Mathematical Puzzles 

Workings

Hint
Going down and across the floor isn't the only route.

Answer
40 feet.

Explanation Diagram
If you imagine the room to be a cardboard box, you can 'unfold' the room in various ways, and each route gives a different answer.



We can use Pythagoras' theorem (a² + b² = c²) to calculate the distances:
   distance² = horizontal² + vertical²
   distance = √(horizontal² + vertical²)

Route #1
   distance = 1 + 30 + 11 = 42 feet.

Route #2
   horizontal = 6 + 30 + 6 = 42 feet.
   vertical = 10 feet.
   distance = √(42² + 10²) ≈ 43.174 feet.

Route #3
   horizontal = 1 + 30 + 6 = 37 feet.
   vertical = 6 + 11 = 17 feet.
   distance = √(37² + 17²) ≈ 43.178 feet.

Route #4
   horizontal = 1 + 30 + 1 = 32 feet.
   vertical = 6 + 12 + 6 = 24 feet.
   distance = √(32² + 24²) = 40 feet.

Workings

Hint
Does the number of adults and children matter?

Answer
3,672.

Reasoning
The actual number of adults and children doesn't matter.

For example, if all of the people were adults, then half of them (306) would be given 12 sweets:

  306 x 12 = 3672

For example, if all of the people were children, then three quarters of them (459) would be given 8 sweets:

  459 x  8 = 3672

For example, if there were 512 adults (so 256 would get 12 sweets = 3072) and 100 children (so 75 would get 8 sweets = 600):

  256 x 12 + 75 x 8 = 3672

As you can see, we can change the numbers of adults and children, and it doesn't change the answer.

The reason for this lies in the fact that
   ¹⁄₂ adults x 12 sweets
and
   ³⁄₄ children x 8 sweets
both equal 6 sweets.

Workings

Hint
There are more than 37, look at the different sizes.

Answer
There are 64 hexagon-type shapes in total.

Reasoning
37 single hexagons
 + 19 hexagons that contain 7 smaller hexagons
 + 7 hexagons that contain 19 smaller hexagons
 + 1 large hexagon that contains all of the smaller hexagons.

Workings

Hint
How many large pipes are required to drain the reservoir in 24 hours?

Answer
21 hours and 36 minutes.

Reasoning
Looking at the first clue:
   in 12 hours, 6 large pipes can drain 1 reservoir
   in 24 hours, 6 large pipes can drain 2 reservoirs
   in 24 hours, 3 large pipes can drain 1 reservoir (*)

Looking at the second clue:
   in 8 hours, 3 large + 9 small pipes can drain 1 reservoir
   in 24 hours, 3 large + 9 small pipes can drain 3 reservoirs

But, by (*), we know that in those 24 hours, 3 large pipes can drain 1 of those reservoirs.

Therefore, the other 2 reservoirs can be drained by the small pipes on their own:
   in 24 hours, 9 small pipes can drain 2 reservoirs
   in 24 hours, 1 small pipe can drain ²⁄₉ reservoirs

multiply the hours by 9:
   in 216 hours, 1 small pipe can drain 2 reservoirs
   in 216 hours, 5 small pipes can drain 10 reservoirs

divide the hours by 10:
   in 21.6 hours, 5 small pipes can drain 1 reservoir

21.6 hours = 21 hours and 36 minutes.