Puzzle for June 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 = 22 eq.2) D + E = B + C eq.3) C = A + F eq.4) B – E = A – F eq.5) A = D – F eq.6) B – C – D = A + C + F

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 D – F (from eq.5): C = D – F + F which makes C = D


  

Hint #2


In eq.2, replace D with C: C + E = B + C Subtract C from each side of the above equation: C + E – C = B + C – C which makes E = B


  

Hint #3


In eq.4, substitute B for E: B – B = A – F which becomes 0 = A – F Add F to both sides of the equation above: 0 + F = A – F + F which makes F = A


  

Hint #4


Substitute A for F in eq.3: C = A + A which makes C = 2×A and also makes D = C = 2×A


  

Hint #5


Substitute 2×A for C and D, and A for F in eq.6: B – 2×A – 2×A = A + 2×A + A which becomes B – 4×A = 4×A Add 4×A to both sides of the above equation: B – 4×A + 4×A = 4×A + 4×A which makes B = 8×A and also makes E = B = 8×A


  

Solution

Substitute 8×A for B and E, 2×A for C and D, and A for F in eq.1: A + 8×A + 2×A + 2×A + 8×A + A = 22 which simplifies to 22×A = 22 Divide both sides of the above equation by 22: 22×A ÷ 22 = 22 ÷ 22 which means A = 1 making B = E = 8×A = 8 × 1 = 8 C = D = 2×A = 2 × 1 = 2 F = A = 1 and ABCDEF = 182281