I will post answers. So I start now.

Quote

1. Solve this cryptic equation, realizing of course that values for M and E could be interchanged. No leading zeros are allowed.

I guess no body want to spend time on a silly question like this. The answer is 777589 - 188106 == 589483 or 777589 - 188103 == 589486.

Quote

3.
1
1 1
2 1
1 2 1 1
1 1 1 2 2 1

What's the next line?

This is the look and say sequence. Every line is obtained by reading out the line before it.

Quote

16. Given a triangle ABC, how would you use only a compass and straight edge to find a point P such that triangles ABP, ACP and BCP have equal perimeters? (Assume that ABC is constructed so that a solution does exist.)

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.