Puzzle for August 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 + F = C eq.3) D = C + F eq.4) C + E = D + F eq.5) A + B – C = C – E eq.6) E + F = A + C – E – F

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

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Hint #1


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


  

Hint #2


Add C to both sides of eq.5: A + B – C + C = C – E + C which becomes A + B = 2×C – E In the above equation, replace E with 2×F: A + B = 2×C – 2×F which may be written as eq.5a) A + B = 2×(C – F)


  

Hint #3


Subtract F from both sides of eq.2: B + F – F = C – F which becomes eq.2a) B = C – F   In eq.5a, substitute B for C – F (from eq.2a): A + B = 2×(B) Subtract B from both sides: A + B – B = 2×(B) – B which means A = B


  

Hint #4


Substitute 2×F for E, and B for A in eq.6: 2×F + F = B + C – 2×F – F which becomes 3×F = B + C – 3×F Add 3×F to both sides of the equation above: 3×F + 3×F = B + C – 3×F + 3×F which becomes eq.6a) 6×F = B + C


  

Hint #5


Substitute B + F for C (from eq.2) in eq.6a: 6×F = B + B + F which becomes 6×F = 2×B + F Subtract F from both sides of the above equation: 6×F – F = 2×B + F – F which makes 5×F = 2×B Divide both sides by 2: 5×F ÷ 2 = 2×B ÷ 2 which makes 2½×F = B and also makes A = B = 2½×F


  

Hint #6


Substitute 2½×F for B in eq.2: 2½×F + F = C which makes 3½×F = C


  

Hint #7


Substitute 3½×F for C in eq.3: D = 3½×F + F which makes D = 4½×F


  

Solution

Substitute 2½×F for A and B, 3½×F for C, 4½×F for D, and 2×F for E in eq.1: 2½×F + 2½×F + 3½×F + 4½×F + 2×F + F = 32 which simplifies to 16×F = 32 Divide both sides of the above equation by 16: 16×F ÷ 16 = 32 ÷ 16 which means F = 2 making A = B = 2½×F = 2½ × 2 = 5 C = 3½×F = 3½ × 2 = 7 D = 4½×F = 4½ × 2 = 9 E = 2×F = 2 × 2 = 4 and ABCDEF = 557942