Puzzle for May 12, 2020  ( )

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

eq.1) A + B + C + D + E + F = 18 eq.2) A + F = C eq.3) D + E = A + B + C eq.4) B + E = A eq.5) C + F = A + B + D eq.6) D – A = A + C

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 + E (from eq.4): D + E = B + E + B + C which becomes D + E = 2×B + E + C Subtract E from both sides of the above equation: D + E – E = 2×B + E + C – E which becomes eq.3a) D = 2×B + C


  

Hint #2


In eq.6, replace D with 2×B + C (from eq.3a): 2×B + C – A = A + C In the equation above, subtract C from each side, and add A to both sides: 2×B + C – A – C + A = A + C – C + A which simplifies to 2×B = 2×A Divide both sides by 2: 2×B ÷ 2 = 2×A ÷ 2 which makes B = A


  

Hint #3


In eq.4, substitute B for A: B + E = B Subtract B from each side of the equation above: B + E – B = B – B which makes E = 0


  

Hint #4


Substitute B for A, and 2×B + C for D (from eq.3a) in eq.5: C + F = B + B + 2×B + C which becomes C + F = 4×B + C Subtract C from both sides of the equation above: C + F – C = 4×B + C – C which makes F = 4×B


  

Hint #5


Substitute B for A, and 4×B for F in eq.2: B + 4×B = C which makes 5×B = C


  

Hint #6


Substitute 5×B for C in eq.3a: D = 2×B + 5×B which makes D = 7×B


  

Solution

Substitute B for A, 5×B for C, 7×B for D, 0 for E, and 4×B for F in eq.1: B + B + 5×B + 7×B + 0 + 4×B = 18 which simplifies to 18×B = 18 Divide both sides of the equation above by 18: 18×B ÷ 18 = 18 ÷ 18 which means B = 1 making A = B = 1 C = 5×B = 5 × 1 = 5 D = 7×B = 7 × 1 = 7 F = 4×B = 4 × 1 = 4 and ABCDEF = 115704