Puzzle for July 8, 2020  ( )

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

eq.1) A + B + C + D + E + F = 27 eq.2) B + F = A eq.3) E + F = A + C – E eq.4) D – E = A + B – C – D + F eq.5) F = B + C eq.6) A – D = D – A – E

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

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Hint #1


In eq.3, replace A with B + F (from eq.2): E + F = B + F + C – E In the equation above, subtract F from each side, and add E to each side: E + F – F + E = B + F + C – E – F + E which becomes eq.3a) 2×E = B + C


  

Hint #2


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


  

Hint #3


Add C and D to both sides of eq.4: D – E + C + D = A + B – C – D + F + C + D which becomes 2×D – E + C = A + B + F In the above equation, substitute A for B + F (from eq.2): 2×D – E + C = A + A which becomes eq.4a) 2×D – E + C = 2×A


  

Hint #4


Add D and A to both sides of eq.6: A – D + D + A = D – A – E + D + A which becomes eq.6a) 2×A = 2×D – E   Substitute 2×A for 2×D – E (from eq.6a) in eq.4a: 2×A + C = 2×A Subtract 2×A from each side of the equation above: 2×A + C – 2×A = 2×A – 2×A which makes C = 0


  

Hint #5


In eq.5, substitute 0 for C: F = B + 0 which makes F = B and also makes B = F = 2×E


  

Hint #6


Substitute 2×E for B and F in eq.2: 2×E + 2×E = A which makes 4×E = A


  

Hint #7


Substitute (4×E) for A in eq.6a: 2×(4×E) = 2×D – E which becomes 8×E = 2×D – E Add E to each side of the above equation: 8×E + E = 2×D – E + E which makes 9×E = 2×D Divide both sides by 2: 9×E ÷ 2 = 2×D ÷ 2 which makes 4½×E = D


  

Solution

Substitute 4×E for A, 2×E for B and F, 0 for C, and 4½×E for D in eq.1: 4×E + 2×E + 0 + 4½×E + E + 2×E = 27 which simplifies to 13½×E = 27 Divide both sides of the equation above by 13½: 13½×E ÷ 13½ = 27 ÷ 13½ which means E = 2 making A = 4×E = 4 × 2 = 8 B = F = 2×E = 2 × 2 = 4 D = 4½×E = 4½ × 2 = 9 and ABCDEF = 840924