Solution:

12

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.

Solution:

Short

(shorter - er = short)

Solution:

For this answer is 3^0, 3^1, 3^2... That is 1,3,9,27,81,243 and 729.

Solution:

Side Show.

Since the word Show is written on a side, it is quite clear.

Solution:

23

Solution:

The task was simple as you can see in the picture. This is how you will cut and join the boards and then you will be able to cover the hole completely.

Solution:

Rising Mercury / Temperature Is Rising

Solution:

The only possible way to do it is to form an equilateral triangle pyramid and then place the balls at the corners of the figure. Yes, you will have to use 3 dimensional figure.

Solution:

A cone consists of just two surfaces.

Solution:

It is pointing towards the phrase - Fat Chance. It is quite evident with the fat letters.