Brain Teasers and Puzzles 

Workings

Hint
Start on any square, and see how far you get.

Answer
Start from the 3U on the 5^th row down and the 6^th column across.

Workings

Hint
Pages 1-9 are one digit each.

Answer
There are 65 pages and the number 9 appears six times.

Reasoning
The 65 pages are (with the six 9's highlighted):

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65.

If we remove the spaces, we get:

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465. Which is 121 digits.

Workings

Hint
Column 4 contains a D.

Answer

Across: CADS, YOLK, DATE, PIGS
Down  : GLAD, STAY, COPE, SKID
Boxes : LADY, SOCK, STAG, PIED

Other anagrams of these words are OK as long as they don't change the answer grid.

Workings

Hint
Is a million grains of sand a heap?

Answer
We can probably agree on two things:

A million grains of sand is a heap.
If we remove one grain of sand from this heap, we will still have a heap.
We can now keep repeating (2) until we only have a single grain of sand remaining.

Is this a heap? Clearly not. But what went wrong with our thinking?

This is called the Sorites paradox (soros being Greek for "heap") and is a classic paradox that has no real answer.

Both (1) and (2) are true, and we can indeed keep removing one grain of sand until we have a single grain remaining. If we remove one more grain, we're left with nothing, is this still a heap?

When does the heap become a non-heap?