Find the Vault code consisting of 5 different numbers (1-9), Sum of first digit and third digit is 10...

Question

Please find the Vault code consisting of 5 different numbers … (1-9)
1. The sum of the first digit and the third digit is 10
2. The first digit is a square of the fifth digit
3. The fourth digit is equal to the second digit plus 1
4. The sum of the digits is 22

What is the Vault code?

Answer 1

94153
.
The sum of the # is 22. The sum of digit 1+3 = 10, thus the sum of # 2,4,5 = 12.
The sum of # 2,4 is 2 * # 2 +1 = uneven, so # 5 must be uneven as well.
.# 1 is the square of # 5, thus # 5 must be 1 or 3.
If # 5 = 1, then # 1 = 1, but the code consists of different numbers.
Thus # 5 = 3, # 1 = 9, # 3 = 1
That leaves # 2,4, whose sum is 12 - 3 = 9
Thus # 2 = 4 and # 4 = 5
And the code is 94153

Answer 2

1. The sum of the first digit and the third digit is 10
   (1, 9 ; 2,8 ; 3,7 ; 4,6 ; 6,4 ; 7,3 ; 8,2 ; 9,1 ) are the possible combinations.
2. The first digit is a square of the fifth digit
   Only combination 4, 6 and 9,1 are valid above.
   and also 5th digit can be 2 or 3
   Since the sum of digits is 22 and the two remaining possible combinations of digit #1, #3 and #5 add upto 12 and 13 respectively.
3. The fourth digit is equal to the second digit plus 1
   Meaning these are consecutive numbers adding to 10 or 9 ( deducted from above)

No way two consecutive numbers add upto an even number hence only possible solution for 2nd and 4th digit is 4 and 5 adding to 9.

Thereby the Vault code should be 9 4 1 5 3

Answer 3

Let the number be VWXYZ

Conditions:

(1) V + X = 10
(2) V = Z^2 => V = 4 / 9 and Z = 2 / 3 (V = 1 => Z = 1 which is NOT allowed.)
(3) Y = W + 1
(4) V + W + X + Y + Z = 22

Applying (1) and (3) in (4) => 10 + 2W + 1 + Z = 22

=> 2W + Z = 11 (5)

Exploring possible values from (2):

(a) If V = 4, Z = 2 then (5) is 2W = 9 which is Impossible.
(b) If V = 9, Z = 3 then X = 1, W = 4, Y = 5.

Hence the code is 94153.