Help Franky fix the water system!

Puzzle 27 Water Pipes.pdf

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Answer

Explanation

The differences between any two pipe joints will be called D.

Notice that, because we are using natural number from 1 to 9, the maximum difference is 8 and minimum difference is 1 => 1 ≤ D ≤ 8

There are 8 connections in total, so all value of D will be used.

For D = 8, the only pair is 1 and 9 => 1 and 9 must be connected to each other

This is the reasoning baseline that we'll use: High D value can only be created by a few number pair. We'll exploit this to bound certain number together.

For D = 7, there are two possibilites: 1-8 or 2-9

Because 1 and 9 are connected, there must be a 9-1-8 or 2-9-1 chain

Let's look at 2-9-1 first

There are 4 possible placement for this chain, we'll look at them one by one.

Placement 1

2-9 for D=7 but 1-8 also create 7. This scenario is wrong.

Placement 2

9-1 for D=8, 9-2 for D=7, 8-2 for D=6

For D=5, we can only use 7-2

(9-4, 8-3, 6-1 are all impossible because at least one number in each pair can't make anymore connection)

Now, how do we get D=4?

9-5, 8-4, 7-3, 6-2, and 5-1 are all impossible

=> This scenario is incorrect.

Placement 3

In this placement, to create D=6, we can only use 7-1

But then it's impossible to make D=5

At least one number in each of these pair isn't available for connecting: 9-4, 8-3, 7-2, 6-1.

=> This scenario is false

Placement 4

You can reason this one out by yourself ദ്ദി(˵ •̀ ᴗ – ˵ )

Alright, so all 2-9-1 placements are wrong, so 9-1-8 must be correct. There are two ways to place a 9-1-8 chain

Placement 1

To make D=6, there is only 7-1

Then, we only have 9-4 for D=5

The remaining numbers are 2 and 6. However, no matter how you place them, the pair a-5 and b-3 will always have the same difference.

2-5 and 3-6, D=3

6-5 and 3-2, D=1

Placement 2

To make D=6, there is only 7-1

Afterward, there is only 7-2 for D=5

6-5 and 4-3 both makes D=1

=> 6-3 for D=3 and 4-5 for D=1

Free talk

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EA is finally over, I have sometime to breathe before RD 😮‍💨