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Take a normal piece of paper, exactly 0.1 mm thick.

Fold it in half, and then in half again, and again, and again.

Do this a total of 50 times.

How thick would the final paper be (if this could be done)?

[Ref: ZOPB]

Hint: We'll assume that we can actually fold it this many times.

Answer:
Very thick indeed! The paper doubles in thickness with each fold. If we could fold it 50 times, it would be around 70 million miles thick!

1 fold would be 0.1 + 0.1 = 0.1 x 2 ^ 1 = 0.2 mm
2 folds would be 0.1 + 0.1 + 0.1 + 0.1 = 0.1 x 2 ^ 2 = 0.4 mm
.
.
.
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10 folds would be 0.1 x 2 ^ 10 = 102.4 mm
.
.
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50 folds would be 0.1 x 2 ^ 50 = 112,589,990,684,262.4 mm = 112,589,990.7 km (around 70 million miles).

Puzzle 2

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, Charles 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?

Hint: How many apples were left after Charles had taken some?

Answer: 162.

Alan hid 54, leaving 108. Brain hid 36, leaving 72. Charles hid 24, leaving 48. 48 apples were then available to share in the morning.

To work these numbers out, we start at the end and work backwards.

At the end there were 16 apples each, therefore there were 48 apples.

Charles removed 1÷3 leaving 48, so there must have been 72 apples before he did this.

Brian removed 1÷3 leaving 72, so there must have been 108 apples before he did this.

Alan removed 1÷3 leaving 108, so there must have been 162 apples before he did this. QED.

Puzzle 3

A haulage contractor did not have room in his garage for 8 of his trucks.

He therefore increased the size of his garage by 50 percent, which gave him room for 8 more trucks than he owned altogether.

How many trucks did he own?

[Ref: ZKFQ]

Hint: A little algebra might help.

Answer: 40 trucks.

His original garage could hold 32 trucks. By increasing the size by 50%, the new garage could then hold 48 trucks - which is 8 more than he currently owned.

If we say he owned T trucks, then the original garage held T - 8 trucks, so:

G = T - 8 [1]

Increasing the garage by 50% gave him 1.5 times the amount of space the original garage had, he had 8 spare spaces, so:

Odd numbers expressed in base 8, 11 in base 8 >> 13.

Puzzle 5

Lemon Twist: how many times does the word LEMON appear in this grid?

N O M E L L E M O N L E M O N
N O M E L N O M E L E L E M O
L E M O N N O M E L L E M O M
L O E L O L L M M L E M O N E
N N O M E L O E L E M O N E L
O L E M L N L L M M L N L L E
M L O E L N L E M O N E E E M
E N M E O E E M N N N M M M O
L O M M N N M N L O O L L O N
N O E L O M O O N E M E L N N
N L N E M E N M N O M E L O N
O N O M E L O E E O M O L M O
M L M O L E M L N L E E N E M
E E E N O M E L E M O N L L E
L M L O M E L E M O N O M E L

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