Puzzle for August 5, 2020  ( )

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

eq.1) A + B + C + D + E + F = 19 eq.2) C + F = A + B eq.3) E + F = B + C + D eq.4) A + E = C + D eq.5) F – D = B – A eq.6) B – D – E = A + D + E

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

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Hint #1


Subtract E and D from both sides of eq.3: E + F – E – D = B + C + D – E – D which becomes F – D = B + C – E In eq.5, replace F – D with B + C – E: B + C – E = B – A In the above equation, subtract B from both sides, and add E and A to both sides: B + C – E – B + E + A = B – A – B + E + A which simplifies to eq.5a) A + C = E


  

Hint #2


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


  

Hint #3


In eq.5, substitute 2×A for D: F – 2×A = B – A Add 2×A to both sides of the above equation: F – 2×A + 2×A = B – A + 2×A which becomes eq.5b) F = B + A


  

Hint #4


Substitute B + A for F (from eq.5b) in eq.2: C + B + A = A + B Subtract A and B from each side of the equation above: C + B + A – A – B = A + B – A – B which makes C = 0


  

Hint #5


Substitute 0 for C in eq.5a: A + 0 = E which makes A = E


  

Hint #6


Substitute 2×A for D, and A for E in eq.6: B – 2×A – A = A + 2×A + A which becomes B – 3×A = 4×A Add 3×A to both sides of the equation above: B – 3×A + 3×A = 4×A + 3×A which makes B = 7×A


  

Hint #7


Substitute 7×A for B in eq.5b: F = 7×A + A which becomes F = 8×A


  

Solution

Substitute 7×A for B, 0 for C, 2×A for D, A for E, and 8×A for F in eq.1: A + 7×A + 0 + 2×A + A + 8×A = 19 which simplifies to 19×A = 19 Divide both sides of the above equation by 19: 19×A ÷ 19 = 19 ÷ 19 which means A = 1 making B = 7×A = 7 × 1 = 7 D = 2×A = 2 × 1 = 2 E = A = 1 F = 8×A = 8 × 1 = 8 and ABCDEF = 170218