Mathematical Puzzles 

Workings

Hint
Think of the pages from 1 to 9, and then 10 to 99.

Answers
There are 147 pages. The number 3 appears 35 times.
Answer #1
For pages 1 to 9, there are 9 pages, which is 9 digits.

For pages 10 to 99, there are 90 pages, which is 180 digits.

This is a total so far of 189, therefore we require another 333 − 189 = 144 digits, which is another 144 ÷ 3 = 48 pages.

Taking us to 9 + 90 + 48 = 147 pages in total.

Answer #2
From page 1 to 147 we have 15 pages that end in 3:

   3, 13, 23, …, 143

We also have the ten pages that start with 30:

   30, 31, 32, …, 39

Plus the ten pages that start with 130:

   130, 131, 132, …, 139

For a total of 15 + 10 + 10 = 35 number 3's.

Workings

Hint
Can the fractions be simplified?

Answer
667.

Reasoning
 3        8        1
———  of  ———  of  ———  of 2001
 4        9        2

which simplifies to:
24
——  of 2001
72

which simplifies to:
 1
———  of 2001
 3

Giving 667.

Workings

Hint
How many 2 x 2 squares are there?

Answer
There are 128 squares without a rook.

Reasoning
There are 62 squares of size 1 x 1.
There are 41 squares of size 2 x 2.
There are 18 squares of size 3 x 3.
There are  6 squares of size 4 x 4.
There is   1 square  of size 5 x 5.

Giving a total of 62 + 41 + 18 + 6 + 1 = 128 squares.

Workings

Hint
How many tyre miles were travelled?

Answer
1,005 miles.

Reasoning
A total of 2,345 miles were travelled and at any one time, three tyres were on the tricycle.

Therefore, 3 x 2,345 = 7,035 tyre miles were travelled, which was shared equally by the 7 tyres.

And 7,035 ÷ 7 = 1,005.