Puzzle 1
I travelled from London to Glasgow last week at 60 miles per hour, I had filled my petrol tank just before I left, so it was full with 25 gallons.
Unfortunately, my petrol tank sprang a leak immediately, and I only managed to drive 300 miles before I ran out of petrol.
My car does 30 miles per gallon, how fast was I losing petrol through the hole?
Puzzle Copyright © Kevin Stone
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Hint
How much petrol was used for the journey itself?
Answer
3 gallons each hour.
Reasoning
I drove 300 miles at 30 miles per gallon, so I used 10 gallons. Which means I lost the other 15 gallons.
I drove the 300 miles at 60mph, which took me 5 hours.
So I lost 15 gallons of petrol in 5 hours = 3 gallons per hour.
Puzzle 2
Here we have a rectangular room, measuring 30 feet by 12 feet, and 12 feet high.
There is a spider in the middle of one of the end walls, 1 foot from the ceiling (A).
There is a fly in the middle of the opposite wall, 1 foot from the floor (B).
What is the shortest distance that the spider must crawl in order to reach the fly?
The Spider and the Fly, The Canterbury Puzzles, Henry Ernest Dudeney.
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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.
Puzzle 3
A Proof That 2 = 1. Let A = B:
A = B
multiply by A on both sides:
A x A = A x B
subtract B x B from both sides:
A x A – B x B = A x B – B x B
factorise:
(A + B)(A – B) = B(A – B)
divide by (A – B) on both sides:
(A + B) = B
replace A with B:
B + B = B
factorise:
2B = B
divide by B on both sides:
2 = 1
Where lies the flaw in the logic?
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Answer
On line 4 we are dividing both sides by (A – B).
But we know that A = B, so A – B = 0.
Dividing by zero is undefined, so we are not allowed to do this.
Puzzle 4
A parent recently arranged a children's party, and had 187 sweets to give out equally to the children.
Each child had more than one sweet, and there were more children than there were sweets per child.
How many sweets did each child end up with?
Puzzle Copyright © Kevin Stone
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Hint
Find two numbers that multiply to 187.
Answer
There were 17 children, with 11 sweets each.
Reasoning
If we number the clues:
each child had more than one sweet
there were more children than there were sweets per child
187 only has the factors 1, 11, 17, 187. Therefore, the only possible answers are:
1 child, with 187 sweets each, which is not allowed by (2)
11 children, with 17 sweets each, which is not allowed by (2)
17 children, with 11 sweets each (*)
187 children, with 1 sweet each, which is not allowed by (1)
Only one possible answer (*) isn't eliminated.
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