Puzzle for April 9, 2020  ( )

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

eq.1) A + B + C + D + E + F = 26 eq.2) E + F = A + B eq.3) B + D = C + E eq.4) D – F = A + B + E + F eq.5) C – A – E = A + B + F eq.6) E – F = A + F

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

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Hint #1


Add A and E to both sides of eq.5: C – A – E + A + E = A + B + F + A + E which becomes C = 2×A + B + E + F In the equation above, replace E + F with A + B (from eq.2): C = 2×A + B + A + B which becomes eq.5a) C = 3×A + 2×B


  

Hint #2


Add F to both sides of eq.4: D – F + F = A + B + E + F + F which becomes D = A + B + E + 2×F In eq.3, replace D with A + B + E + 2×F: B + A + B + E + 2×F = C + E which becomes A + 2×B + E + 2×F = C + E Subtract E from each side of the above equation: A + 2×B + E + 2×F – E = C + E – E which becomes eq.3a) A + 2×B + 2×F = C


  

Hint #3


In eq.3a, substitute 3×A + 2×B for C (from eq.5a): A + 2×B + 2×F = 3×A + 2×B Subtract A and 2×B from each side of the equation above: A + 2×B + 2×F – A – 2×B = 3×A + 2×B – A – 2×B which simplifies to 2×F = 2×A Divide both sides by 2: 2×F ÷ 2 = 2×A ÷ 2 which makes F = A


  

Hint #4


Substitute A for F in eq.2: E + A = A + B Subtract A from both sides of the above equation: E + A – A = A + B – A which makes E = B


  

Hint #5


Substitute A for F in eq.6: E – A = A + A which becomes E – A = 2×A Add A to each side of the equation above: E – A + A = 2×A + A which makes E = 3×A and also makes B = E = 3×A


  

Hint #6


Substitute B for E in eq.3: B + D = C + B Subtract B from each side of the above equation: B + D – B = C + B – B which makes D = C


  

Hint #7


Substitute (3×A) for B in eq.5a: C = 3×A + 2×(3×A) which is the same as C = 3×A + 6×A which makes C = 9×A and also makes D = C = 9×A


  

Solution

Substitute 3×A for B and E, 9×A for C and D, and A for F in eq.1: A + 3×A + 9×A + 9×A + 3×A + A = 26 which simplifies to 26×A = 26 Divide both sides of the equation above by 26: 26×A ÷ 26 = 26 ÷ 26 which means A = 1 making B = E = 3×A = 3 × 1 = 3 C = D = 9×A = 9 × 1 = 9 F = A = 1 and ABCDEF = 139931