Puzzle for March 3, 2020  ( )

Scratchpad

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

eq.1) A + B + C + D + E + F = 15 eq.2) F – A = C eq.3) B + D + E = C – D eq.4) C + D = A + F eq.5) E – C = D – F eq.6) A + E = B + D

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 C with F – A (from eq.2): F – A + D = A + F In the above equation, add A to both sides, and subtract F from both sides: F – A + D + A – F = A + F + A – F which makes D = 2×A


  

Hint #2


Add C and F to both sides of eq.5: E – C + C + F = D – F + C + F which becomes E + F = D + C which may be written as E + F = C + D In eq.4, replace C + D with E + F: E + F = A + F Subtract F from both sides of the equation above: E + F – F = A + F – F which makes E = A


  

Hint #3


Substitute A for E, and 2×A for D in eq.6: A + A = B + 2×A which becomes 2×A = B + 2×A Subtract 2×A from each side of the equation above: 2×A – 2×A = B + 2×A – 2×A which makes 0 = B


  

Hint #4


Substitute 0 for B, 2×A for D, and A for E in eq.3: 0 + 2×A + A = C – 2×A which becomes 3×A = C – 2×A Add 2×A to both sides of the equation above: 3×A + 2×A = C – 2×A + 2×A which makes 5×A = C


  

Hint #5


Substitute 5×A for C, and 2×A for D in eq.4: 5×A + 2×A = A + F which becomes 7×A = A + F Subtract A from both sides of the above equation: 7×A – A = A + F – A which makes 6×A = F


  

Solution

Substitute 0 for B, 5×A for C, 2×A for D, A for E, and 6×A for F in eq.1: A + 0 + 5×A + 2×A + A + 6×A = 15 which simplifies to 15×A = 15 Divide both sides of the equation above by 15: 15×A ÷ 15 = 15 ÷ 15 which means A = 1 making C = 5×A = 5×1 = 5 D = 2×A = 2×1 = 2 E = A = 1 F = 6×A = 6×1 = 6 and ABCDEF = 105216