Random Puzzles 

Workings

Hint
The first word begins with T.

Answer
Too many cooks spoil the broth.

Reasoning
The saying has been broken down into blocks of 4, the last block has been filled with dummy letters. Each block has then been reversed.

Double-Checking
Start with:
   TOO MANY COOKS SPOIL THE BROTH

Group together:
   TOOMANYCOOKSSPOILTHEBROTH

Split into groups of 4 letters:
   TOOM ANYC OOKS SPOI LTHE BROT H

Add 3 random characters:
   TOOM ANYC OOKS SPOI LTHE BROT HBCY

Reverse each group of 4 letters:
   MOOT CYNA SKOO IOPS EHTL TORB YCBH

Workings

Hint
You don't need to use a calculator.

Answer
Cube root of 2.

Reasoning
We could use a calculator to see that the tenth root of 10 = 1.2589 and the cube root of 2 = 1.2599.

But, if you've not got your calculator handy...

We have:

   10^1/10 vs 2^1/3

Raise each to the power of 30 (since 10 x 3 = 30) to give:

   (10^1/10)³⁰ vs (2^1/3)³⁰

   10^{1/10 x 30} vs 2^{1/3 x 30}

   10³ vs 2¹⁰

   1000 vs 1024

Therefore, the cube root of 2 is larger.

Workings

Hint
The answers are related to time.

Answer
60.

The numbers represent 100 years in a century, 365 days in a year, 24 hours in a day, 60 minutes in an hour and 60 seconds in an minute.

Workings

Hint
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...

Answer
71.

The first twenty prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71.