考考你: $10哪去了? Where is $10?

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John borrows $500 from dad and $500 from mom. He uses $970 to buy a pair of shoes, balance $30.
He returns $10 to dad, $10 to mom, keeps $10 for himself.
So he owes $490 to dad, $490 to mom,
$490 + $490 = $980,
with his $10 = $990.

Where is the missing $10 (= $500 + $500 – $990) ?

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PUZZLE

SOLVE PUZZLE……

I am a 5 letter word.
I am normally below u
If u remove my 1st letter
u’ll find me above u
If u remove my 1st & 2nd letters, u cant see me

Answer ?????
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Answer: chair

500 Doors

Five hundreds closed doors along a corridor are numbered from 1 to 500.
A person walks through the corridor and opens each door.
A 2nd person walks through the corridor and closes every alternate door.

Continuing in this manner, the i-th person comes and toggles (opened becomes closed, vice-versa) the position of every i-th door starting from door i.

Question: Which of the 500 doors are open after the 500-th person has walked through.

[HINT]: Solving such abstract problem, it helps to visualize on small sample (eg. 10 doors) to find the pattern.

Initial doors: (All closed)
■■■■■■■■■■
After 1st person (open all doors)
□□□□□□□□□□

After 2nd person
□■ □ ■□ ■ □■ □ ■

After 3rd person (toggles all 3n doors)
□■[■]■□[□]□■[■]■

After 4th person (toggles 4n)
□■ ■[□]□□□[□]■■

After 5th person (toggles 5n)
□■ ■ □[■]□□□■[□]

After 6th person (toggles 6n)
□■ ■ □ ■[■]□□■□

After 7th person (toggles 7n)
□■ ■ □ ■ ■[■]□■□

After 8th person (toggles 8n)
□■ ■ □ ■ ■ ■[■]■□

After 9th person (toggles 9n)
□■ ■ □ ■ ■ ■ ■[□]□

After 10th person (toggles 10n)
□■ ■ □ ■ ■ ■ ■ □[■]
Notice the 3 open doors are: {1, 4, 9}.

Do they give you any clue ?

They are all perfect square:
1 = 1^{2}
4 = 2^{2}
9 = 3^{2}

For 20 doors, we can get the 4 open doors : {1, 4, 9, 16}.
16 = 4^{2}

Therefore,
for 500 doors there will be N open doors:
N = \sqrt {500} = 22.36
22^2 = 484

Answer: the open doors are:
{1, 4, 9, 16, 25, 36, 49, …, 484}

Proof:
Perfect square has odd number of divisors.
4 : {1, 2, 4} (odd) 3 divisors
9: {1, 3, 9} (odd) 3 divisors
16: {1, 2, 4, 8, 16} (odd) 5 divisors
but
8: {1, 2, 4, 8} (even) 4 divisors

For an open door, after even number of toggles will close it, but odd number of toggles will open it again, hence all doors of perfect square will be open.

Phone Number Tells Your Age

(1)Take the last digit number of your telephone number

(2)Multiply it by 2

(3)Add 5

(4)Multiply by 50

(5)Add 1763

[Note: Next year (2014) you add 1764, and so on …]

(6)Last step: subtract your year of birth

Now you should get a 3-digit number

The first digit is your number in (1), followed by next 2 digits give your age.

Prof Su Buqing Problem

Prof Su 苏步青, the founding pioneer Math professor of the China’s top universities (Zhejiang 浙江大学 and Fudan 复旦大学), was one of the few mathematicians who had longevity above 100 years old (the other was French Mathematician Hadammard).

http://en.m.wikipedia.org/wiki/Su_Buqing

Two men A and B are 100 km apart, walking towards each other, A at speed 6 km/hour and B at 4 km/hour.
A brings a dog which runs at 10 km/hour between them,  starting from A towards B, upon reaching B it runs back to reach A, then back to B again, and so on…

Find total distance the dog has covered when A and B finally meet ?