At a party, everyone shook hands with everybody else. There were 66 handshakes. How many people were at the party?
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Answer:

With two people (A and B), there is one handshake
   (A with B).
With three people (A, B, and C), there are three handshakes 
  (A with B and C;   B with C).
With four people (A, B, C, and D), there are six handshakes 
  (A with B, C, and D;   B with C and D;   C with D).
In general, with n+1 people, the number of handshakes is the sum of the first n consecutive numbers: 1+2+3+ ... + n.
Since this sum is n(n+1)/2, we need to solve the equation n(n+1)/2 = 66.
This is the quadratic equation n2+ n -132 = 0. Solving for n, we obtain 11 as the answer and deduce that there were 12 people at the party.
Since 66 is a relatively small number, you can also solve this problem with a hand calculator. Add 1 + 2 = + 3 = +... etc. until the total is 66. The last number that you entered (11) is n.
Carl Friedrich Gauss (1777-1855) is credited with finding the formula for computing the sum of the first n consecutive numbers when he was an elementary school student, at age 8. The teacher had asked the students to compute the sum (S) of the first 100 integers. To the teacher's astonishment, Gauss was able to do it very quickly by noticing that the sum of the sequence and the reverse sequence produced a series of constants.

  S =   1 +  2 +  3 + ... + 100
  S = 100 + 99 + 98 + ... +   1
 2S = 101 +101 +101 + ... + 101 = 100×101

  S = (100×101)/2 = 5,050

The symmetry of the solution can also be observed with a graphical representation:

Scientific Psychic

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FAH

All students in the physics class also study mathematics. Half of those who study literature also study mathematics. Half of the students in the mathematics class study physics. Thirty students study literature and twenty study physics. Nobody who studies literature studies physics. How many students in the mathematics class study neither physics nor literature?

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Answer:

If there are 20 physics students who all take mathematics, and half of the mathematics students study physics, there must be 40 students in the mathematics class. If half of the 30 literature students take mathematics, then 15 of them take mathematics. Since none of the literature students study physics, only five students in the mathematics class study neither physics nor literature.




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Scientific psychic

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FAH

Three students checked into a hotel and paid the clerk $30 for a room ($10 each). When the hotel manager returned, he noticed that the clerk had incorrectly charged $30 instead of $25 for the room. The manager told the clerk to return $5 to the students. The clerk, knowing that the students would not be able to divide $5 evenly, decided to keep $2 and to give them only $3.
The students were very happy because they paid only $27 for the room ($9 each). However, if they paid $27 and the clerk kept $2, that adds up to $29. What happened to the other Dollar?

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Answer:
There is a saying that you cannot add apples and oranges. If you have 3 apples and 2 oranges do you have 5 apples? No. Do you have 5 oranges? No. You have five fruits, but the number of apples and oranges has not changed. Similarly, you cannot add real money and "what they think they paid".
When we count only real money, the students have $3, the clerk has $2, and the manager has $25. That is $30 total.

 Accounting of Real Money 

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FAH