There are 62 squares of size 1x1. There are 41 squares of size 2x2. There are 18 squares of size 3x3. There are 6 squares of size 4x4. There is 1 square of size 5x5.

A total of 62 + 41 + 18 + 6 + 1 = 128.

Puzzle 2

Last week I decided to cycle to my Grandmother's house.

On the first day, I cycled half of the distance.

On day 2, I cycled one half of the remaining distance.

On day 3, I cycled three quarters of the remaining distance.

On day 4, I cycled 10 miles.

On day 5 I cycled two thirds of the remaining distance and on the final day I cycled the remaining 5 miles.

On the final day I cycled the remaining 5 miles
On Day 5 I started with 15 miles (as I cycled two thirds, leaving 5 miles).
On Day 4 I started with 25 miles (as I cycled 10 miles).
On Day 3 I started with 100 miles (as I cycled three quarters, leaving 25 miles).
On Day 2 I started with 200 miles (as I cycled half, leaving 100 miles).
On Day 1 I started with 400 miles (as I cycled half, leaving 200 miles).

Puzzle 3

During a recent expedition, three intrepid adventurers were left stranded in the middle of the desert with only a crate full of apples.

During the night, Alan woke up and decided to hide his share of the apples and hid a third, then promptly fell asleep again.

Brian woke up shortly after and also decided to hide a third of the remaining apples and he also dozed back to sleep.

Finally, Charlie woke up and seeing the others were asleep, took a third of what was left.

Of course none of the adventurers knew of the other's antics, so, in the morning, they shared the remaining apples, each receiving sixteen. How many apples were in the crate originally?