Solution:

They used the planks in the manner that is shown in the figure.

Solution:

It can be solve as shown below.

Solution:

The attached picture shows you how you can do it. If you were able to deduce any other way, feel free to comment.

Solution:

LOTTO

Explanation:
Just flip the two rows 1 and 2 vertically, you will find the word "Lotto".

Solution:

The equation can be solved by substituting
B = 7
A = 8
S = 8
E = 3
L = 5

Therefore the equation now looks as:
7483 +
7455
-------
14938
-------

Solution:

Never

Explanation:
In the question, the distance of x miles is given at a particular moment. The middle car is running at that time at 50 MPH. The distance will keep increasing with time as three are running at different speeds. It will increase by x miles every hour till a certain point of time and then it will start decreasing till the fastest car meets the slowest again.

As per question, all the cars stop then. Thus, the distance will never be x miles again.

Solution:

The node can be connected as shown.

Solution:

Q = 1
U = 9
I = 6
C = 8
K = 3.

Explanation:
Replace values in( Q + U + I + C + K)^3 = QUICK
(1+9+8+6+3)^3 = 19863
27^3 = 19863
19863 = 19863

Solution:

23rd day

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 = 276

Thus our anniversary falls on the 23rd day of the month.

You can apply other formulas to shorten the process but that is the simplest way to do it.

Solution:

Playing chess

Game needs two players so Brother-8 is playing chess with Brother-2

Solution:

Put one red marble in one box and all other marbles in the 2nd box.

Probability :
50 + 24/49