Random Puzzles 

Workings

Hint
How many miles were left at the start of yesterday?

Answer
336 miles.

Reasoning
To work these numbers out, it is perhaps easier to start at the end and work backwards.

Yesterday I travelled ¹/₃ of the distance to leave 21 miles, therefore there were 31.5 miles (=21 ÷ ²/₃) at the start of yesterday.

On day 3 I travelled ³/₄ of the distance to leave 31.5 miles, therefore there were 126 miles (=31.5 ÷ ¹/₄) at the start of day 3.

On day 2 I travelled ¹/₂ of the distance to leave 126 miles, therefore there were 252 miles (=126 ÷ ¹/₂) at the start of day 2.

On day 1, I travelled ¹/₄ of the distance to leave 252 miles, therefore there were 336 miles (=252 ÷ ³/₄) at the start of day 1.

Double-Checking
On day one, I travelled one quarter of the distance (84, leaving 252).

On day two, I travelled one half of the remaining distance (126, leaving 126).

On day three, I travelled three quarters of the remaining distance (94.5, leaving 31.5).

Yesterday I travelled one third of the remaining distance (10.5).

I now have 21 miles left to travel.

Workings

Hint
Try working backwards.

Answer
120 miles.

Reasoning
I now have 5 miles left to travel, so I must have travelled 5 miles yesterday.

At the end of Day 3, I had 10 miles left to travel, so I must have had 40 miles left at the start of the Day 3.

At the end of Day 2, I had 40 miles left to travel, so I must have had 60 miles left at the start of the Day 2.

At the end of Day 1, I had 60 miles left to travel, so I must have had 120 miles left at the start of the Day 1.

Double-Checking
On the Day 1, I travelled 60 miles, leaving 60 miles.

On Day 2, I travelled 20 miles, leaving 40 miles.

On Day 3, I travelled 30 miles, leaving 10 miles.

Yesterday, I travelled 5 miles, leaving 5 miles.

Workings

Hint
Start by filling the 7 gallon container.

Answer
Here are the contents of the containers after each stage:
A  B
0  7    fill B
3  4    fill A from B
0  4    empty A
3  1    fill A from B
0  1    empty A
1  0    fill A from B
1  7    fill B
3  5    fill A from B

This leaves 5 gallons in container B after 8 steps.

Workings

Hint
Make sure you answer the question asked.

Answer
10,001.