Demystefying The Fourth Mystery Number

What was the challenge? 

Given are hints to identify number ABCDEFGHIJ.

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1) B + C + F + J = E + G + H + I = AD

2) B - H = J - G = 3

3) C - F = E - I = 5

4) B * I = AJ

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STEPS : 

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STEP 1 :

The sum of digits from 0 to 9 is 45.

Maximum value of AD = 98 and 

Minimum value of AD = 10 (A can't be 0 as leading 0's not allowed).

If AD = 98 then sum of rest of digits B + C + F + J + E + G + H + I must be 
45 - (9 + 8) = 28.

If AD = 10 then sum of rest of digits B + C + F + J + E + G + H + I must be 
45 - (1 + 0) = 44.

The sum of such 8 digits is divided into 2 parts in form of
B + C + F + J  and E + G + H + I which in turn must be equal to AD.

Therefore, each group of four digits must sum to one of the following: 14, 15, 16, 17, 18, 19, 20, 21, 22 (with AD varying from 28 to 44).

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STEP 2 :

Using Trial And Error method to get possible values of AD.

If AD = 14 then B + C + F + J + E + G + H + I = 45 - (1 + 4) = 40 and

B + C + F + J  = E + G + H + I = 40/2 = 20 = AD but AD = 14 assumed.

Hence, this value of AD is invalid.

Similarly, 15, 16, 17, 19, 20, 22 are invalid values of AD leaving behind only 18 and 21 as possible values.

Hence, A must be either 1 or 2 and D must be either 1 or 8.

That is either A or D takes 1.

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STEP 3 : 

As per Hint 2, B and J > H and G by 3 respectively and since 1 already taken by A or D,

Possible Values of B and J - 3, 5, 6, 7, 8, 9.

Possible Values of H and G - 2, 3, 4, 5, 6.

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STEP 4 : 

As per Hint 3, C and E > F and I by 3 respectively and since 1 already taken by A or D,

Possible Values of C and E - 5, 7, 8, 9.

Possible Values of F and I -  0, 2, 3, 4.

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STEP 5 : 

So with B having possible values as 3, 5, 6, 7, 8, 9 and I having possible values as 0, 2, 3, 4 the equation B * I has following 24 possibilities - 

(3 x 0 = 0) (3 x 2 = 6) (3 x 3 = 9) (3 x 4 = 12)

(5 x 0 = 0) (5 x 2 = 10) (5 x 3 = 15) (5 x 4 = 20) 
  
(6 x 0 = 0) (6 x 2 = 12) (6 x 3 = 18) (6 x 4 = 24)

(7 x 0 = 0) (7 x 2 = 14) (7 x 3 = 21) (7 x 4 = 28) 

(8 x 0 = 0) (8 x 2 = 16) (8 x 3 = 24) (8 x 4 = 32) 

(9 x 0 = 0) (9 x 2 = 18) (9 x 3 = 27) (9 x 4 = 36).

B * I can't be 0 as AJ can't be 0. Also, since A can't be 0 the product B*I can't be single digit like 6 or 9. 

Moreover, A has to be either 1 or 2 as deduced in STEP 2 and J must be among 3, 5, 6, 7, 8, 9 as deduced in STEP 3. 

Now the equation B * I = AJ has possibilities as - 

(5 x 3 = 15) (6 x 3 = 18)
  
(7 x 4 = 28) (8 x 2 = 16) 

(9 x 2 = 18) (9 x 3 = 27)  

Since (5 x 3 = 15) suggests that B = J = 5 which is against the rule that no 2 alphabets can take same digit. Hence, that possibility is eliminated.

Revised Possible Values of B - 6, 7, 8, 9.

Revised Possible Values of I - 2, 3, 4.

Revised Possible Values of J - 6, 7, 8. 

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STEP 6 :  

Since I can't be 0, E can't be 5 as E - I = 5.

Revised Possible Values of E - 7, 8, 9.

Since J - G = 3, and if J is among 6, 7, 8

Revised Possible Values of G - 3, 4, 5.

Since B - H = 3, and if B is among 6, 7, 8, 9

Revised Possible Values of H - 3, 4, 5, 6.

So letters A, B, D, E, G, H, I and J together takes digits 1, 2, 3, 4, 6, 7, 8 and 9 not in order though.

This leaves behind only possible value of C = 5 and F = 0. 

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STEP 7 :

Now, since H can't be 5 hence B can't be 8. Also, G too can't be 5 so J can't be 8 too. So both B and J can't be 8. 

Now, revising B * I = AJ possibilities deduced in STEP 4 as - 

 (9 x 3 = 27)  

Leaves only possible valid combination thereby.

So, we get, B = 9, I = 3, A = 2 and J = 7.

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STEP 8 : 

If B = 9 then H = 6.

If I = 3 then E = 8.

If J = 7 then G = 4.

The equation B + C + F + J = 9 + 5 + 0 + 7 = 21 = AD gives A = 2 and D = 1.

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CONCLUSION :

A = 2, B = 9, C = 5, D = 1, E = 8, F = 0, G = 4, H = 6, I = 3, J = 7.

Hence, the mystery number ABCDEFGHIJ is 2951804637.

 Verifying the given hints - 

1) B + C + F + J = E + G + H + I = AD     
    9 + 5 + 0 + 7 = 8 + 4 + 6 + 3 = 21

2) B - H = J - G = 3
    8 - 3 = 5 - 0 = 5 

3) C - F = E - I = 5
    7 - 4 = 9 - 6 = 3 

4) B * I = AJ 
    9 * 3 = 27