MHB Crack the Code: A Puzzle Challenge!

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A puzzle challenge was shared involving a man who forgot his building access code but remembered five clues. The clues relate to the relationships and sums of five numbers, ultimately leading to a solution where the numbers must add up to 30. Participants are encouraged to solve the puzzle based on the provided clues, which include equations linking the numbers together. The discussion invites engagement and problem-solving from the community. The goal is to determine the correct sequence of the five numbers.
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I posted this on my website a couple days ago. I like puzzles a lot, so I figured I would share it here and see how many people could figure it out; enjoy!;)
A man wanted to get into his work building, but he had forgotten his code. However, he did remember five clues. These are what those clues were:

The fifth number plus the third number equals fourteen.

The fourth number is one more than the second number.

The first number is one less than twice the second number.

The second number plus the third number equals ten.

The sum of all five numbers is 30.

What were the five numbers and in what order?
 
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Hello, Farmtalk!

A man wanted to get into his work building, but he had forgotten his code.
However, he did remember five clues. These are what those clues were:

[1] The fifth number plus the third number equals fourteen.

[2] The fourth number is one more than the second number.

[3] The first number is one less than twice the second number.

[4] The second number plus the third number equals ten.

[5] The sum of all five numbers is 30.

What were the five numbers and in what order?
Let a,b,c,d,e . . be the five numbers, in that order.We are given:

. . [1] \;e+c \,=\,14

. . [2]\;d \,=\,b+1

. . [3]\;a \,=\,2b-1

. . [4]\;b+c \:=\:10

. . [5]\;a+b+c+d+e \,=\,30From [5], we have: .a + b + d + (e+c) \:=\:30

Since e+c\,=\,14 . , we have: .a+b+d+14 \:=\:30
. . Hence: .a+b+d \:=\:16

Substitute [3] and [2]: .(2b-1) + b+ (b+1) \:=\:16

. . 4b \:=\:16 \quad\Rightarrow\quad b \,=\,4Substitute into [3]: .a \:=\:2b-1 \quad\Rightarrow\quad a \,=\,7

Substitute into [4]: .c \:=\:10-b \quad\Rightarrow\quad c \,=\,6

Substitute into [2]: .d \:=\:b+1 \quad\Rightarrow\quad d \,=\,5

Substitute into [3]: .e \:=\:14-c \quad\Rightarrow\quad e \,=\,8Therefore, the code is 74658.
 
That's the number I got! CORRECT! ;)
 
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