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.
??
Puzzle 34
A large fresh water reservoir has two types of drainage system: small pipes and large pipes.
6 large pipes, on their own, can drain the reservoir in 12 hours.
3 large pipes and 9 small pipes, at the same time, can drain the reservoir in 8 hours.
How long will 5 small pipes, on their own, take to drain the reservoir?
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.
?
Puzzle 35
A very long time ago, at the beginning of the week, I was given some money for my birthday.
On Monday, I spent a quarter of the money on clothes.
On Tuesday, I spent one third of the remaining money on music.
On Wednesday, I spent half of the remaining money on food.
Finally, on Thursday, I spent the last £1.25 on a book.
On Wednesday, I had £1.25 x (1 ÷ one half) = £2.50.
On Tuesday, I had £2.50 x (1 ÷ two thirds) = £3.75.
On Monday, I had £3.75 x (1 ÷ three quarters) = £5.00.
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Puzzle 36
Three teachers were discussing how long they had been teaching.
Alex and Billie had been teaching for a total of 36 years. Charlie and Billie had been teaching for a total of 22 years. Charlie and Alex had been teaching for a total of 28 years.
How long had each been teaching?
Hint
If we look at all of the clues, each person appears exactly twice.
Answer
Alex had been teaching for 21 years.
Billie had been teaching for 15 years.
Charlie had been teaching for 7 years.
Reasoning
If we look at all of the clues, each person appears exactly twice.
So the total of the three clues gives us two lots of Alex + Billie + Charlie = 36 + 22 + 28 = 86.
So, Alex + Billie + Charlie = 43 years.
In each clue we are given two of the people, so we can use this to find the missing person:
Charlie and Billie had been teaching for a total of 22 years, which means that Alex must be: 43 – 22 = 21 years.
Charlie and Alex had been teaching for a total of 28 years, which means that Billie must be: 43 – 28 = 15 years.
Alex and Billie had been teaching for a total of 36 years, which means that Charlie must be: 43 – 36 = 7 years.
Reasoning With Algebra
Let Alex = A, Billie = B and Charlie = C, then:
[1] A + B = 36 [2] C + B = 22 [3] C + A = 28
If we use [3] – [2] we have:
[4] A – B = 6
If we use [1] + [4] we have:
2A = 42 A = 21
By [1] B = 15. By [3] C = 7.
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