Random Puzzles 

Workings

Hint
Start by filling the 9 gallon container, and then use that to fill the 4 gallon container.

Answer
Here are the contents of the containers after each stage:
A   B
9   0      fill A
5   4      fill B from A
5   0      empty B
1   4      fill B from A
1   0      empty B
0   1      fill B from A
9   1      fill A
6   4      fill B from A
6   0      empty B

This leaves 6 gallons in container A.

Workings

Hint
There are 13 loaves in a baker's dozen.

Answer
Loaves in a dozen baker's dozen.

Reasoning
A baker's dozen is 13, so a dozen baker's dozen is:

   12 x 13 = 156 loaves

There are 15 red balls in snooker, and each of these can be followed by the black ball, and then the colours in order gives:

   15 x (1 + 7) + 2 + 3 + 4 + 5 + 6 + 7 = 147 points

This can rise to 155 if the break follows a foul.

Workings

Hint
The cube root of the given number is a good starting point.

Answer
71, 73, 79.

Reasoning
The cube root of 409,457 is around 74.

The closest consecutive prime numbers to 74 are: 61, 67, 71, 73, 79, 83, 89, 97.

Using a calculator, it is quickly seen that the answer can only be found using 71, 73 and 79.

Workings

Hint
How many squares of size 1 x 1 are there?

Answer
There are 116 squares without a rook.

Reasoning
There are 61 squares of size 1 x 1.
There are 37 squares of size 2 x 2.
There are 15 squares of size 3 x 3.
There are  3 squares of size 4 x 4.

Giving a total of 61 + 37 + 15 + 3 = 116 squares.