Puzzle for January 6, 2020  ( )

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Find the 6-digit number ABCDEF by solving the following equations:

eq.1) A + B + C + D + E + F = 32 eq.2) B = A + D eq.3) C = D + F eq.4) D + E = A + F eq.5) E = C + D eq.6) F = A + B

A, B, C, D, E, and F each represent a one-digit non-negative integer.

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Hint #1


In eq.5, replace C with D + F (from eq.3): E = D + F + D which becomes eq.5a) E = 2×D + F


  

Hint #2


In eq.4, replace E with 2×D + F (from eq.5a): D + 2×D + F = A + F which becomes 3×D + F = A + F Subtract F from each side of the equation above: 3×D + F – F = A + F – F which makes 3×D = A


  

Hint #3


In eq.2, substitute 3×D for A: B = 3×D + D which makes B = 4×D


  

Hint #4


Substitute 3×D for A, and 4×D for B in eq.6: F = 3×D + 4×D which makes F = 7×D


  

Hint #5


Substitute 7×D for F in eq.3: C = D + 7×D which makes C = 8×D


  

Hint #6


Substitute 7×D for F in eq.5a: E = 2×D + 7×D which makes E = 9×D


  

Solution

Substitute 3×D for A, 4×D for B, 8×D for C, 9×D for E, and 7×D for F in eq.1: 3×D + 4×D + 8×D + D + 9×D + 7×D = 32 which simplifies to 32×D = 32 Divide both sides of the equation above by 32: 32×D ÷ 32 = 32 ÷ 32 which means D = 1 making A = 3×D = 3 × 1 = 3 B = 4×D = 4 × 1 = 4 C = 8×D = 8 × 1 = 8 E = 9×D = 9 × 1 = 9 F = 7×D = 7 × 1 = 7 and ABCDEF = 348197